위상에서의 필터

수학의 하위 분야인 위상의 필터는 위상학적 공간을 연구하고 융합, 연속성, 콤팩트성 등과 같은 모든 기본 위상학적 개념을 정의하는 데 사용될 수 있다.필터는 특정 집합의 하위 집합의 특수 패밀리인 것으로서, 좌/우, 무한대, 점 또는 집합, 그리고 그 밖의 여러 가지 유형의 기능 한계를 정의하기 위한 공통 프레임워크도 제공한다.울트라필터라고 불리는 특수한 유형의 필터는 많은 유용한 기술적 특성을 가지고 있고 그것들은 종종 임의의 필터 대신에 사용될 수 있다.

필터는 프리필터(필터 베이스라고도 함)와 필터 서브베이스라는 일반화를 가지며, 이 모든 것은 위상 전체에 걸쳐 자연스럽고 반복적으로 나타난다.예를 들어 근린 필터/베이스/하위 데이터베이스 및 통일성을 들 수 있다.모든 필터는 프리필터고 둘 다 필터 서브베이스다.모든 프리필터와 필터 서브베이스는 고유한 가장 작은 필터에 포함되어 있으며, 필터 서브베이스는 생성된다고 한다.이는 필터와 프리필터 사이의 관계를 설정하는데, 이 두 개념 중 기술적으로 더 편리한 개념 중 하나를 사용할 수 있도록 종종 이용될 수 있다.notion , {\$displaystyle$\,\$leq ,\}$로 표시된 세트 제품군에는 하나의 개념(필터, 프리필터 등)이 다른 개념 대신 사용될 수 있거나 사용할 수 없는 정확한 시기와 방법을 결정하는 데 도움이 되는 특정$$ 사전 주문이 있다.This preorder's importance is amplified by the fact that it also defines the notion of filter convergence, where by definition, a filter (or prefilter) ${\displaystyle {\mathcal {B}}}$ converges to a point if and only if ${\displaystyle {\mathcal {N}}\leq {\mathcal {B}},}$ where ${\displaystyle {\mathca$$l{{N}}}$는 그 지점의 근린 필터다.결과적으로, 종속성은 클러스터 포인트와 기능의 한계와 같이 융합과 관련된 많은 개념에서도 중요한 역할을 한다.In addition, the relation ${\displaystyle {\mathcal {S}}\geq {\mathcal {B}},}$ which denotes ${\displaystyle {\mathcal {B}}\leq {\mathcal {S}}}$ and is expressed by saying that ${\displaystyle {\mathcal {S}}}$ is subordinate to ${\displaystyle {\mathcal {B}},}$ also establishes a$S{\displaystyle \mathcal{}}$$$ ${\$이(가) $B{\displaystyle \mathcal{}}$ ${\$에 대한 관계(즉$$, 후순위라고 하는 관계$$ $\ge {\displaystyle }$, ${\displaystyle \geq ,})$는 "후순위"의 아날로그 필터에 대한 것이다.

필터는 1937년^[1][2] 앙리 카르탄에 의해 소개되었고 이후 부르바키가 1922년 E. H. 무어와 H. L. 스미스에 의해 개발된 그물의 유사한 개념의 대안으로 그들의 책 Topologie Généale에 의해 사용되었다.필터는 또한 시퀀스 및 네트 수렴의 개념을 특성화하는 데 사용될 수 있다.하지만 unlike[노트 1]순서와 네트 통합 필터 통합 전적으로 위상 공간 X{X\displaystyle}의 하위 집합의 조건이 융합의 완전히 위상 공간 고유의 것이다 생각을 제공한다. 실제로, 위상 공간의 범주 동등하게 전적으로 측면에서 정의될 수 있고 정의된다.필터s. 모든 네트는 표준 필터를 유도하고, 한 번에 모든 필터는 표준 그물을 유도하며, 이 유도 그물(resp. 유도 필터)이 원래 필터(resp. net)와 동일한 경우에만 한 지점으로 수렴된다.이러한 특성화는 클러스터 지점과 같은 다른 많은 정의에도 적용된다.이러한 관계들은 필터와 그물 사이를 전환할 수 있게 해주며, 또한 종종 이 두 가지 개념(필터 또는 그물) 중 어떤 것이 당면한 문제에 더 편리한지를 선택할 수 있게 해준다.그러나 "하위 네트워크"가 가장 널리 사용되는 정의(Willard와 Kelley가 제공한 정의) 중 하나를 사용하여 정의된다고 가정하면, 일반적으로 이 관계는 하위 필터와 서브넷으로 확장되지 않는다. 왜냐하면 아래에 자세히 설명했듯이 필터/하위-필터 관계를 설명할 수 없는 하위 필터가 존재하기 때문이다.그러나, 해당 네트워크/서브넷 관계의 측면에서, 이 문제는 AA-서브넷의 정의인 "서브넷"의 덜 흔하게 접하는 정의를 사용함으로써 해결될 수 있다.

따라서 필터/프리필터와 이 단일 사전 주문서 ${\displaystyle \{}$ ${\$은(는) 위상학적 공간(근린 필터 사용), 근린 기반, 수렴, 다양한 기능 제한, 연속성, 콤팩트함, 시퀀스(순차 필터 사용), th와 같은 기본 위상학적 개념을 원활하게 결합하는 프레임워크를 제공한다$$.e 필터 "변형성"(변형성), 균일한 공간 등에 해당하며, 그렇지 않으면 상대적으로 이질적인 것처럼 보이고 관계가 덜 명확한 개념이다.

동기

필터의 전형적인 예

필터의 전형적 예로는 위상학적 ${\displaystyle \mathrm{공간}(X}$ , ${\displaystyle )}$) ,$${\$displaystyle$ x}의 점$$x {\$displaystyle x$}에 $있는{\displaystyle \mathcal{}}$ N$($$x$$)$ {\$mathcal{N$}(x$)}$이$(가)$ 주변 필터인데$$, 정의상${\displaystyle }$x .$${\$displaystystyle$x}의 모든 인접 영역으로 구성된 집합이다.d of some given point ${\displaystyle x}$ is any subset ${\displaystyle B\subseteq X}$ whose topological interior contains this point; that is, such that ${\displaystyle x\in \operatorname {Int} _{X}B.}$ Importantly, neighborhoods are not required to be open sets; those are called open neighborhoods.아래 열거된 근린필터가 공유하는 근본적인 속성은 결국 '필터'의 정의가 되었다.$X{\displaystyle }$ ${\displaystyle X}$의 필터는 $다음{\displaystyle }$ 조건을 모두 충족하는$$ X ${\displaystyle$ ${\mathcal{B}$의 하위 집합 B ${\$displaystyle $X}$ 집합이다$$.

1. 비어$$ 있지 않음: $X{\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle B\mathcal{}}$ ${\$ $$– $X{\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle N}$( x${\displaystyle }$)${\displaystyle }$, ${\displaystyle$ X$\in {\mathcal {N}(x)$과 마찬가지로 X ${\displaysty X}$은 항상$$ $x{\displaystyle }$ ${\displaystysty x}($및 그 밖의 모든 것)의 이웃);
2. $빈{\displaystyle }$ 세트를 포함하지 않음: ∅ ${\displaystyle \notin }$ ${\displaystyle B\mathcal{}}$ ${\$ $$- x ${\displaystyle x}$의 인접 항목이 비어 있지$$ 않은 것처럼,
3. 유한 교차로에 따라 폐쇄됨:$만약{\displaystyle }$ $B{\displaystyle }$, ${\displaystyle C}$ ${\displaystyle \in }$ ${\displaystyle B\mathcal{}}$, 그 ${\displaystyle \text{다음}}$에 ${\displaystyle B}$ ${\displaystyle \cap }$ ${\displaystyle C\mathcal{}}$ B {\$displaystyle$ B ${B}{\mathcal {B$}{\$text{}}:{}B\cap C\in {\mathcal {B}}$이$($가) 교차하는 것이 다시$$ $x{\displaystyle }$ ${\displaystystystyle$ x$\}$의 주변인 것처럼 말이다.$$
4. 위쪽 닫힘:If ${\displaystyle B\in {\mathcal {B}}{\text{ and }}B\subseteq S\subseteq X}$ then ${\displaystyle S\in {\mathcal {B}}}$ – just as any subset of ${\displaystyle X}$ that contains a neighborhood of ${\displaystyle x}$ will necessarily be a neighborhood of ${\displaystyle x$$}}($${\displaystyle {\mathrm{Int}}_{}}$ ${\displaystyle X_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle B}$ ${\displaystyle \subseteq }$ ${\displaystyle {Int}_{}}$ X ${\displaystyle {}_{}}$ ${\displaystyle S}$ ${\displaystyle \operatorname {Int} _{X}B\subseteq \operatorname {Int} _{X}S})$ 및$$ "$x{\displaystyle }${\$displaysty x}$의 정의에 따른다.$$

세트를 사용하여 시퀀스 수렴 일반화 - 시퀀스 없이 시퀀스 수렴 결정

$X{\displaystyle }$ ${\displaystyle X}$의 순서는 정의상 자연수에서 $공간$ X .$${\$displaystyle X}$까지의 지도 ${\displaystyle N}$→ X ${\displaystyle \mathb {N} \to X}$에 대한 원래 위상학적 공간에서의 수렴 개념은 메트릭 공간과 같은 공간의 특정 지점으로 수렴되는 순서의 개념이었다.측정 가능한 공간(또는 더 일반적으로 계산 가능한 공간 또는 Fréchet-Uryson 공간)의 경우 시퀀스는 대개 하위 집합의 폐쇄나 기능의 연속성과 같은 대부분의 위상학적 특성을 특징짓거나 "설명"하기에 충분하다.그러나 폐쇄나 연속성과 같은 기본적인 위상학적 특성조차 설명하기 위해 시퀀스를 사용할 수 없는 공간이 많다.이러한 시퀀스의 실패는 그물이나 필터와 같은 개념을 정의하는 동기였으며, 위상학적 성질을 특징 짓는데 실패하지 않았다.

Nets directly generalize the notion of a sequence since nets are, by definition, maps ${\displaystyle I\to X}$ from an arbitrary directed set ${\displaystyle (I,\leq )}$ into the space ${\displaystyle X.}$ A sequence is just a net whose domain is ${\displaystyle I=\mathbb {N} }$ with the natural o갈기갈기 찢기는그물에는 그들만의 융합 개념이 있는데, 이것은 시퀀스 융합의 직접적인 일반화다.

필터는 시퀀스 값만 고려하여 다른 방식으로 시퀀스 수렴을 일반화한다.To see how this is done, consider a sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X,}$ which is by definition just a function ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ whose value at ${\displaystyle i\in \mathb$$b {N} }$은(는) 임의 함수에 일반적으로 사용되는 일반적인$$ 괄호 $표기법{\displaystyle {}_{}}$ x ${\displaystyle {\bullet }_{}}$ (${\displaystyle {}_{}i}$) ${\displaystyle x_{\bullet }(i)}$이(가) 아닌$$ $x{\displaystyle {}_{}}$ ${\displaystyle i_{}}$ ${\${\bullet }($i)$로 표시된다$$.Knowing only the image (sometimes called "the range") ${\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{i}:i\in \mathbb {N} \right\}=\left\{x_{1},x_{2},\ldots \right\}}$ of the sequence is not enough to characterize its convergence; multiple sets are needed.필요한 세트는 다음과 같은 것으로 밝혀졌는데,^{[note 2]} 이를 $sequence{\displaystyle {}_{}}$ x ${\displaystyle {\bullet }_{}}$ ${\$x_{\$bullet }}$의 꼬리라고 한다$.$

$이{\displaystyle }$ 세트는 이 시퀀스의 수렴(또는 비융합)을 완전히 결정하는데, 이 시퀀스는 모든 주변 $U{\displaystyle }$ ${\displaystyle$ U$}($이 지점의$)$에 대해 U {\$displaystyle$ U$}$이(${\displaystyle {}_{}}$) 모든 ${\displaystyle {포인트}_{}}$를 포함하는$$ 정수$$ n ${\displaystyle n}$이(가) 있는 경우에만 수렴되기 때문이다${\displaystyle {}_{}}$${\displaystyle n_{}}$+ ${\displaystyle 1_{}}$,${\displaystyle }$… ${\displaystyle x_{n},x_{n+1},\ldots .}$이(가) 다음과 같이 다시 쓰일 수 있다.

모든 이웃 $U{\displaystyle }$ ${\displaystyle U}$은(는) 하위$$ 집합으로${\displaystyle }${$xn$ ${\displaystyle x_{}}$ ${\displaystyle {}_{}}$+ 1${\displaystyle }$,$$… {\$displaystyle \{$x_${n$},$x_{n+1$},\ldots $\}$의 일부 형식을 포함해야 한다$$.

It is the above characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence ${\displaystyle x_{\bullet }:\mathbb {N} \to X.}$ Specifically, with these sets in hand, the function ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ is no이 시퀀스의 수렴을 결정하는 데 더 오래 필요함($X{\displaystyle }$ ${\displaystyle X}$에 어떤 토폴로지를 배치하든 상관 없음).$$이러한 관찰을 일반화함으로써, 「융합」의 개념은 기능에서 집합의 패밀리로 확장될 수 있다.

위의 일련의 꼬리는 일반적으로 필터가 아니라 위쪽으로 닫힘으로써 필터를 "생성"한다.일반적으로 필터가 아니라 상향 폐쇄를 통해 필터를 생성한다(특히, 해당 지점에서 근린 필터를 생성함).이들 가족이 공유하는 속성은 필터 베이스(prefilter)라는 개념으로 이어졌는데, 정의상으로는 필터의 상향 폐쇄만을 통해 필터를 생성하기에 필요한 최소한의 속성을 가지고 있는 모든 제품군이다.

- 장단점 - - 장 - 점

필터와 그물은 각자 장점과 단점을 가지고 있으며, 한 가지 개념을 다른 개념보다 독점적으로 사용할 이유가 없다.^{[note 3]}증명되고 있는 것에 따라, 증거는 다른 개념 대신에 이러한 개념들 중 하나를 사용함으로써 상당히 쉽게 만들어질 수 있다.^[3]필터와 그물 모두 주어진 토폴로지를 완전히 특성화하는 데 사용될 수 있다.그물은 시퀀스의 직접적인 일반화로서 종종 시퀀스와 유사하게 사용될 수 있기 때문에, 그물에 대한 학습 곡선은 일반적으로 필터에 대한 그것보다 훨씬 덜 가파르다.그러나 필터, 특히 울트라필터는 세트이론, 수학논리학, 모델이론(예를 들어 초고성능), 추상대수학,^[4] 순서이론, 일반화된 수렴공간, 코우치공간, 초현실수의 정의와 사용 등 위상 외의 용도가 더 많다.

순서와 마찬가지로, 그물망도 기능이기 때문에 기능의 장점을 가지고 있다.예를 들어, 시퀀스와 마찬가지로, 네트는 다른 기능에 "플러그인"될 수 있으며, 여기서 "플러그인"은 기능 구성일 뿐이다.기능 및 기능 구성과 관련된 이론은 그물에 적용될 수 있다.한 예로 역한계의 보편적 특성이 있는데, 이는 세트보다는 함수 구성의 관점에서 정의되며 필터와 같은 세트보다는 네트와 같은 기능에 더 쉽게 적용된다(역한계의 두드러진 예는 데카르트 제품이다).$필터$는 공간 X$${\$displaystyle$ X$}$의 필터와 밀도가 높은 ${\displaystyle \mathrm{하위}}$ 공간 S${\displaystyle \subseteq }$ X$.$ {\$displaystyle$S\$subseteq$X.}의 필터 간 전환과 같은 특정 상황에서 사용하기 어려울 수 있다$.$

그물망과 대조적으로 필터(및 프리필터)는 세트 제품군이기 때문에 세트 장점이 있다.예를 들어, $f{\displaystyle }$ ${\displaystyle f}$이(${\displaystyle 가\mathcal{}}$) 돌출적인 경우$$, 이미지 $f{\displaystyle {}^{}}$- 1${\displaystyle {}^{}}$( B${\displaystyle }$)${\displaystyle }$: {${\displaystyle f{}^{}}$- ${\displaystyle 1^{}}$ (${\displaystyle {}^{}B}$)${\displaystyle }$: ${\displaystyle \text{B}}$ ${\displaystyle \in }$ ${\displaystyle B}$} ${\$ f$^{\mathcal{B}}:$$=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}}$ under ${\displaystyle f^{-1}}$ of an arbitrary filter or prefilter ${\displaystyle {\mathcal {B}}}$ is both easily defined and guaranteed to be a prefilter on ${\displaystyle f}$'s domain, whereas it is less clear how to pullback (unambiguously/without choice) an임의 시퀀스(또는 net) $y{\displaystyle {}_{}}$∙$${\$displaystyle y_{\displaystyle }}}$을(를) 도메인에서 시퀀스나 그물을 얻을 수 있도록$$ $($f {\$displaystyle$ f$}$도 주입되어$$ 결과적으로 바이어싱이 되는 경우는 제외되며, 이는 엄격한 요구 사항이다.필터는 고려 중인 매우$$ 위상학적 공간 $X{\displaystyle }$ ${\displaystyle$ X$}$의 하위 집합으로 구성되기 때문에 위상학적 세트 작업(폐쇄 또는 내부 등)을 필터를 구성하는 세트에 적용할 수 있다.예를 들어 필터의 모든 세트를 닫는 것이 기능 분석에 유용할 수 있다.기능에 따른 세트의 이미지 또는 사전 이미지에 대한 이론과 결과는 필터를 구성하는 세트에도 적용될 수 있다. 그러한 결과의 예는 개방/폐쇄 세트의 사전 이미지 또는 내부/폐쇄 운영자의 측면에서 연속성 특성 중 하나가 될 수 있다.울트라필터라고 불리는 특수한 유형의 필터는 결과 입증에 크게 도움이 될 수 있는 유용한 특성을 많이 가지고 있다.그물의 한 가지 단점은 도메인을 구성하는 방향 집합에 의존하는 것인데, 일반적으로는 $공간{\displaystyle }$X와 전혀 무관할 수 있다$$$.$ 실제로 주어진 $집합{\displaystyle }$ X$${\$displaystyle X}$의 그물 등급은 세트(적절한 등급)조차 되기에는 너무 크다$$. $이$는 X ${\displaysty$}의 그물 등급이 너무 크기 때문이다.$레$ X$}$은(는) 카디널리티의 도메인을 가질 수 있다$$.In contrast, the collection of all filters (and of all prefilters) on ${\displaystyle X}$ is a set whose cardinality is no larger than that of ${\displaystyle \wp (\wp (X)).}$ Similar to a topology on ${\displaystyle X,}$ a filter on ${\displaystyle X}$ is "intrinsic to ${\displaystyle X}$"두 구조물이 $모두{\displaystyle }$ X$${\$displaystyle X$}의 하위 집합으로 구성되며$$, 어떤 정의도 $X{\displaystyle }$ ${\$$displaystyle X}$에서 구성할 수 없는 집합($$$예{\displaystyle \mathbb{}}$: N {\displaysty $\mathb {N})$ 또는 시퀀스$$ 및 네트가 필요로 하는 기타 방향 집합)을 필요로 하지 않는다는 점에서.

예단, 표기법 및 기본 개념

In this article, upper case Roman letters like ${\displaystyle S{\text{ and }}X}$ denote sets (but not families unless indicated otherwise) and ${\displaystyle \wp (X)}$ will denote the power set of ${\displaystyle X.}$ A subset of a power set is called a family of sets (or simply, a family) where it is over ${\displaystyle X}$ if it is a subset of ${\displaystyle \wp (X).}$ Families of sets will be denoted by upper case calligraphy letters such as ${\displaystyle {\mathcal {B}},{\mathcal {C}},{\text{ and }}{\mathcal {F}}.}$ Whenever these assumptions are needed, then it should be assumed$X{\displaystyle }$ ${\displaystyle X}$이(가) 비어 있지 않고 $B{\displaystyle \mathcal{},}$ ${\displaystyle F,}$ ${\displaystyle {B},{\mathcal {F}$ 등이$$ $X{\displaystyle }$ . ${\displaystyle$ X}을$($를) 넘는 집합 집합 집합 집합입니다${\displaystyle .}$

"프리필터"와 "필터 베이스"라는 용어는 동의어로, 상호간에 사용될 것이다.

경쟁 정의 및 표기법에 대한 경고

불행히도 필터 이론에는 다른 작가들에 의해 다르게 정의되는 몇 가지 용어가 있다.여기에는 "필터"와 같은 가장 중요한 용어들이 포함되어 있다.동일한 용어의 서로 다른 정의는 대개 상당한 중첩을 가지지만, 필터(및 포인트 설정 위상)의 기술적 특성 때문에, 그럼에도 불구하고 정의의 이러한 차이는 종종 중요한 결과를 초래한다.수학적 문헌을 읽을 때 필터와 관련된 용어가 저자에 의해 어떻게 정의되는지를 독자들이 확인하는 것이 좋다.이 때문에 이 글에는 이 글에서 사용되는 모든 정의가 명쾌하게 기술되어 있을 것이다.불행히도 필터와 관련된 모든 표기법이 잘 확립되어 있는 것은 아니며 일부 표기법은 문헌에 걸쳐 크게 달라지기 때문에(예를 들어, 한 세트의 모든 프리필터 집합에 대한 표기법) 이러한 경우 이 글은 가장 자기적으로 설명하거나 쉽게 기억되는 표기법을 사용한다.

필터와 프리필터의 이론은 잘 발달되어 있고 정의와 명언이 많으며, 그 중 많은 것들이 현재 이 글이 장황하게 되는 것을 막고 표기법과 정의의 쉬운 룩업을 가능하게 하기 위해 약식으로 나열되어 있다.그들의 중요한 성질은 나중에 설명된다.

설정 작업 정정

세트 $B{\displaystyle \mathcal{}}$ ${\displaystyle \subseteq }$$\$ ( X ){\displaystyle {B$}\subseteq \wp(X)}$의^[6][7] $X{\displaystyle }$ {\$displaystyle$ $X$${\displaystyle \}}$의 상향 마감 또는$$ 이소톤화는

${\displaystyle {\mathcal {B}}^{\uparrow X}:=\{S\subseteq X~:~B\subseteq S{\text{ for some }}B\in {\mathcal {B}}\,\}=\bigcup _{B\in {\mathcal {B}}}\{S~:~B\subseteq S\subseteq X\}}$

and similarly the downward closure of ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}^{\downarrow }:=\{S\subseteq B~:~B\in {\mathcal {B}}\,\}=\bigcup _{B\in {\mathcal {B}}}\wp (B).$$}$

과		이름
$displaystyle \ker {\mathcal{B}=\bigcap_{B\in {\mathcal {B}B}$		$B{\displaystyle \mathcal{}}$ ${\$ {\mathcal {B}의$커널$
$displaystystyle S\setminus {B}{B}:=\\{S\setminus B~:~B\in {\mathcal {B}\}\(\setminus )\,{\mathcal {B}}}}}$	${\displaystyle S}$ ${\displaystyle S}$은(는) 집합임	${\displaystyle S}$ ${\displaystyle {\mathcal{B}{\text{{{}S}$에 $B$의 이중 표시[8]
${\displaystyle {\mathcal {B}{B}}{B}}{\big \vert }_{S}=\{B\cap S~:~B\in {\mathcal {B}\}={\mathcal {B}\, (\cap )\,\{S\}}}}}}}}}}}}}.$	${\displaystyle S}$ ${\displaystyle S}$은(는) 집합임	${\displaystyle S}$ ${\displaystyle {\mathcal {B}{\text{{}}$에 ${\displaystyle \text{대한}}$ $B$의 추적$$[8] $또는{\displaystyle \mathcal{}}$ ${\displaystyle B}$에서 ${\displaystyle \text{S}}$로 제한$${\$displaystyle${\$mathcal{B$}{\$text$에 $가끔{\displaystyle \mathcal{}}$ B${\displaystyle \mathrm{display}}$ S$${\$displaystystyle${\$mathcal {B}\cap S}$에 의해 표시됨
${\displaystyle {\mathcal {{B}\,(\cap )\,{\mathcal {{C}=\{B\cap C~:~B\in {\mathcal{B}{\text}, }}}}}C\in {\mathcal}\}}}}}}}}$[9]		요소별(세트) 교차로(${\displaystyle B\mathcal{}}$ ${\displaystyle \cap }$ C ${\$는 일반적인 교차로임을 나타낸다.
$mathcal}, (컵 )\{mathcal}, {mathcal}B}=\{B\컵 C~:~B\in {\mathcal{B}{\text}, }}}}}C\in {\mathcal}\}}}}}}}}$[9]		요소별(세트) 결합(${\displaystyle B\mathcal{}}$ ${\displaystyle \cup }$ C ${\$}컵 ${\mathcal {C})$은 일반적인 결합을 나타낸다.
${\displaystyle {\mathcal {{B}\,(\setminus )\,{\mathcal {{C}=\{B\setminus C~:~B\in {\mathcal {{B}{\text}, }}C\in {\mathcal}}}}}}$		요소별(세트) 뺄셈(${\displaystyle B\mathcal{}}$ ${\displaystyle \setminus }$ C ${\${\$mathcal {C})$은 일반적인 세트 뺄셈을 나타낸다.
${\displaystyle {\mathcal {B}^{\#X}={\mathcal {B}^{\b}=\{S\seteq X~:~S\cap B\neq \varnoth{\text{}}}}}모든 }B\mathcal {B}\in}}}}}}}}}}}}}}}}}}$		${\displaystyle X}$ ${\displaystyle {\mathcal{B}{\text{{{}X}$의 B 그릴[10]
$s\ Xdisplaystyle \wp(X)=\{S~~S\subseteq X\}}}}$		집합 $X$의 전원 세트 ${\displaystyle X}$

전체적으로 $f{\displaystyle }$ ${\displaystyle f}$은(는) 지도가 된다$$.

과		이름
$displaystyle f^{\mathcal}({\mathcal})B})=\왼쪽\{f^{-1}(B)~:~B\in {\mathcal {B}\오른쪽\}}}$[14]		${\displaystyle f{}^{}}$- ${\displaystyle 1^{}}$,${\displaystyle {\mathcal{B}{\mathcal{-1}}$ ${\displaystyle 아래}$의 B $이미지$ 또는$$$$$f$ {\$displaystyle${\$mathcal{B$}{\$f}} 아래$의 B $사전{\displaystyle \mathcal{}}$ 이미지
$f f Sdisplaystyle f^{-1(S)=\x\in \domain}f~f(x)\in S\}$	${\displaystyle S}$ ${\displaystyle S}$은(는) 임의 집합이다.	${\displaystyle f{}^{}}$- ${\displaystyle 1^{}}$,${\displaystyle S{\text{{} 아래$의 S{\$displaystyle$ S${-1},$ ${\displaystyle 또는}$ f {\$displaystyle S{\text{} 아래$의 $S{\displaystyle }$ 사전 ${\displaystyle \text{이미지}}$
$(B)~{displaystyle f({\mathcal {B}=\f(B)~:\b\}}}}}}$[14]		${\displaystyle f}$ ${\displaystyle {\mathcal {B}{\text{{}f}$ ${\displaystyle \text{아래}}$의 $B{\displaystyle \mathcal{}}$ 이미지
$)=\{f(sdisplaystyle f(S)=\{f(s)~:~s\in S\cap \operatorname {domain} f\}}}}}$	${\displaystyle S}$ ${\displaystyle S}$은(는) 임의 집합이다.	${\displaystyle f}$ ${\displaystyle$ S${\text{}} 아래$의 S $이미지{\displaystyle }$
$domain}f \f=f(\domainname {domain}f)}$		$f{\displaystyle }$ ${\displaystyle f}$의 이미지

위상 표기법

$X{\displaystyle }$ ${\displaystyle \mathrm{by}\mathrm{Top})}$( X$$) . {\$displaystyle X{\text${by $}\\operatorname {Top}($X)를 ${\displaystyle \text{기준}}$으로 집합의 모든 위상 집합을 나타낸다$.$$}$ $\tau {\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle \mathrm{Top}}$ ${\displaystyle }$( X${\displaystyle }$)${\displaystyle }$. ${\displaystyle \tau \in \operatorname {Top} ($X)를$$ 가정해 보십시오$.$$}$

과		이름
$Odisplaystyle \tau=\{O\in \tau ~:~S\subseteq O\}}}}}$	$S\subseteq X}$	${\displaystyle (X}$, ${\displaystyle \tau }$) ${\displaystyle S{\text{{{}}(X,\tau )}$에서 $S$의 열린 인접 영역 설정 또는 프리필터[note 4]
$Odisplaystyle \tau(x)=\{O\in \tau ~:~x\in O\}$	$에 있는 스 x\ X$	$x{\displaystyle }$ ${\displaystyle \text{in}}$ (${\displaystyle X}$ ,${\displaystyle \tau }$) ${\displaystyle$ x${\text{{{}{{}}(X,\$tau $)}$의 열린 영역 설정 또는 프리필터
${\displaystyle {\mathcal{N}_{\tau }={\mathcal{N}(S): Xtau(S)^{\uparrow X}}$	$S\subseteq X}$	${\displaystyle (X}$ ,${\displaystyle \tau }$) ${\displaystyle S{\text{{}}(X,\tau )}$에서 S $인접{\displaystyle }$ 영역의 설정 또는 필터[note 4]
${\displaystyle {\mathcal{N}_{\tau }(x)={\mathcal{N}(x):=\tau (x)^{\uparrow X}}$	$X}에 표시 스타일 x\in X}$	${\displaystyle (X}$ ,${\displaystyle \tau }$) ${\displaystyle x{\text{{{}}(X,\tau )}$의 주변 $환경{\displaystyle }$ 설정 또는 필터

If ${\displaystyle \varnothing \neq S\subseteq X}$ then ${\displaystyle \tau (S)=\bigcap _{s\in S}\tau (s){\text{ and }}{\mathcal {N}}_{\tau }(S)=\bigcap _{s\in S}{\mathcal {N}}_{\tau }(s).$$}$

그물과 꼬리

나는 함께 preorder과{\displaystyle 1세}는≤{\displaystyle \,\leq\와 같이,}(지 않으면 명시적으로 별도 표시한)에 의해 표시된(나는, ≤)(위로)를에{\displaystyle(I,\leq)}는 직접적인 세트는 세트라[15]이것은 모두에게 나는, j∈이 전처{\displaystyle i,j\in 나는,}을 의미한다.ists일부 k∈ 나는{\displaystyle k\in 나는}가 나는 ≤ k와 j≤ k.{\displaystylei\leq k{\text{과}}j\leq k.}에게 어떤 지수 i와 j,{\displaystyle 나는{\text{과}}j,}기호 j≥ 나는{\displaystyle j\geq 나는}은 정의될 생각은 나는 ≤ j{\displaystyle i\leq j}는 동안 나는 <, j{\displaystyle i<, j.}나는s는 $i{\displaystyle }$ ${\displaystyle \le }$ ${\displaystyle j}$ ${\displaystyle i\leq$ $j$$}$을(를) 보유한다는 의미로 정의되었지만 ${\displaystyle j}$ ${\displaystyle \le }$ i ${\displaystyle j\\leq \}($만약 ${\displaystyle \le }$ ${\$$displaystyle$ $\,\leq$$\}$이(가$$) 대칭이라면 i ${\displaystyle \le }$ ${\displaystyle j}$와 i ${\displaystyle \{}$ ${\displaystyle j}$ ${\$leq j {\$leq$ j ${\l\j${{{}에$해당된다.$

$X$^[15] ${\displaystyle X}$의 그물은 $X{\displaystyle }$. ${\displaystyle X}$에 설정된 비어 있지 않은 상태의 지도입니다.

표기법과 정의	가정	이름
${\displaystyle I_{\geq i}=\{j\in I~:~i\leq j\}}$	${\displaystyle \in }$ ${\displaystyle I}$ ${\displaystyle \text{및}}$(${\displaystyle I}$, ${\displaystyle \le }$) ${\displaystyle I\in I{\text{$및$}}}(I,\leq )}$은 지시된 집합이다.	$I{\displaystyle }$ ${\displaystyle I}$에서 시작하는 I ${\displaystyle I}$의 꼬리 또는 섹션
${\displaystyle x_{\geq i}=\좌측\{x_{j}~:~i\leq j{\text{\cext{ 및 }}j\in I\right\}}}}$	${\displaystyle \in }$ ${\displaystyle \text{I}}$과 ${\displaystyle x{}_{}}$ ${\displaystyle {\bullet }_{}}$=${\displaystyle {}_{}}$( ${\displaystyle {x_{}}_{}}$ ${\displaystyle {{}_{}}_{}}$) ${\displaystyle {}_{}}${ ${\displaystyle I_{}}$ ${\$ I${\text{ 및 }x_{\bullet }=\왼쪽(x_{i}\오른쪽)_{i\in I}}}$는 그물망이다.	$i{\displaystyle }$ ${\displaystyle i}$에서 시작되는 $x{\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$ ${\$의 꼬리 또는 단면
${\displaystyle \operatorname {Details} \좌측(x_{\bullet }\우측)=\좌측\{x_{\geq i}~:~i\in I\right\}}}}$	${\displaystyle x_{}}$ ${\displaystyle {\bullet }_{}}$=${\displaystyle {}_{}}$( ${\displaystyle {x_{}}_{}}$ i${\displaystyle {{}_{}}_{}}$) ${\displaystyle i_{}}$ ${\displaystyle {}_{}}$ {\$displaystyle$x_{\$bullet$}=\$왼쪽(x_{i$}\$오른쪽)_{i\$in I}}}는$$ 그물망이다.	Set or prefilter of tails/sections of ${\displaystyle x_{\bullet }.}$ Also called the eventuality filter base generated by (the tails of) ${\displaystyle x_{\bullet }.}$ If ${\displaystyle x_{\bullet }}$ is a sequence then ${\displaystyle \operatorname {Tails} \left(x_{\bullet$$}\\right)}$을(를) 대신 순차 필터 베이스라고 한다$$.[16]
${\displaystyle \operatorname {TailsFilter} \left(x_{\bullet }\right)=\Detailsname {Details} \left(x_{\bullet }\right)^{\uparrow X}}$	${\displaystyle x_{}}$ ${\displaystyle {\bullet }_{}}$=${\displaystyle {}_{}}$( ${\displaystyle {x_{}}_{}}$ i${\displaystyle {{}_{}}_{}}$) ${\displaystyle i_{}}$ ${\displaystyle {}_{}}$ {\$displaystyle$x_{\$bullet$}=\$왼쪽(x_{i$}\$오른쪽)_{i\$in I}}}는$$ 그물망이다.	(사건성) x ${\displaystyle {\bullet }_{}}$($꼬리{\displaystyle {}_{}}$){\$displaystyle x_{\bullet }}$의 필터
${\displaystyle f\left(I_{\geq i}\오른쪽)=\{f(j)~:~i\leq j{\text{ 및 }}}j\in I\}}}$	${\displaystyle \in }$ ${\displaystyle I}$ 및 ${\displaystyle f}$:${\displaystyle }$( ${\displaystyle I}$, ${\displaystyle \le }$)${\displaystyle }$→ ${\displaystyle X}$ ${\displaystyle i\in$ I${\text{$및$}}f~:(I,\leq )\to X}$는 그물이다.	$i{\displaystyle }$ ${\displaystyle i}$에서 시작하는 f ${\displaystyle f}$의 꼬리 또는 섹션

엄격한 비교 사용에 대한 경고

만약 x ∙)()나는)나는 나는 세트를}와 나는 ∈ 순 그 때 그것을 가능하다{\displaystyle i\in 나는})을{\displaystyle x_{\bullet}(x_{나는}\right)_{나는 i\in}, 나는{)j∈ 나는:나는입니다.;j와 j∈ 1세}, I~{\displaystyle x_{>나는}=\left\{x_{j}\in:~i&gt원;j{\text{과}}j\in I\right\},}은 c은 ∈는 분명 alled$i{\displaystyle }$ ${\displaystyle i}$ 이후$$$$x ${\displaystyle {\bullet }_{}}$ ${\$의 일(예$$: i ${\displaystyle$ $i}$이(가) 방향 $집합{\displaystyle }$ I ${\$displaystyle I}의 상한인 경우).$$이 경우{> i: ${\$은(는) 빈 세트가 포함되어 프리필터가 되지 못하게 된다(나중에 정의됨).{)≥ 나는: 난 ∈}{\displaystyle \left\{x_{\geq 나는}~로 테일즈 ⁡()∙)(\left(x_{\bullet}\right)}를 정의하는 이것은(중요한)이유:~i\in I\right\}}보다는{)>i: 난 ∈}{\displaystyle \left\{x_{>나는}~:~i\in I\right\}}또는{. )>나는:나는}나는 ∈ ∪{)≥ 나는: 난 ∈}{\displaystyle \left\{x_{>나는}~:~i\in I\right\}\cup \left\{x_{\geq 나는}~:~i\in I\right\}}와 그것도 이 이유에서 일반적으로 때, 그물의 꼬리의 prefilter를 다루는데 있어서 엄격한 불평등<>{\displaystyle \,<, \,}호환 inequali과 함께 사용할 못할 수도 있다.ty${\displaystyle \,\leq .}$

필터 및 프리필터

제품군 $F{\displaystyle \mathcal{}}$ ${\$$mathcal$${F}$$Ω$ 이상 세트
$반드시{\displaystyle \mathcal{}}$F :$${\$displaystyle${\$mathcal {F}\colon}$에 해당됨 $또는{\displaystyle \mathcal{}}$ F ${\$이(가) 다음 조건에 따라 닫힘:	연출된 by ${\displaystyle \,\supseteq }$	$A\cap B}$	$A\컵 B}$	$B\setminus A}$	$\displaystyle \Oomega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\컵 A_{2}\컵 \cdots }$	${\displaystyle \Oomega \in {\mathcal {F}}$	${\displaystyle \varnothing \in {\mathcal {F}}$	F.I.P.
π-시스템
세미닝										결코 하지 않다
세미날게브라 (세미필드)										결코 하지 않다
모노톤급						${\displaystyle A_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle A_{i}\searrow }$인 경우에만 해당됨	$A{\displaystyle {}_{}}$ ${\displaystyle i_{}}${$${\$displaystyle A_{i}\nearrow }$인 경우에만 해당됨
𝜆-시스템 (Dynkin 시스템				할 때만 $A\subseteq B}$			$A{\displaystyle {}_{}}$ ${\displaystyle i_{}}$ ${\displaystyle {}_{}}$ ${\displaystyle A_{i}\nearrow }$ 또는$$ 그들은 서로 단절되어 있다.			결코 하지 않다
반지(순서가론)
링(측정 이론)										결코 하지 않다
Δ-링										결코 하지 않다
𝜎-링										결코 하지 않다
대수(필드)										결코 하지 않다
𝜎알게브라 (필드-필드)										결코 하지 않다
이중 이상
필터				결코 하지 않다	결코 하지 않다				${\displaystyle \varnot \in {\mathcal {F}}$
프리필터 (필터 베이스)				결코 하지 않다	결코 하지 않다				${\displaystyle \varnot \in {\mathcal {F}}$
필터 서브베이스				결코 하지 않다	결코 하지 않다				${\displaystyle \varnot \in {\mathcal {F}}$
위상							(임의의 조합도)			결코 하지 않다
$반드시{\displaystyle \mathcal{}}$F :$${\$displaystyle${\$mathcal {F}\colon}$에 해당됨 $또는{\displaystyle \mathcal{}}$ F ${\$이(가) 다음 조건에 따라 닫힘:	연출된 아래쪽으로	유한한 교차점	유한한 조합	상대적 보완물	보완물 $\Omega {\displaystyle \mathrm{}}$ ${\displaystyle \Oomega }$	셀 수 있는 교차점	셀 수 있는 조합	$\Omega {\displaystyle \mathrm{}}$ ${\displaystyle \Oomega}$ 포함	포함 ${\displaystyle \varnothing }$	유한한 교차로 재산.
또한 의미 부여는 모든 보완 $B{\displaystyle }$ ${\displaystyle \setminus }$ ${\displaystyle A}$ ${\displaystyle B\setminus A$}이(가)${\displaystyle }$F .$${\$displaystyle${\$mathcal {F}$에서 집합의 유한 분리 결합과 동일한 π-시스템이다.$}$ 세미메이지브라(semialgebra)는$\Omega {\displaystyle \mathrm{}}$ . {\$displaystyle \Oomega .}$을(를) 포함하는 기호다. ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle {}_{}}$ ${\displaystyle {1\mathrm{}}_{}}$,${\displaystyle {}_{}}$A ${\displaystyle {2\mathrm{}}_{}}$,${\displaystyle }$… ${\displaystyle A,B,A_{1},A_{2},\ldots }$은(는) $F{\displaystyle \mathcal{}}$ ${\$의 임의적인 요소이며$$$$, $≠.{\displaystystyle. {\\f}\neqvarnot.}$라고 가정한다.

다음은 제품군$$ $B{\displaystyle \mathcal{}}$ ${\$이(가) 소유할 수 있는 속성 목록이며 필터, 프리필터 및 필터 하위베이스의 정의 속성을 형성한다.필요할 때마다 $B{\displaystyle \mathcal{}}$ ${\displaystyle \subseteq \mathrm{\wp }}$ ( X${\displaystyle }$)${\displaystyle }$. ${\displaystyle {\mathcal {B}\subseteq \wp (X$)라고 가정해야 한다$.$$}$

$B{\displaystyle \mathcal{}}$ ${\$ 집합의 제품군은 다음과$$ 같다.

1. ${\displaystyle \varnothing }$ ∉ ${\displaystyle B}$. ${\displaystyle \varnot \not \in$ {\mathcal ${B}}.}$ 또는$$ otherwise ${\displaystyle \in }$ ${\displaystyle B}$, ${\displaysty \varnot$\varnot \$in {\$mathcal {$B}}$인 경우 부적절하거나^[17] 퇴보한다고 한다.
2. $A$^[15],${\displaystyle B}$ ${\displaystyle \in }$ ${\displaystyle B\mathcal{}}$ ${\${$B$$}$이(가) 있을 때마다 $C{\displaystyle }$ ${\displaystyle \in }$ ${\displaystyle A}$$$${\displaystyle \cap }$ ${\displaystyle B.}$ ${\displaystyle C\\subseteq$A\$cap$ B}이$(가$) 있는$$ $경우{\displaystyle }$ 아래로 향한다$.$$}$
   • This property can be characterized in terms of directedness, which explains the word "directed": A binary relation ${\displaystyle \,\preceq \,}$ on ${\displaystyle {\mathcal {B}}}$ is called (upward) directed if for any two ${\displaystyle A{\text{ and }}B,}$ there is some ${\displaystyle C}$ satisfying ${\displaystyle A\preceq C{\text{ and }}B\preceq C.}$ Using ${\displaystyle \,\supseteq \,}$ in place of ${\displaystyle \,\preceq \,}$ gives the definition of directed downward whereas using ${\displaystyle \,\subseteq \,}$ instead gives the definition of directed upward.Explicitly, ${\displaystyle {\mathcal {B}}}$ is directed downward (resp. directed upward) if and only if for all ${\displaystyle A,B\in {\mathcal {B}},}$ there exists some "greater" ${\displaystyle C\in {\mathcal {B}}}$ such that ${\displaystyle A\supseteq C{\text{ and }}B\$$supseteq C}$ (resp. such that ${\displaystyle A\subseteq C{\text{ and }}B\subseteq C}$) − where the "greater" element is always on the right hand side,^{[note 5]} − which can be rewritten as ${\displaystyle A\cap B\supseteq C}$ (resp. as ${\displaystyle A\cup B\subseteq C}$).
   • $B{\displaystyle \mathcal{}}$ ${\$ 계열이 ${\displaystyle \supseteq }$ ${\$\},(예: ${\displaystyle \varnothing }$ {$$B {\$displaysty \varnothing \$in {\$mathcal$ {$B$에 대해 가장 큰 요소를 가지고$$ 있으면 반드시 아래로 향한다.$$
3. 유한 교차점(resp) 아래 폐쇄됨.교차로(resp)인 경우 조합.$B{\displaystyle \mathcal{}}$ ${\$의 두 요소 중 조합)은 B${\displaystyle }$. ${\displaystyle${\$mathcal {B}$의 요소다.$}$
   • $B{\displaystyle \mathcal{}}$ ${\$이(가) 유한 교차로에서 닫히면 B ${\$ {$B}$이(가) $반드시{\displaystyle \mathcal{}}$ 아래로 향한다$$.그 역은 대체로 거짓이다.
4. 만약 B및 B=B↑ X,{\displaystyle{{B\mathcal}}℘(X)⊆ 상향 또는 동중 성자 원소 X에서{X\displaystyle}[6]\subseteq\wp(X){\text{과}}{{B\mathcal}}={{B\mathcal}}^{X\uparrow},}또는 동등하게, 만약 때마다 B∈ B{\displaystyle B\in{{B\mathcal}}}과 일부 세트 C{C\displaystyle} 앉아 마감했다.isfies ${\displaystyle B\subseteq C\subseteq X,{\text{ then }}C\in {\mathcal {B}}.}$ Similarly, ${\displaystyle {\mathcal {B}}}$ is downward closed if ${\displaystyle {\mathcal {B}}={\mathcal {B}}^{\downarrow }.}$ An upward (respectively, downward) closed set is also called상위 세트 또는 이체 세트(하위 세트 또는 하위 세트)
   • The family ${\displaystyle {\mathcal {B}}^{\uparrow X},}$ which is the upward closure of ${\displaystyle {\mathcal {B}}{\text{ in }}X,}$ is the unique smallest (with respect to ${\displaystyle \,\subseteq }$) isotone family of sets over ${\displaystyle X}$ having ${\displaysty$$l$ {\$mathcal{B}}$을(를) 하위 집합으로$$.

"property" 및 "direction downward"와 같이 위아래로 정의된$$ $B{\displaystyle \mathcal{}}$ ${\$의 속성 중 상당수는 $X{\displaystyle }$, ${\displaystyle$ X$}$에 의존하지 않으므로$$ $이러한{\displaystyle }$ 용어를 사용할 때 X ${\displaystyle X}$ 집합에 대한 언급은 선택 사항이다$$."$X{\displaystyle }$, ${\displaystyle X$의 필터와 같은 "$X{\displaystyle }$, ${\$displaystyle X$,}$에서 위쪽 닫힘$"$과 관련된 정의는 $X{\displaystyle }$ ${\displaystyle$ X$}$에 따라 달라지므로$$ 컨텍스트에서 명확하지 않은 $경우{\displaystyle }$ X ${\displaystyle$ X}을$($를)로 설정해야 한다$$.

$A{\displaystyle \mathcal{}}$ 계열 B ${\$ {$B}$은(는$$) a(n):

1. $B{\displaystyle \mathcal{}}$ $≠{\displaystyle {\mathcal {B}\neq \varnothing}}$이(가) 유한 조합에서 아래로 닫히고 닫힌 경우에 이상적이다^[17][18].
2. $B{\displaystyle \mathcal{}}$ ${\displaystyle \ne }$ ∅{\$displaystyle {\mathcal {B}\neq \varnothing$ }이$($가) X ${\displaystyle X}$에서 위쪽으로 닫히고$$ 유한 교차로에서도 닫힌 경우$$^[19] $X{\displaystyle }$ ${\displaysty$ X$}$에 이중 이상적임.Equivalently, ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a dual ideal if for all ${\displaystyle R,S\subseteq X,}$${\displaystyle R\cap S\in {\mathcal {B}}\;{\text{ if and only if }}\;R,S\in {\mathcal {B}}.$$}$^[10]
   • "dual"이라는 단어의 설명: $가족{\displaystyle \mathcal{}}$ B$${\$displaystyle${\$mathcal{B$$}}$은(는) 가족인${\displaystyle \text{X}}$${\$$displaystyle$$X}$의 이중 $이상{\displaystyle \mathcal{}}$(resp. 이상)이다$.$
     $${\displaystyle X\setminus {\mathcal{B}:=\{X\setminus B~:~B\in {\mathcal {B}\}}$$

     
     $X{\displaystyle .}$ ${\displaystyle X.}$의 이상(이중 이상)이다. 즉$$, 이중 이상은 "이상적 이중"을 의미한다.The family ${\displaystyle X\setminus {\mathcal {B}}}$ should not be confused with ${\displaystyle \wp (X)\setminus {\mathcal {B}}=\{S\subseteq X~:~S\notin {\mathcal {B}}\}}$ because these two sets are not equal in general; for instance, ${\displaystyle X\setminus \mathcal{B}=\mathrm{\wp }(X)\text{if and only if}}$${\displaystyle X\setminus {\mathcal{B}}=\wp(X){{\mathcal{B}=\wp(X$$}$ The dual of the dual is the original family, meaning ${\displaystyle X\setminus (X\setminus {\mathcal {B}})={\mathcal {B}}.}$ The set ${\displaystyle X}$ belongs to the dual of ${\displaystyle {\mathcal {B}}}$ if and only if ${\displaystyle \varnothing \in {\mathcal {B}}.$$}$^[17]
3. 필터 X에서{X\displaystyle}[19][8]X에서 만약 B{\displaystyle{{B\mathcal}}}은properdual 이상이다.℘(X)∖{∅}{\displaystyle \wp(X)\setminus){\varnothing)}의 유한 교차로 아래 위쪽과 닫혀 있{X\displaystyle}즉, X{X\displaystyle}에 대한 필터는non−empty 부분 집합}. 폐쇄in ${\displaystyle X.}$ Equivalently, it is a prefilter that is upward closed in ${\displaystyle X.}$ In words, a filter on ${\displaystyle X}$ is a family of sets over ${\displaystyle X}$ that (1) is not empty (or equivalently, it contains ${\displaystyle X}$), (2) is closed under finite intersections, (3) $X{\displaystyle }$, ${\displaystyle$ X$,}$에서 위쪽으로 닫히고, (4) 빈 세트를 요소로 가지고 있지 않다.
   • 경고: 어떤 작가들, 특히 알제브리스트들은 이중 이상을 의미하기 위해 "필터"를 사용하고, 다른 작가들, 특히 토폴로지스트들은 적절한/비감속적인 이중 이상을 의미하기 위해 "필터"를 사용한다.^[20]수학적 문학을 읽을 때 항상 '필터'가 어떻게 정의되는지 확인하는 것이 좋다.그러나 "초광필터", "프리필터", "필터 서브베이스"의 정의는 항상 비지연성을 요구한다.이 글은 비감소성이 필요했던 앙리 카르탄의 필터에 대한 원래 정의를 사용한다.
   • A dual filter on ${\displaystyle X}$ is a family ${\displaystyle {\mathcal {B}}}$ whose dual ${\displaystyle X\setminus {\mathcal {B}}}$ is a filter on ${\displaystyle X.}$ Equivalently, it is an ideal on ${\displaystyle X}$ that does not contain ${\displaystyle X}$ as an element.
   • 전원 세트 $\wp {\displaystyle \mathrm{}}$( X${\displaystyle }$) ${\displaystyle \wp (X)}$은(는) $X{\displaystyle }$ ${\displaystyle$ X$}$에서 $필터$가 아닌$$ 유일한 이중 이상이다.위상에서의 "필터"의$$ 정의에서 $\wp {\displaystyle \mathrm{}}$(${\displaystyle X}$ ${\displaystyle )}${\$displaystyle \wp ($X)}을 제외하는 것은 "prime number"의 $정의$에서$$1 {\$displaystyle$ 1$}$을 제외하는 것과 같은 이점이 있다: 많은 중요한 결과에서 "비-디제너제너레이션"(" 또는 "$비-1"$의 아날로그)$$을 지정할 필요가 없으므로, 결과적으로 maki가 된다.ng 그들의 진술이 덜 어색하다.
4. $B{\displaystyle \mathcal{}}$ ${\displaystyle \ne }$ ${\displaystyle {\mathcal {B}\neq \varnothing$}이$($가) 적절하고 아래쪽으로 향할 경우 프리필터 또는 필터 베이스^[8][21].마찬가지로 $B{\displaystyle \mathcal{}}$ ${\$의 위쪽 닫힘 $B{\displaystyle {\mathcal{}}^{}}$ ${\displaystyle {\uparrow }^{}}$ ${\displaystyle X^{}}$ ${\$이(가) 필터라면$$ 프리필터라고 불린다$$.또한 일부 필터에 해당하는 모든 패밀리로 정의할 수 있다$({\displaystyle \leq$ $\}$ 참조).^[9]A proper family ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a prefilter if and only if ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {B}}\leq {\mathcal {B}}.}$^[9] A family is a prefilter if and only if the same is true of its upward closure.
   • If ${\displaystyle {\mathcal {B}}}$ is a prefilter then its upward closure ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is the unique smallest (relative to ${\displaystyle \subseteq }$) filter on ${\displaystyle X}$ containing ${\displaystyle {\mathcal {B}}}$ and it is called the filter generated by ${\displaystyle {\mathcal {B}}.}$ A filter ${\displaystyle {\mathcal {F}}}$ is said to be generated by a prefilter ${\displaystyle {\mathcal {B}}}$ if ${\displaystyle {\mathcal {F}}={\mathcal {B}}^{\uparrow X},}$ in which ${\displaystyle {\mathcal {B}}}$ is$F{\displaystyle \mathcal{}}$. ${\displaystyle {\mathcal {F}$에 대한 필터 베이스를 호출함.$}$
   • 필터와 달리 프리필터는 한정된 교차로에서 반드시 닫히지 않는다.
5. π-system B ${\displaystyle {\mathcal {B}\neq \varnothing$}이$($가) 유한 교차로에서 폐쇄된$$ 경우.비어 있지 않은 모든 $제품군{\displaystyle \mathcal{}}$ B ${\$은(는) ${\displaystyle \mathcal{}}$, ${\displaystyle {\mathcal {B}$이(는) 생성되는 π–system이라고 하는 고유한 가장 작은 π-system에 포함되어$$ 있으며$$, 이 $\pi {\displaystyle }$(${\displaystyle 은)}$. ${\displaystypi({\mathcal {B}).$$}$ $B$ {\$displaystyle${\$mathcal {B}$을(를) 포함하는 모든 ––systems의 교차점$$ 및 $B{\displaystyle \mathcal{}}${\$displaystyle${\$mathcal {B}$에서 집합의 가능한 모든 유한 교차점 집합의 집합과 동일함$:$
   $${\displaystyle \pi({\mathcal{B}}}=\왼쪽\{B_{1}\cap \cdots \cap B_{n}~:n\geq 1{\text{},\ldots ,B_{n}\mathcal {B}\}\rig}.}$$

   
   • π-시스템은 적절한 경우에만 프리필터다.모든 필터는 적절한 π-시스템이고 모든 적절한 π-시스템은 프리필터지만 대화 내용은 일반적으로 유지되지 않는다.
   • 프리필터는${\displaystyle }$프리필터가 생성한 π-system과 동등하며$,$ 이러한 두 패밀리는 $모두{\displaystyle }$ X${\displaystyle }$. ${\displaystyle X.}$에 동일한 필터를 생성한다.
6. $B$ $≠{\displaystyle {\mathcal {B}\neq \varnothing$} 및 $B{\displaystyle \mathcal{}}$ ${\$이(가) 다음과 같은 동등한 조건 중 하나를 충족하면$$ 하위 베이스를^[8][22] 필터링하고 중심을 맞춘다^[9].
   1. ${\displaystyle {\mathcal {B}}}$ has the finite intersection property, which means that the intersection of any finite family of (one or more) sets in ${\displaystyle {\mathcal {B}}}$ is not empty; explicitly, this means that whenever ${\displaystyle n\geq 1{\text{ and }}B_{1}$$,\ldots ,B_{n}\in {\mathcal{B}},$ 그 다음$$ ${\displaystyle \varnothing }$ ${\displaystyle {}_{}}$ ${\displaystyle {1\mathrm{}}_{}}$ ${\displaystyle \cap }$ ${\displaystyle B_{}}$ n${\displaystyle {}_{}}$. ${\displaystyle \varnoth \neq B_{1}\cap \cdots \cap B_{n}.$$}$
   2. $B{\displaystyle \mathcal{}}$ ${\$에서 생성된 ${\displaystyle \mathrm{\pi -system}}$이 적절함$$, 즉 ∉${\displaystyle \notin }$(${\displaystyle B\mathcal{})}$ .$${\$displaystyle \varnot \not \in \pi({\mathcal {B}).$$}$
   3. $B{\displaystyle \mathcal{}}$ ${\$이(가) 생성한 π–시스템은 프리필터다$$.
   4. ${\displaystyle \mathcal{B}}$ ${\$은(는) 일부 프리필터의 하위 집합이다.
   5. ${\displaystyle \mathcal{B}}$ ${\$은(는) 일부 필터의 하위 집합이다.
   • $B{\displaystyle \mathcal{}}$ ${\$이(가) 필터 하위 베이스라고$$ 가정하십시오.Then there is a unique smallest (relative to ${\displaystyle \subseteq }$) filter ${\displaystyle {\mathcal {F}}_{\mathcal {B}}{\text{ on }}X}$ containing ${\displaystyle {\mathcal {B}}}$ called the filter generated by ${\displaystyle {\mathcal {B}}}$, and ${\displaystyle {\ma$$thcal {B}}$은(는) 이 필터의 필터 하위 베이스라고 한다$$.This filter is equal to the intersection of all filters on ${\displaystyle X}$ that are supersets of ${\displaystyle {\mathcal {B}}.}$ The π–system generated by ${\displaystyle {\mathcal {B}},}$ denoted by ${\displaystyle \pi ({\mathcal {B}}),}$ will be a prefilter and a subset of ${\displaystyle {\mathcal{F}}_{}}$${\displaystyle {\mathcal {F}}_{\mathcal {B}}.}$ Moreover, the filter generated by ${\displaystyle {\mathcal {B}}}$ is equal to the upward closure of ${\displaystyle \pi ({\mathcal {B}}),}$ meaning ${\displaystyle \pi ({\mathcal {B}})^{\uparrow X}={\mathcal {F}}_${{B\mathcal}}.}[9]그러나, B↑ X)FB{\displaystyle{{B\mathcal}}^{X\uparrow}={{F\mathcal}}_{{B\mathcal}}}F에 만일 B{\displaystyle{{B\mathcal}}}은 prefilter(비록 B↑ X{\displaystyle{{B\mathcal}}^{X\uparrow}}은 항상 위쪽 문을 닫필터 subbase${\displaystyle {\mathcal{B}}_{}}$ ${\$
   • 필터 하위 베이스 $B{\displaystyle \mathcal{}}$ ${\displaystyle$ $\subseteq }$ $$– 가장 작은$([\displaystyle \$$subseteq$ $}}$에 비해 가장 작은$){\displaystyle }$$$ 프리필터는 특정 상황에서만 존재할 수 있다$$$.$예를 들어 필터 하위 베이스 $B{\displaystyle \mathcal{}}$ ${\$도 프리필터인 경우 이 값이 존재한다.It also exists if the filter (or equivalently, the π–system) generated by ${\displaystyle {\mathcal {B}}}$ is principal, in which case ${\displaystyle {\mathcal {B}}\cup \{\ker {\mathcal {B}}\}}$ is the unique smallest prefilter containing ${\displaystyle {\mathcal {B}}.}$ Otherwise, in general$.{\displaystyle }$ B ${\$$displaystyle \subseteq }$ $$–$B{\displaystyle \mathcal{}}$ {\$displaystyle {\mathcal {B}$을(를) 포함하는 가장 작은 프리필터가 없을 수 있음$$.For this reason, some authors may refer to the π–system generated by ${\displaystyle {\mathcal {B}}}$ as the prefilter generated by ${\displaystyle {\mathcal {B}}.}$ However, as shown in an example below, if a ${\displaystyle \subseteq }$ –smallest prefilter does exist (say it is denoted by ${\$$displaystyle \operatorname {minPre} {\mathcal {B}}}$) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by ${\displaystyle {\mathcal {B}}}$" (that is, ${\displaystyle \operatorname {minPre} {\mathcal {B}}\neq \pi ({\mathcal {B}})}$ is possible).And if the filter subbase ${\displaystyle {\mathcal {B}}}$ happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning ${\displaystyle \pi ({\mathcal {B}})}$) will not be ${\displaystyle {\mathcal {B}}=\operatorname {minPre} {\m$$athcal {B}}}$ (that is, ${\displaystyle \pi ({\mathcal {B}})\neq {\mathcal {B}}}$ is possible even when ${\displaystyle {\mathcal {B}}}$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ${\displaystyle {\mathcal {B}}}$".
7. B{\displaystyle{{B\mathcal}}}[17][23]의 Subfilter 필터 F의{\displaystyle{{F\mathcal}}}과 F{\displaystyle{{F\mathcal}}}은 superfilter 만약}\subseteq{{F\mathcal}B{\displaystyle{{B\mathcal}}}은 필터와 B⊆ F{\displaystyle{{B\mathcal}}}이 filte.개발,${\displaystyle {\mathcal {B}\subseteq {\mathcal$$인 경우에만 해당$$}$
   • 중요한 것은 "superfilter is a superfilter"라는 표현은 "superfilter is a filters of"의 아날로그 필터에 대한 것이다.그래서 공통적으로 "sub"라는 접두사를 가지고 있음에도 불구하고, "subfilter is a subfilter"는 사실 "subpilter is a subpilter"의 역순이다.However, ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}}$ can also be written ${\displaystyle {\mathcal {F}}\vdash {\mathcal {B}}}$ which is described by saying "${\displaystyle {\mathcal {F}}}$ is subordinate to ${\displaystyle {\mathcal {B}}.}$" With this terminology, "is subor"dinate to"는 필터(그리고 프리필터의 경우에도)의 아날로그인 " is serpendence of"^[24]를 사용하게 되는데, 이는 "dinate"와 기호$$symbol $displaystyle \,\vdash \,}$을(를) 사용하는 것이 도움이 될 수 있는 하나의$$ 상황을 만든다.

$X{\displaystyle }$= ${\displaystyle \varnothing }$ ${\displaystyle$ X$=\varnothing }($또한 $\varnothing {\displaystyle }$ ${\displaystyle \varnothing }$에 가치 있는 그물도 없음)$$에 대한 프리필터가 없기 때문에 대부분의 저자와 마찬가지로 이 글은 이 가정이 필요할 때마다$$ $X{\displaystyle }$ ${\displaystyle \ne }$ ${\displaystyle \varnothing }$ ${\displaystyle$ X$\neq \varnothannothannothing }$을 자동으로 가정한다.

기본 예시

명명된 예제

• X{X\displaystyle}에[25][11]그것은 독특한 최소한의 필터 X{X\displaystyle}에 모든 필터의 하위 집합 그 singleton}X에서 또는 사소한 밀착한 필터라고 불린다.{X\displaystyle}모든 prefilter 하지만, 그것일 필요는 없다 하위 집합에 B){X}{\displaystyle{{B\mathcal}}=\{X\}을 세웠다.x$(\displaystyle X.}$
• 이중 이상적${\displaystyle \mathrm{}((X}$)$${\$displaystyle \wp(X)}$은 $X{\displaystyle }${\$displaystyle X}($^[10]실제로 필터가 아님에도 불구하고)의 퇴행적 필터라고도 불린다.$X{\displaystyle }$ ${\displaystyle X}$에서 X${\displaystyle }$. ${\displaystyle X}$의 필터가 아닌$$ 유일한 이중 이상이다.
• If ${\displaystyle (X,\tau )}$ is a topological space and ${\displaystyle x\in X,}$ then the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ at ${\displaystyle x}$ is a filter on ${\displaystyle X.}$ By definition, a family ${\displaystyle {\mathcal {B}}\$$B{\displaystyle \mathcal{}}$ ${\$$$${\mathcal {B}$이(가) 프리필터(resp)인$$ 경우에만${\displaystyle }$(${\displaystyle X}$,$){\displaystyle$ x${\text{}$에 ${\displaystyle \text{대한}}$ $x$에서 subrony bases($resp$. neighbase)라고$$ 한다.${\displaystyle \mathcal{B}}$ ${\$은$($는) 필터 하위 기반이며, $B{\displaystyle \mathcal{}}$ ${\$이(${\displaystyle 가}$) 생성하는$$$$ X ${\displaystyle X}$의 필터는 주변 $필터{\displaystyle \mathcal{}}$ N${\displaystyle (}$ x${\displaystyle }$)${\displaystyle }$. ${\displaystystyle {\mathcal}(x$)과 동일하다.$}}$ 열린 동네의 하위$$ 패밀리 ${\displaystyle \tau \mathcal{}}$( ${\displaystyle x}$) ${\displaystyle \subseteq }$ N${\displaystyle }$( ${\displaystyle x}$) ${\displaystyle \tau (x)\subseteq {\mathcal{N}(x)}}$는 N${\displaystyle }$( ${\displaystyle x}$)${\displaystyle }$. ${\displaystyle {\mathcal}(x$)의 필터 베이스다$.$$}$ Both prefilters ${\displaystyle {\mathcal {N}}(x){\text{ and }}\tau (x)}$ also form a bases for topologies on ${\displaystyle X,}$ with the topology generated ${\displaystyle \tau (x)}$ being coarser than ${\displaystyle \tau .}$ This example immediately generalizes from neighb비어 있지 않은 하위 집합 $S{\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle X.}$ ${\displaystyle S\subseteq$ X$.}$의 인접 지역에 대한 점의 변화
• ${\displaystyle {\mathcal {B}}}$ is an elementary prefilter^[26] if ${\displaystyle {\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)}$ for some sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X.$$}$
• ${\displaystyle }$$B{\displaystyle \mathcal{}}$ ${\$$displaystyle {\mathcal {B}}$이(가) 일부 기본 프리필터에 의해 $생성$된 X ${\$$displaystyle {\mathcal {B}$의 필터인 경우$$^[27] $X{\displaystyle }$ ${\$$displaystyle X}$에 대한 기본 필터 또는 순차 필터로$$,결국 일정하지 않은 시퀀스에 의해 생성되는 꼬리의 필터는 반드시 초필터가 아니다.^[28]카운트할 수 있는 세트의 모든 주 필터는 카운트할 수 있는 무한 세트의 모든 코피나이트 필터와 마찬가지로 순차적이다.^[10]정밀하게 많은 순차 필터들의 교차점은 다시 순차적이다.^[10]
• $X{\displaystyle }$ ${\$$displaystyle X}$의 모든 코피나이트 하위 집합(X {\displaystyle $X}$에서 보수가 유한한$$ 집합) 중 $설정$된 F ${\$ {F$}$이(가$$) 무한($또는{\displaystyle }$ $동등$하게$,$ X {\$displaysty X}$인 경우에만)이 적절하며, 이 경우 $F$가 무한하다.$${\displaystyle{{F\mathcal}}}은 필터에서 X{X\displaystyle}은 프레셰 필터 또는cofinite 필터에서 X.{X\displaystyle}[11][25]만약 X{X\displaystyle}은 유한한 다음 F{\displaystyle{{F\mathcal}}}와 이중 이상적인 ℘(X),{\displaystyle \wp(X),}이 아닌 filte.rIf ${\displaystyle X}$ is infinite then the family ${\displaystyle \{X\setminus \{x\}~:~x\in X\}}$ of complements of singleton sets is a filter subbase that generates the Fréchet filter on ${\displaystyle X.}$ As with any family of sets over ${\displaystyle X}$ that contains ${\displaystyle \{X\setminus \{x\}}$${\displaystyle :}$ ${\displaystyle \text{X}}$$$${\displaystyle \in }$ ${\displaystyle X\},}$ ${\displaystyle \{X\setminus \{x\}~:~x\$$in$ X$\}}$에 있는 Frechet 필터의$$ 커널은 비어 있는 집합$$: ${\displaystyle \ker f}$F =${\displaystyle }$∅${\displaystyle }$.$${\$displaystyle \ker${\$mathcal}=\varnothot.}}}}}}$
• The intersection of all elements in any non–empty family ${\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X)}$ is itself a filter on ${\displaystyle X}$ called the infimum or greatest lower bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X$$),}$ which is why it may be denoted by ${\displaystyle \bigwedge _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ Said differently, ${\displaystyle \ker \mathbb {F} =\bigcap _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}\in \operatorname {Filters} (X).$$}}$ $X{\displaystyle }$ ${\displaystyle X}$의 모든 필터에는 $\{{\displaystyle X\}}$ ${\displaystyle \{X\}$이 부분 집합으로 포함되어$$ 있기$$ 때문에$$ 이 교차로는 절대 비어 있지 않다.정의에 의해, 하한은finest/largest(그리고 ≤ ⊆에 비례하여{\displaystyle\,\subseteq \,{\text{과}}\,\leq\,})필터 F.{\displaystyle \mathbb{F}의 각 멤버에 대한 기본이 된다.}[11]만약 B형과 F{\displaystyle{{B\mathcal}}{\text{과}}{{F\mathcal}}}은 필터 그 다음에 그들의 i.nfi엄마 필터에(X){\displaystyle \operatorname{필터}(X)⁡}은 필터가 B(∪)F.{\displaystyle{{B\mathcal}}\,(\cup)\,{{F\mathcal}}.}[9]만약 B형과 F{\displaystyle{{B\mathcal}}{\text{과}}{{F\mathcal}}}은 prefilters 그 다음에 B(∪)F{\displaystyle{{B\mathcal}}\,(\cup).
  • $\,{\mathcal {F}}}$ is a prefilter that is coarser (with respect to ${\displaystyle \,\leq }$) than both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ (that is, ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal$
  • ${B}}{\text{ and }}{\mathcal {B}}\,(\cup )\,{\mathcal {F}}\leq {\mathcal {F}}}$); indeed, it is one of the finest such prefilters, meaning that if ${\displaystyle {\mathcal {S}}}$ is a prefilter such that ${\displaystyle {\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal$
  {F}}}그때 반드시 S≤ B(∪)F.{\displaystyle{{S\mathcal}}\leq{{B\mathcal}}\,(\cup)\,{{F\mathcal}}.}[9]더 일반적으로, 만약 B형과 F{\displaystyle{{B\mathcal}}{\text{과}}{{F\mathcal}}}은non−empty 가족들과 S다음과 같은 경우={SB와 S≤ F≤℘(X):S⊆}.
  • ${\displaystyle \mathbb {S} :=\{{\mathcal {S}}\subseteq \wp (X)~:~{\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}\}}$ then ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}\in \mathbb {S} }$ and ${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {F}}}$
  • $S{\displaystyle \mathbb{}}$ . ${\displaystyle$$\leq$ $\}$에 대한 가장 큰 요소(S$.$ ${\displaystyle$ \$leq }).$
• Let ${\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {DualIdeals} (X)}$ and let ${\displaystyle \cup \mathbb {F} =\bigcup _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ The supremum or least upper bound of ${\disp$$laystyle \mathbb {F} {\text{ in }}\operatorname {DualIdeals} (X),}$ denoted by ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}},}$ is the smallest (relative to ${\displaystyle \subseteq }$) dual ideal on ${\displaystyle X}$ containing every element of ${\displaystyle \$부분 집합으로$$ $mathbb {F}$}. 즉, 부분 ${\displaystyle 집합}$으로$∪$F {\$displaystyle$$$\$subseteq }$을(를$$) $포함$하는$$X {\$displaystyle$ X$}$에서 가장 작은 이상이다.This dual ideal is ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X},}$ where ${\displaystyle \pi$$\left(\cup \cup \mathb {F} \right):$$=\왼쪽\{$$F_{1}\cap \cdots \cap F_{n}~:~n\in \mathbb {N} {\text{ and every }}F_{i}{\text{ belongs to some }}{\mathcal {F}}\in \mathbb {F} \right\}}$ is the π–system generated by ${\displaystyle \cup \mathbb {F} .}$ As with any non–empty family of sets, ${\displaystyle \cup \mathbb {F} }$ is contained in some filter on ${\di$$splaystyle X}$ if and only if it is a filter subbase, or equivalently, if and only if ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$ is a filter on ${\displaystyle X,}$ in which case this family is the smallest$({\displaystyle }$$F{\displaystyle \mathbb{}}${\$displaystyle$\$subseteq$에 대한 $)$ F {\$displaystyle$ \$mathb$ {$F}$의 모든 요소를 하위 집합으로$$ 포함하고$$ 반드시 F${\displaystyle }f{\displaystyle \mathbb{}}$ ${\displaystyle \mathrm{필터}}$ ( ${\displaystyle X}$)${\displaystyle }$. ${\displaystyle \mathb {F} \subseteq \operatorname {Filters}(X$)에 대한 필터$.$$}$
• Let ${\displaystyle \varnothing \neq \mathbb {F} \subseteq \operatorname {Filters} (X)}$ and let ${\displaystyle \cup \mathbb {F} =\bigcup _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ The supremum or least upper bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),}$ denoted by ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}}$ if it exists, is by definition the smallest (relative to ${\displaystyle \subseteq }$) filter on ${\displaystyle X}$ containing every element of ${\displaystyle \mathbb {F} }$ as a subset. If it exists then necessarily ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$^[11] (as defined above) and ${\displaystyle \bigvee _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}}$ will also be equal to the intersection of all filters on ${\displaystyle X}$ containing ${\displaystyle \cup \mathbb {F} .}$ This supremum of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)}$ exists if and only if the dual ideal ${\displaystyle \pi \left(\cup \mathbb {F} \right)^{\uparrow X}}$ is a filter on ${\displaystyle X.}$ The least upper bound of a family of filters ${\displaystyle \mathbb {F} }$ may fail to be a filter.^[11] Indeed, if ${\displaystyle X}$ contains at least 2 distinct elements then there exist filters ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}{\text{ on }}X}$ for which there does not exist a filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ that contains both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ If ${\displaystyle \cup \mathbb {F} }$ is not a filter subbase then the supremum of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X)}$ does not exist and the same is true of its supremum in ${\displaystyle \operatorname {Prefilters} (X)}$ but their supremum in the set of all dual ideals on ${\displaystyle X}$ will exist (it being the degenerate filter ${\displaystyle \wp (X)}$).^[10]
  • If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters (resp. filters on ${\displaystyle X}$) then ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}}$ is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filter) on ${\displaystyle X}$ (with respect to ${\displaystyle \,\leq }$) that is finer (with respect to ${\displaystyle \,\leq }$) than both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}};}$ this means that if ${\displaystyle {\mathcal {S}}}$ is any prefilter (resp. any filter) such that ${\displaystyle {\mathcal {B}}\leq {\mathcal {S}}{\text{ and }}{\mathcal {F}}\leq {\mathcal {S}}}$ then necessarily ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}\leq {\mathcal {S}},}$^[9] in which case it is denoted by ${\displaystyle {\mathcal {B}}\vee {\mathcal {F}}.}$^[10]
• Let ${\displaystyle I{\text{ and }}X}$ be non−empty sets and for every ${\displaystyle i\in I}$ let ${\displaystyle {\mathcal {D}}_{i}}$ be a dual ideal on ${\displaystyle X.}$ If ${\displaystyle {\mathcal {I}}}$ is any dual ideal on ${\displaystyle I}$ then ${\displaystyle \bigcup _{\Xi \in {\mathcal {I}}}\;\;\bigcap _{i\in \Xi }\;{\mathcal {D}}_{i}}$ is a dual ideal on ${\displaystyle X}$ called Kowalsky's dual ideal or Kowalsky's filter.^[17]

Other examples

• Let ${\displaystyle X=\{p,1,2,3\}}$ and let ${\displaystyle {\mathcal {B}}=\{\{p\},\{p,1,2\},\{p,1,3\}\},}$ which makes ${\displaystyle {\mathcal {B}}}$ a prefilter and a filter subbase that is not closed under finite intersections. Because ${\displaystyle {\mathcal {B}}}$ is a prefilter, the smallest prefilter containing ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}.}$ The π–system generated by ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle \{\{p,1\}\}\cup {\mathcal {B}}.}$ In particular, the smallest prefilter containing the filter subbase ${\displaystyle {\mathcal {B}}}$ is not equal to the set of all finite intersections of sets in ${\displaystyle {\mathcal {B}}.}$ The filter on ${\displaystyle X}$ generated by ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}^{\uparrow X}=\{S\subseteq X:p\in S\}=\{\{p\}\cup T~:~T\subseteq \{1,2,3\}\}.}$ All three of ${\displaystyle {\mathcal {B}},}$ the π–system ${\displaystyle {\mathcal {B}}}$ generates, and ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ are examples of fixed, principal, ultra prefilters that are principal at the point ${\displaystyle p;{\mathcal {B}}^{\uparrow X}}$ is also an ultrafilter on ${\displaystyle X.}$
• Let ${\displaystyle (X,\tau )}$ be a topological space, ${\displaystyle {\mathcal {B}}\subseteq \wp (X),}$ and define ${\displaystyle {\overline {\mathcal {B}}}:=\left\{\operatorname {cl} _{X}B~:~B\in {\mathcal {B}}\right\},}$ where ${\displaystyle {\mathcal {B}}}$ is necessarily finer than ${\displaystyle {\overline {\mathcal {B}}}.}$^[29] If ${\displaystyle {\mathcal {B}}}$ is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of ${\displaystyle {\overline {\mathcal {B}}}.}$ If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle {\overline {\mathcal {B}}}}$ is a prefilter but not necessarily a filter on ${\displaystyle X}$ although ${\displaystyle \left({\overline {\mathcal {B}}}\right)^{\uparrow X}}$ is a filter on ${\displaystyle X}$ equivalent to ${\displaystyle {\overline {\mathcal {B}}}.}$
• The set ${\displaystyle {\mathcal {B}}}$ of all dense open subsets of a (non–empty) topological space ${\displaystyle X}$ is a proper π–system and so also a prefilter. If ${\displaystyle X=\mathbb {R} ^{n}}$ (with ${\displaystyle 1\leq n\in \mathbb {N} }$) then the set ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }}$ of all ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B}$ has finite Lebesgue measure is a proper π–system and prefilter that is also a proper subset of ${\displaystyle {\mathcal {B}}.}$ The prefilters ${\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }{\text{ and }}{\mathcal {B}}}$ generate the same filter on ${\displaystyle X.}$
• A filter subbase with no ${\displaystyle \,\subseteq -}$smallest prefilter containing it: In general, if a filter subbase ${\displaystyle {\mathcal {S}}}$ is not a π–system then an intersection ${\displaystyle S_{1}\cap \cdots \cap S_{n}}$ of ${\displaystyle n}$ sets from ${\displaystyle {\mathcal {S}}}$ will usually require a description involving ${\displaystyle n}$ variables that cannot be reduced down to only two (consider, for instance ${\displaystyle \pi ({\mathcal {S}})}$ when ${\displaystyle {\mathcal {S}}=\{(-\infty ,r)\cup (r,\infty )~:~r\in \mathbb {R} \}}$). This example illustrates an atypical class of a filter subbases ${\displaystyle {\mathcal {S}}_{R}}$ where all sets in both ${\displaystyle {\mathcal {S}}_{R}}$ and its generated π–system can be described as sets of the form ${\displaystyle B_{r,s},}$ so that in particular, no more than two variables (specifically, ${\displaystyle r{\text{ and }}s}$) are needed to describe the generated π–system. For all ${\displaystyle r,s\in \mathbb {R} ,}$ let
  $${\displaystyle B_{r,s}=(r,0)\cup (s,\infty ),}$$

  
  where ${\displaystyle B_{r,s}=B_{\min(r,s),s}}$ always holds so no generality is lost by adding the assumption ${\displaystyle r\leq s.}$ For all real ${\displaystyle r\leq s{\text{ and }}u\leq v,}$ if ${\displaystyle s{\text{ or }}v}$ is non-negative then ${\displaystyle B_{-r,s}\cap B_{-u,v}=B_{-\min(r,u),\max(s,v)}.}$^{[note 6]} For every set ${\displaystyle R}$ of positive reals, let^{[note 7]}
  $${\displaystyle {\mathcal {S}}_{R}:=\left\{B_{-r,r}:r\in R\right\}=\{(-r,0)\cup (r,\infty ):r\in R\}\quad {\text{ and }}\quad {\mathcal {B}}_{R}:=\left\{B_{-r,s}:r\leq s{\text{ with }}r,s\in R\right\}=\{(-r,0)\cup (s,\infty ):r\leq s{\text{ in }}R\}.}$$

  
  Let ${\displaystyle X=\mathbb {R} }$ and suppose ${\displaystyle \varnothing \neq R\subseteq (0,\infty )}$ is not a singleton set. Then ${\displaystyle {\mathcal {S}}_{R}}$ is a filter subbase but not a prefilter and ${\displaystyle {\mathcal {B}}_{R}=\pi \left({\mathcal {S}}_{R}\right)}$ is the π–system it generates, so that ${\displaystyle {\mathcal {B}}_{R}^{\uparrow X}}$ is the unique smallest filter in ${\displaystyle X=\mathbb {R} }$ containing ${\displaystyle {\mathcal {S}}_{R}.}$ However, ${\displaystyle {\mathcal {S}}_{R}^{\uparrow X}}$ is not a filter on ${\displaystyle X}$ (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and ${\displaystyle {\mathcal {S}}_{R}^{\uparrow X}}$ is a proper subset of the filter ${\displaystyle {\mathcal {B}}_{R}^{\uparrow X}.}$ If ${\displaystyle R,S\subseteq (0,\infty )}$ are non−empty intervals then the filter subbases ${\displaystyle {\mathcal {S}}_{R}{\text{ and }}{\mathcal {S}}_{S}}$ generate the same filter on ${\displaystyle X}$ if and only if ${\displaystyle R=S.}$ If ${\displaystyle {\mathcal {C}}}$ is a prefilter satisfying ${\displaystyle {\mathcal {S}}_{(0,\infty )}\subseteq {\mathcal {C}}\subseteq {\mathcal {B}}_{(0,\infty )}}$^{[note 8]} then for any ${\displaystyle C\in {\mathcal {C}}\setminus {\mathcal {S}}_{(0,\infty )},}$ the family ${\displaystyle {\mathcal {C}}\setminus \{C\}}$ is also a prefilter satisfying ${\displaystyle {\mathcal {S}}_{(0,\infty )}\subseteq {\mathcal {C}}\setminus \{C\}\subseteq {\mathcal {B}}_{(0,\infty )}.}$ This shows that there cannot exist a minimal/least (with respect to ${\displaystyle \subseteq }$) prefilter that both contains ${\displaystyle {\mathcal {S}}_{(0,\infty )}}$ and is a subset of the π–system generated by ${\displaystyle {\mathcal {S}}_{(0,\infty )}.}$ This remains true even if the requirement that the prefilter be a subset of ${\displaystyle {\mathcal {B}}_{(0,\infty )}=\pi \left({\mathcal {S}}_{(0,\infty )}\right)}$ is removed; that is, (in sharp contrast to filters) there does not exist a minimal/least (with respect to ${\displaystyle \subseteq }$) prefilter containing the filter subbase ${\displaystyle {\mathcal {S}}_{(0,\infty )}.}$

Ultrafilters

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.

A non–empty family ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ of sets is/is an:

8. Ultra^[8][30] if ${\displaystyle \varnothing \not \in {\mathcal {B}}}$ and any of the following equivalent conditions are satisfied:
   1. For every set ${\displaystyle S\subseteq X}$ there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\subseteq S{\text{ or }}B\subseteq X\setminus S}$ (or equivalently, such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing }$).
   2. For every set ${\displaystyle S\subseteq \bigcup _{B\in {\mathcal {B}}}B}$ there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing .}$
      • This characterization of "${\displaystyle {\mathcal {B}}}$ is ultra" does not depend on the set ${\displaystyle X,}$ so mentioning the set ${\displaystyle X}$ is optional when using the term "ultra."
   3. For every set ${\displaystyle S}$ (not necessarily even a subset of ${\displaystyle X}$) there exists some set ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B\cap S{\text{ equals }}B{\text{ or }}\varnothing .}$
      • If ${\displaystyle {\mathcal {B}}}$ satisfies this condition then so does every superset ${\displaystyle {\mathcal {F}}\supseteq {\mathcal {B}}.}$ For example, if ${\displaystyle T}$ is any singleton set then ${\displaystyle \{T\}}$ is ultra and consequently, any non–degenerate superset of ${\displaystyle \{T\}}$ (such as its upward closure) is also ultra.
9. Ultra prefilter^[8][30] if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter ${\displaystyle {\mathcal {B}}}$ is ultra if and only if it satisfies any of the following equivalent conditions:
   1. ${\displaystyle {\mathcal {B}}}$ is maximal in ${\displaystyle \operatorname {Prefilters} (X)}$ with respect to ${\displaystyle \,\leq ,\,}$ which means that
      $${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Prefilters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$$

      
   2. ${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Filters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$
      • Although this statement is identical to that given below for ultrafilters, here ${\displaystyle {\mathcal {B}}}$ is merely assumed to be a prefilter; it need not be a filter.
   3. ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is ultra (and thus an ultrafilter).
   4. ${\displaystyle {\mathcal {B}}}$ is equivalent (with respect to ${\displaystyle \leq }$) to some ultrafilter.
   • A filter subbase that is ultra is necessarily a prefilter. A filter subbase is ultra if and only if it is a maximal filter subbase with respect to ${\displaystyle \,\leq \,}$ (as above).^[17]
10. Ultrafilter on ${\displaystyle X}$^[8][30] if it is a filter on ${\displaystyle X}$ that is ultra. Equivalently, an ultrafilter on ${\displaystyle X}$ is a filter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ that satisfies any of the following equivalent conditions:
    1. ${\displaystyle {\mathcal {B}}}$ is generated by an ultra prefilter.
    2. For any ${\displaystyle S\subseteq X,S\in {\mathcal {B}}{\text{ or }}X\setminus S\in {\mathcal {B}}.}$^[17]
    3. ${\displaystyle {\mathcal {B}}\cup (X\setminus {\mathcal {B}})=\wp (X).}$ This condition can be restated as: ${\displaystyle \wp (X)}$ is partitioned by ${\displaystyle {\mathcal {B}}}$ and its dual ${\displaystyle X\setminus {\mathcal {B}}.}$
       • The sets ${\displaystyle {\mathcal {B}}{\text{ and }}X\setminus {\mathcal {B}}}$ are disjoint whenever ${\displaystyle {\mathcal {B}}}$ is a prefilter.
    4. ${\displaystyle \wp (X)\setminus {\mathcal {B}}=\{S\in \wp (X):S\not \in {\mathcal {B}}\}}$ is an ideal.^[17]
    5. For any ${\displaystyle R,S\subseteq X,}$ if ${\displaystyle R\cup S=X}$ then ${\displaystyle R\in {\mathcal {B}}{\text{ or }}S\in {\mathcal {B}}.}$
    6. For any ${\displaystyle R,S\subseteq X,}$ if ${\displaystyle R\cup S\in {\mathcal {B}}}$ then ${\displaystyle R\in {\mathcal {B}}{\text{ or }}S\in {\mathcal {B}}}$ (a filter with this property is called a prime filter).
       • This property extends to any finite union of two or more sets.
    7. For any ${\displaystyle R,S\subseteq X,}$ if ${\displaystyle R\cup S\in {\mathcal {B}}{\text{ and }}R\cap S=\varnothing }$ then either ${\displaystyle R\in {\mathcal {B}}{\text{ or }}S\in {\mathcal {B}}.}$
    8. ${\displaystyle {\mathcal {B}}}$ is a maximal filter on ${\displaystyle X}$; meaning that if ${\displaystyle {\mathcal {C}}}$ is a filter on ${\displaystyle X}$ such that ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {C}}}$ then necessarily ${\displaystyle {\mathcal {C}}={\mathcal {B}}}$ (this equality may be replaced by ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {B}}{\text{ or by }}{\mathcal {C}}\leq {\mathcal {B}}}$).
       • If ${\displaystyle {\mathcal {C}}}$ is upward closed then ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ if and only if }}{\mathcal {B}}\subseteq {\mathcal {C}}.}$ So this characterization of ultrafilters as maximal filters can be restated as:
         $${\displaystyle {\text{For all }}{\mathcal {C}}\in \operatorname {Filters} (X),\;{\mathcal {B}}\leq {\mathcal {C}}\;{\text{ implies }}\;{\mathcal {C}}\leq {\mathcal {B}}.}$$

         
       • Because subordination ${\displaystyle \,\geq \,}$ is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA–subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from ${\displaystyle X}$" in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example),^{[note 9]} which is an idea that is actually made rigorous by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

Any non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property. The trivial filter ${\displaystyle \{X\}{\text{ on }}X}$ is ultra if and only if ${\displaystyle X}$ is a singleton set.

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).^[31]

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.^{[11][proof 1]} Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

Kernels

The kernel is useful in classifying properties of prefilters and other families of sets.

The kernel^[6] of a family of sets ${\displaystyle {\mathcal {B}}}$ is the intersection of all sets that are elements of ${\displaystyle {\mathcal {B}}:}$

$${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$$

If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then for any point ${\displaystyle x,x\not \in \ker {\mathcal {B}}{\text{ if and only if }}X\setminus \{x\}\in {\mathcal {B}}^{\uparrow X}.}$

Properties of kernels

If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle \ker \left({\mathcal {B}}^{\uparrow X}\right)=\ker {\mathcal {B}}}$ and this set is also equal to the kernel of the π–system that is generated by ${\displaystyle {\mathcal {B}}.}$ In particular, if ${\displaystyle {\mathcal {B}}}$ is a filter subbase then the kernels of all of the following sets are equal:

(1) ${\displaystyle {\mathcal {B}},}$ (2) the π–system generated by ${\displaystyle {\mathcal {B}},}$ and (3) the filter generated by ${\displaystyle {\mathcal {B}}.}$

If ${\displaystyle f}$ is a map then ${\displaystyle f(\ker {\mathcal {B}})\subseteq \ker f({\mathcal {B}}){\text{ and }}f^{-1}(\ker {\mathcal {B}})=\ker f^{-1}({\mathcal {B}}).}$ If ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ then }}\ker {\mathcal {C}}\subseteq \ker {\mathcal {B}}}$ while if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are equivalent then ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are principal then they are equivalent if and only if ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}.}$

Classifying families by their kernels

A family ${\displaystyle {\mathcal {B}}}$ of sets is/is an:

1. Free^[7] if ${\displaystyle \ker {\mathcal {B}}=\varnothing ,}$ or equivalently, if ${\displaystyle \{X\setminus \{x\}~:~x\in X\}\subseteq {\mathcal {B}}^{\uparrow X};}$ this can be restated as ${\displaystyle \{X\setminus \{x\}~:~x\in X\}\leq {\mathcal {B}}.}$
   • A filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ is free if and only if ${\displaystyle X}$ is infinite and ${\displaystyle {\mathcal {F}}}$ contains the Fréchet filter on ${\displaystyle X}$ as a subset.
2. Fixed if ${\displaystyle \ker {\mathcal {B}}\neq \varnothing }$ in which case, ${\displaystyle {\mathcal {B}}}$ is said to be fixed by any point ${\displaystyle x\in \ker {\mathcal {B}}.}$
   • Any fixed family is necessarily a filter subbase.
3. Principal^[7] if ${\displaystyle \ker {\mathcal {B}}\in {\mathcal {B}}.}$
   • A proper principal family of sets is necessarily a prefilter.
4. Discrete or Principal at${\displaystyle x\in X}$^[25] if ${\displaystyle \{x\}=\ker {\mathcal {B}}\in {\mathcal {B}}.}$
   • The principal filter at ${\displaystyle x{\text{ on }}X}$ is the filter ${\displaystyle \{x\}^{\uparrow X}.}$ A filter ${\displaystyle {\mathcal {F}}}$ is principal at ${\displaystyle x}$ if and only if ${\displaystyle {\mathcal {F}}=\{x\}^{\uparrow X}.}$
5. Countably deep if whenever ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}}$ is a countable subset then ${\displaystyle \ker {\mathcal {C}}\in {\mathcal {B}}.}$^[10]

If ${\displaystyle {\mathcal {B}}}$ is a principal filter on ${\displaystyle X}$ then ${\displaystyle \varnothing \neq \ker {\mathcal {B}}\in {\mathcal {B}}}$ and

where ${\displaystyle \{\ker {\mathcal {B}}\}}$ is also the smallest prefilter that generates ${\displaystyle {\mathcal {B}}.}$

Family of examples: For any non–empty ${\displaystyle C\subseteq \mathbb {R} ,}$ the family ${\displaystyle {\mathcal {B}}_{C}=\{\mathbb {R} \setminus (r+C)~:~r\in \mathbb {R} \}}$ is free but it is a filter subbase if and only if no finite union of the form ${\displaystyle \left(r_{1}+C\right)\cup \cdots \cup \left(r_{n}+C\right)}$ covers ${\displaystyle \mathbb {R} ,}$ in which case the filter that it generates will also be free. In particular, ${\displaystyle {\mathcal {B}}_{C}}$ is a filter subbase if ${\displaystyle C}$ is countable (for example, ${\displaystyle C=\mathbb {Q} ,\mathbb {Z} ,}$ the primes), a meager set in ${\displaystyle \mathbb {R} ,}$ a set of finite measure, or a bounded subset of ${\displaystyle \mathbb {R} .}$ If ${\displaystyle C}$ is a singleton set then ${\displaystyle {\mathcal {B}}_{C}}$ is a subbase for the Fréchet filter on ${\displaystyle \mathbb {R} .}$

For every filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ there exists a unique pair of dual ideals ${\displaystyle {\mathcal {F}}^{*}{\text{ and }}{\mathcal {F}}^{\bullet }{\text{ on }}X}$ such that ${\displaystyle {\mathcal {F}}^{*}}$ is free, ${\displaystyle {\mathcal {F}}^{\bullet }}$ is principal, and ${\displaystyle {\mathcal {F}}^{*}\wedge {\mathcal {F}}^{\bullet }={\mathcal {F}},}$ and ${\displaystyle {\mathcal {F}}^{*}{\text{ and }}{\mathcal {F}}^{\bullet }}$ do not mesh (that is, ${\displaystyle {\mathcal {F}}^{*}\vee {\mathcal {F}}^{\bullet }=\wp (X)}$). The dual ideal ${\displaystyle {\mathcal {F}}^{*}}$ is called the free part of ${\displaystyle {\mathcal {F}}}$ while ${\displaystyle {\mathcal {F}}^{\bullet }}$ is called the principal part^[10] where at least one of these dual ideals is filter. If ${\displaystyle {\mathcal {F}}}$ is principal then ${\displaystyle {\mathcal {F}}^{\bullet }:={\mathcal {F}}{\text{ and }}{\mathcal {F}}^{*}:=\wp (X);}$ otherwise, ${\displaystyle {\mathcal {F}}^{\bullet }:=\{\ker {\mathcal {F}}\}^{\uparrow X}}$ and ${\displaystyle {\mathcal {F}}^{*}:={\mathcal {F}}\vee \{X\setminus \left(\ker {\mathcal {F}}\right)\}^{\uparrow X}}$ is a free (non–degenerate) filter.^[10]

Finite prefilters and finite sets

If a filter subbase ${\displaystyle {\mathcal {B}}}$ is finite then it is fixed (that is, not free); this is because ${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$ is a finite intersection and the filter subbase ${\displaystyle {\mathcal {B}}}$ has the finite intersection property. A finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.

If ${\displaystyle X}$ is finite then all of the conclusions above hold for any ${\displaystyle {\mathcal {B}}\subseteq \wp (X).}$ In particular, on a finite set ${\displaystyle X,}$ there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on ${\displaystyle X}$ are principal filters generated by their (non–empty) kernels.

The trivial filter ${\displaystyle \{X\}}$ is always a finite filter on ${\displaystyle X}$ and if ${\displaystyle X}$ is infinite then it is the only finite filter because a non–trivial finite filter on a set ${\displaystyle X}$ is possible if and only if ${\displaystyle X}$ is finite. However, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters). If ${\displaystyle X}$ is a singleton set then the trivial filter ${\displaystyle \{X\}}$ is the only proper subset of ${\displaystyle \wp (X)}$ and moreover, this set ${\displaystyle \{X\}}$ is a principal ultra prefilter and any superset ${\displaystyle {\mathcal {F}}\supseteq {\mathcal {B}}}$ (where ${\displaystyle {\mathcal {F}}\subseteq \wp (Y){\text{ and }}X\subseteq Y}$) with the finite intersection property will also be a principal ultra prefilter (even if ${\displaystyle Y}$ is infinite).

Characterizing fixed ultra prefilters

If a family of sets ${\displaystyle {\mathcal {B}}}$ is fixed (that is, ${\displaystyle \ker {\mathcal {B}}\neq \varnothing }$) then ${\displaystyle {\mathcal {B}}}$ is ultra if and only if some element of ${\displaystyle {\mathcal {B}}}$ is a singleton set, in which case ${\displaystyle {\mathcal {B}}}$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter ${\displaystyle {\mathcal {B}}}$ is ultra if and only if ${\displaystyle \ker {\mathcal {B}}}$ is a singleton set.

Every filter on ${\displaystyle X}$ that is principal at a single point is an ultrafilter, and if in addition ${\displaystyle X}$ is finite, then there are no ultrafilters on ${\displaystyle X}$ other than these.^[7]

The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

Finer/coarser, subordination, and meshing

The preorder ${\displaystyle \,\leq \,}$ that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence",^[24] where "${\displaystyle {\mathcal {F}}\geq {\mathcal {C}}}$" can be interpreted as "${\displaystyle {\mathcal {F}}}$ is a subsequence of ${\displaystyle {\mathcal {C}}}$" (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of ${\displaystyle {\mathcal {B}}}$ meshes with ${\displaystyle {\mathcal {C}},}$ which is closely related to the preorder ${\displaystyle \,\leq ,}$ is used in Topology to define cluster points.

Two families of sets ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh^[8] and are compatible, indicated by writing ${\displaystyle {\mathcal {B}}\#{\mathcal {C}},}$ if ${\displaystyle B\cap C\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ do not mesh then they are dissociated. If ${\displaystyle S\subseteq X{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ are said to mesh if ${\displaystyle {\mathcal {B}}{\text{ and }}\{S\}}$ mesh, or equivalently, if the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S,}$ which is the family

does not contain the empty set, where the trace is also called the restriction of ${\displaystyle {\mathcal {B}}{\text{ to }}S.}$

Declare that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},{\mathcal {F}}\geq {\mathcal {C}},{\text{ and }}{\mathcal {F}}\vdash {\mathcal {C}},}$ stated as ${\displaystyle {\mathcal {C}}}$ is coarser than${\displaystyle {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ is finer than (or subordinate to) ${\displaystyle {\mathcal {C}},}$^{[11][12][13][9][10]} if any of the following equivalent conditions hold:

1. Definition: Every ${\displaystyle C\in {\mathcal {C}}}$contains some ${\displaystyle F\in {\mathcal {F}}.}$ Explicitly, this means that for every ${\displaystyle C\in {\mathcal {C}},}$ there is some ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle F\subseteq C.}$
   • Said more briefly in plain English, ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ if every set in ${\displaystyle {\mathcal {C}}}$ is larger than some set in ${\displaystyle {\mathcal {F}}.}$ Here, a "larger set" means a superset.
2. ${\displaystyle \{C\}\leq {\mathcal {F}}{\text{ for every }}C\in {\mathcal {C}}.}$
   • In words, ${\displaystyle \{C\}\leq {\mathcal {F}}}$ states exactly that ${\displaystyle C}$ is larger than some set in ${\displaystyle {\mathcal {F}}.}$ The equivalence of (a) and (b) follows immediately.
   • From this characterization, it follows that if ${\displaystyle \left({\mathcal {C}}_{i}\right)_{i\in I}}$ are families of sets, then ${\displaystyle \bigcup _{i\in I}{\mathcal {C}}_{i}\leq {\mathcal {F}}{\text{ if and only if }}{\mathcal {C}}_{i}\leq {\mathcal {F}}{\text{ for all }}i\in I.}$
3. ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}^{\uparrow X},}$ which is equivalent to ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}^{\uparrow X}}$;
4. ${\displaystyle {\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}}$;
5. ${\displaystyle {\mathcal {C}}^{\uparrow X}\leq {\mathcal {F}}^{\uparrow X},}$ which is equivalent to ${\displaystyle {\mathcal {C}}^{\uparrow X}\subseteq {\mathcal {F}}^{\uparrow X}}$;

and if in addition ${\displaystyle {\mathcal {F}}}$ is upward closed, which means that ${\displaystyle {\mathcal {F}}={\mathcal {F}}^{\uparrow X},}$ then this list can be extended to include:

6. ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {F}}.}$^[6]
   • So in this case, this definition of "${\displaystyle {\mathcal {F}}}$ is finer than ${\displaystyle {\mathcal {C}}}$" would be identical to the topological definition of "finer" had ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ been topologies on ${\displaystyle X.}$

If an upward closed family ${\displaystyle {\mathcal {F}}}$ is finer than ${\displaystyle {\mathcal {C}}}$ (that is, ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$) but ${\displaystyle {\mathcal {C}}\neq {\mathcal {F}}}$ then ${\displaystyle {\mathcal {F}}}$ is said to be strictly finer than ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {C}}}$ is strictly coarser than ${\displaystyle {\mathcal {F}}.}$

Two families are comparable if one of these sets is finer than the other.^[11]

For example, if ${\displaystyle {\mathcal {B}}}$ is any family then ${\displaystyle \varnothing \leq {\mathcal {B}}\leq {\mathcal {B}}\leq \{\varnothing \}}$ always holds and furthermore, ${\displaystyle \{\varnothing \}\leq {\mathcal {B}}{\text{ if and only if }}\varnothing \in {\mathcal {B}}.}$

Assume that ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ are families of sets that satisfy ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ and }}{\mathcal {C}}\leq {\mathcal {F}}.}$ Then ${\displaystyle \ker {\mathcal {F}}\subseteq \ker {\mathcal {C}},}$ and ${\displaystyle {\mathcal {C}}\neq \varnothing {\text{ implies }}{\mathcal {F}}\neq \varnothing ,}$ and also ${\displaystyle \varnothing \in {\mathcal {C}}{\text{ implies }}\varnothing \in {\mathcal {F}}.}$ If in addition to ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},{\mathcal {F}}}$ is a filter subbase and ${\displaystyle {\mathcal {C}}\neq \varnothing ,}$ then ${\displaystyle {\mathcal {C}}}$ is a filter subbase^[9] and also ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ mesh.^{[19][proof 2]} More generally, if both ${\displaystyle \varnothing \neq {\mathcal {B}}\leq {\mathcal {F}}{\text{ and }}\varnothing \neq {\mathcal {C}}\leq {\mathcal {F}}}$ and if the intersection of any two elements of ${\displaystyle {\mathcal {F}}}$ is non–empty, then ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh.^{[proof 2]} Every filter subbase is coarser than both the π–system that it generates and the filter that it generates.^[9]

If ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}}$ are families such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}},}$ the family ${\displaystyle {\mathcal {C}}}$ is ultra, and ${\displaystyle \varnothing \not \in {\mathcal {F}},}$ then ${\displaystyle {\mathcal {F}}}$ is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily be ultra. In particular, if ${\displaystyle {\mathcal {C}}}$ is a prefilter then either both ${\displaystyle {\mathcal {C}}}$ and the filter ${\displaystyle {\mathcal {C}}^{\uparrow X}}$ it generates are ultra or neither one is ultra. If a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra. A filter subbase ${\displaystyle {\mathcal {B}}}$ that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by ${\displaystyle {\mathcal {B}}}$ to be ultra. If ${\displaystyle S\subseteq X{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ is upward closed in ${\displaystyle X}$ then ${\displaystyle S\not \in {\mathcal {B}}{\text{ if and only if }}(X\setminus S)\#{\mathcal {B}}.}$^[10]

Relational properties of subordination

The relation ${\displaystyle \,\leq \,}$ is reflexive and transitive, which makes it into a preorder on ${\displaystyle \wp (\wp (X)).}$^[32]

Symmetry: For any ${\displaystyle {\mathcal {B}}\subseteq \wp (X),{\mathcal {B}}\leq \{X\}{\text{ if and only if }}\{X\}={\mathcal {B}}.}$ So the set ${\displaystyle X}$ has more than one point if and only if the relation ${\displaystyle \,\leq \,{\text{ on }}\operatorname {Filters} (X)}$ is not symmetric.

Antisymmetry: If ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {C}}{\text{ then }}{\mathcal {B}}\leq {\mathcal {C}}}$ but while the converse does not hold in general, it does hold if ${\displaystyle {\mathcal {C}}}$ is upward closed (such as if ${\displaystyle {\mathcal {C}}}$ is a filter). Two filters are equivalent if and only if they are equal, which makes the restriction of ${\displaystyle \,\leq \,}$ to ${\displaystyle \operatorname {Filters} (X)}$ antisymmetric. But in general, ${\displaystyle \,\leq \,}$ is not antisymmetric on ${\displaystyle \operatorname {Prefilters} (X)}$ nor on ${\displaystyle \wp (\wp (X))}$; that is, ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}}$ does not necessarily imply ${\displaystyle {\mathcal {B}}={\mathcal {C}}}$; not even if both ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {B}}}$ are prefilters.^[13] For instance, if ${\displaystyle {\mathcal {B}}}$ is a prefilter but not a filter then ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}^{\uparrow X}{\text{ and }}{\mathcal {B}}^{\uparrow X}\leq {\mathcal {B}}{\text{ but }}{\mathcal {B}}\neq {\mathcal {B}}^{\uparrow X}.}$

Equivalent families of sets

The preorder ${\displaystyle \,\leq \,}$ induces its canonical equivalence relation on ${\displaystyle \wp (\wp (X)),}$ where for all ${\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X)),}$ ${\displaystyle {\mathcal {B}}}$ is equivalent to ${\displaystyle {\mathcal {C}}}$ if any of the following equivalent conditions hold:^[9][6]

1. ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}.}$
2. The upward closures of ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {B}}}$ are equal.

Two upward closed (in ${\displaystyle X}$) subsets of ${\displaystyle \wp (X)}$ are equivalent if and only if they are equal.^[9] If ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ then necessarily ${\displaystyle \varnothing \leq {\mathcal {B}}\leq \wp (X)}$ and ${\displaystyle {\mathcal {B}}}$ is equivalent to ${\displaystyle {\mathcal {B}}^{\uparrow X}.}$ Every equivalence class other than ${\displaystyle \{\varnothing \}}$ contains a unique representative (that is, element of the equivalence class) that is upward closed in ${\displaystyle X.}$^[9]

Properties preserved between equivalent families

Let ${\displaystyle {\mathcal {B}},{\mathcal {C}}\in \wp (\wp (X))}$ be arbitrary and let ${\displaystyle {\mathcal {F}}}$ be any family of sets. If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are equivalent (which implies that ${\displaystyle \ker {\mathcal {B}}=\ker {\mathcal {C}}}$) then for each of the statements/properties listed below, either it is true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ or else it is false of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$:^[32]

1. Not empty
2. Proper (that is, ${\displaystyle \varnothing }$ is not an element)
   • Moreover, any two degenerate families are necessarily equivalent.
3. Filter subbase
4. Prefilter
   • In which case ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ generate the same filter on ${\displaystyle X}$ (that is, their upward closures in ${\displaystyle X}$ are equal).
5. Free
6. Principal
7. Ultra
8. Is equal to the trivial filter ${\displaystyle \{X\}}$
   • In words, this means that the only subset of ${\displaystyle \wp (X)}$ that is equivalent to the trivial filter is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters).
9. Meshes with ${\displaystyle {\mathcal {F}}}$
10. Is finer than ${\displaystyle {\mathcal {F}}}$
11. Is coarser than ${\displaystyle {\mathcal {F}}}$
12. Is equivalent to ${\displaystyle {\mathcal {F}}}$

Missing from the above list is the word "filter" because this property is not preserved by equivalence. However, if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are filters on ${\displaystyle X,}$ then they are equivalent if and only if they are equal; this characterization does not extend to prefilters.

Equivalence of prefilters and filter subbases

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then the following families are always equivalent to each other:

1. ${\displaystyle {\mathcal {B}}}$;
2. the π–system generated by ${\displaystyle {\mathcal {B}}}$;
3. the filter on ${\displaystyle X}$ generated by ${\displaystyle {\mathcal {B}}}$;

and moreover, these three families all generate the same filter on ${\displaystyle X}$ (that is, the upward closures in ${\displaystyle X}$ of these families are equal).

In particular, every prefilter is equivalent to the filter that it generates. By transitivity, two prefilters are equivalent if and only if they generate the same filter.^{[9][proof 3]} Every prefilter is equivalent to exactly one filter on ${\displaystyle X,}$ which is the filter that it generates (that is, the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.^[9]

A filter subbase that is not also a prefilter cannot be equivalent to the prefilter (or filter) that it generates. In contrast, every prefilter is equivalent to the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot. Every filter is both a π–system and a ring of sets.

Examples of determining equivalence/non–equivalence

Examples: Let ${\displaystyle X=\mathbb {R} }$ and let ${\displaystyle E}$ be the set ${\displaystyle \mathbb {Z} }$ of integers (or the set ${\displaystyle \mathbb {N} }$). Define the sets

All three sets are filter subbases but none are filters on ${\displaystyle X}$ and only ${\displaystyle {\mathcal {B}}}$ is prefilter (in fact, ${\displaystyle {\mathcal {B}}}$ is even free and closed under finite intersections). The set ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }}$ is fixed while ${\displaystyle {\mathcal {C}}_{\operatorname {open} }}$ is free (unless ${\displaystyle E=\mathbb {N} }$). They satisfy ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }\leq {\mathcal {C}}_{\operatorname {open} }\leq {\mathcal {B}},}$ but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal. This conclusion can be reached by showing that the π–systems that they generate are not equivalent. Unlike with ${\displaystyle {\mathcal {C}}_{\operatorname {open} },}$ every set in the π–system generated by ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }}$ contains ${\displaystyle \mathbb {Z} }$ as a subset,^{[note 10]} which is what prevents their generated π–systems (and hence their generated filters) from being equivalent. If ${\displaystyle E}$ was instead ${\displaystyle \mathbb {Q} {\text{ or }}\mathbb {R} }$ then all three families would be free and although the sets ${\displaystyle {\mathcal {C}}_{\operatorname {closed} }{\text{ and }}{\mathcal {C}}_{\operatorname {open} }}$ would remain not equivalent to each other, their generated π–systems would be equivalent and consequently, they would generate the same filter on ${\displaystyle X}$; however, this common filter would still be strictly coarser than the filter generated by ${\displaystyle {\mathcal {B}}.}$

Set theoretic properties and constructions relevant to topology

Trace and meshing

If ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. filter) on ${\displaystyle X{\text{ and }}S\subseteq X}$ then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S,}$ which is the family ${\displaystyle {\mathcal {B}}{\big \vert }_{S}:={\mathcal {B}}(\cap )\{S\},}$ is a prefilter (resp. a filter) if and only if ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh (that is, ${\displaystyle \varnothing \not \in {\mathcal {B}}(\cap )\{S\}}$^[11]), in which case the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is said to be induced by ${\displaystyle S}$. If ${\displaystyle {\mathcal {B}}}$ is ultra and if ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh then the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{S}}$ is ultra. If ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle X}$ then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is a filter on ${\displaystyle S}$ if and only if ${\displaystyle S\in {\mathcal {B}}.}$

For example, suppose that ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X{\text{ and }}S\subseteq X}$ is such that ${\displaystyle S\neq X{\text{ and }}X\setminus S\not \in {\mathcal {B}}.}$ Then ${\displaystyle {\mathcal {B}}{\text{ and }}S}$ mesh and ${\displaystyle {\mathcal {B}}\cup \{S\}}$ generates a filter on ${\displaystyle X}$ that is strictly finer than ${\displaystyle {\mathcal {B}}.}$^[11]

When prefilters mesh

Given non–empty families ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},}$ the family

satisfies ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}}$ and ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}.}$ If ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is proper (resp. a prefilter, a filter subbase) then this is also true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ In order to make any meaningful deductions about ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ from ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}},{\mathcal {B}}(\cap ){\mathcal {C}}}$ needs to be proper (that is, ${\displaystyle \varnothing \not \in {\mathcal {B}}(\cap ){\mathcal {C}},}$ which is the motivation for the definition of "mesh". In this case, ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is a prefilter (resp. filter subbase) if and only if this is true of both ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}.}$ Said differently, if ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ are prefilters then they mesh if and only if ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, ${\displaystyle \,\leq \,}$):

Two prefilters (resp. filter subbases) ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ mesh if and only if there exists a prefilter (resp. filter subbase) ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}}$ and ${\displaystyle {\mathcal {B}}\leq {\mathcal {B}}(\cap ){\mathcal {C}}.}$

If the least upper bound of two filters ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ exists in ${\displaystyle \operatorname {Filters} (X)}$ then this least upper bound is equal to ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {C}}.}$^[28]

Images and preimages under functions

Throughout, ${\displaystyle f:X\to Y{\text{ and }}g:Y\to Z}$ will be maps between non–empty sets.

Images of prefilters

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (Y).}$ Many of the properties that ${\displaystyle {\mathcal {B}}}$ may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.

Explicitly, if one of the following properties is true of ${\displaystyle {\mathcal {B}}{\text{ on }}Y,}$ then it will necessarily also be true of ${\displaystyle g({\mathcal {B}}){\text{ on }}g(Y)}$ (although possibly not on the codomain ${\displaystyle Z}$ unless ${\displaystyle g}$ is surjective):^{[11][14][33][34][35][31]}

• Filter properties: ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate.
• Ideal properties: ideal, closed under finite unions, downward closed, directed upward.

Moreover, if ${\displaystyle {\mathcal {B}}\subseteq \wp (Y)}$ is a prefilter then so are both ${\displaystyle g({\mathcal {B}}){\text{ and }}g^{-1}(g({\mathcal {B}})).}$^[11] The image under a map ${\displaystyle f:X\to Y}$ of an ultra set ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is again ultra and if ${\displaystyle {\mathcal {B}}}$ is an ultra prefilter then so is ${\displaystyle f({\mathcal {B}}).}$

If ${\displaystyle {\mathcal {B}}}$ is a filter then ${\displaystyle g({\mathcal {B}})}$ is a filter on the range ${\displaystyle g(Y),}$ but it is a filter on the codomain ${\displaystyle Z}$ if and only if ${\displaystyle g}$ is surjective.^[33] Otherwise it is just a prefilter on ${\displaystyle Z}$ and its upward closure must be taken in ${\displaystyle Z}$ to obtain a filter. The upward closure of ${\displaystyle g({\mathcal {B}}){\text{ in }}Z}$ is

where if ${\displaystyle {\mathcal {B}}}$ is upward closed in ${\displaystyle Y}$ (that is, a filter) then this simplifies to:

If ${\displaystyle X\subseteq Y}$ then taking ${\displaystyle g}$ to be the inclusion map ${\displaystyle X\to Y}$ shows that any prefilter (resp. ultra prefilter, filter subbase) on ${\displaystyle X}$ is also a prefilter (resp. ultra prefilter, filter subbase) on ${\displaystyle Y.}$^[11]

Preimages of prefilters

Let ${\displaystyle {\mathcal {B}}\subseteq \wp (Y).}$ Under the assumption that ${\displaystyle f:X\to Y}$ is surjective:

${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of ${\displaystyle {\mathcal {B}}.}$

However, if ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle Y}$ then even if ${\displaystyle f}$ is surjective (which would make ${\displaystyle f^{-1}({\mathcal {B}})}$ a prefilter), it is nevertheless still possible for the prefilter ${\displaystyle f^{-1}({\mathcal {B}})}$ to be neither ultra nor a filter on ${\displaystyle X}$ ^[34] (see this^{[note 11]} footnote for an example).

If ${\displaystyle f:X\to Y}$ is not surjective then denote the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}f(X)}$ by ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)},}$ where in this case particular case the trace satisfies:

and consequently also:

This last equality and the fact that the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ is a family of sets over ${\displaystyle f(X)}$ means that to draw conclusions about ${\displaystyle f^{-1}({\mathcal {B}}),}$ the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ can be used in place of ${\displaystyle {\mathcal {B}}}$ and the surjection ${\displaystyle f:X\to f(X)}$ can be used in place of ${\displaystyle f:X\to Y.}$ For example:^[14][11][35]

${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}.}$

In this way, the case where ${\displaystyle f}$ is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).

Even if ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle Y,}$ if ${\displaystyle f}$ is not surjective then it is nevertheless possible that ${\displaystyle \varnothing \in {\mathcal {B}}{\big \vert }_{f(X)},}$ which would make ${\displaystyle f^{-1}({\mathcal {B}})}$ degenerate as well. The next characterization shows that degeneracy is the only obstacle. If ${\displaystyle {\mathcal {B}}}$ is a prefilter then the following are equivalent:^[14][11][35]

1. ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter;
2. ${\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}}$ is a prefilter;
3. ${\displaystyle \varnothing \not \in {\mathcal {B}}{\big \vert }_{f(X)}}$;
4. ${\displaystyle {\mathcal {B}}}$ meshes with ${\displaystyle f(X)}$

and moreover, if ${\displaystyle f^{-1}({\mathcal {B}})}$ is a prefilter then so is ${\displaystyle f\left(f^{-1}({\mathcal {B}})\right).}$^[14][11]

If ${\displaystyle S\subseteq Y}$ and if ${\displaystyle \operatorname {In} :S\to Y}$ denotes the inclusion map then the trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$ is equal to ${\displaystyle \operatorname {In} ^{-1}({\mathcal {B}}).}$^[11] This observation allows the results in this subsection to be applied to investigating the trace on a set.

Bijections, injections, and surjections

All properties involving filters are preserved under bijections. This means that if ${\displaystyle {\mathcal {B}}\subseteq \wp (Y){\text{ and }}g:Y\to Z}$ is a bijection, then ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. ultra, ultra prefilter, filter on ${\displaystyle X,}$ ultrafilter on ${\displaystyle X,}$ filter subbase, π–system, ideal on ${\displaystyle X,}$ etc.) if and only if the same is true of ${\displaystyle g({\mathcal {B}}){\text{ on }}Z.}$^[34]

A map ${\displaystyle g:Y\to Z}$ is injective if and only if for all prefilters ${\displaystyle {\mathcal {B}}{\text{ on }}Y,{\mathcal {B}}}$ is equivalent to ${\displaystyle g^{-1}(g({\mathcal {B}})).}$^[28] The image of an ultra family of sets under an injection is again ultra.

The map ${\displaystyle f:X\to Y}$ is a surjection if and only if whenever ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle Y}$ then the same is true of ${\displaystyle f^{-1}({\mathcal {B}}){\text{ on }}X}$ (this result does not require the ultrafilter lemma).

Subordination is preserved by images and preimages

The relation ${\displaystyle \,\leq \,}$ is preserved under both images and preimages of families of sets.^[11] This means that for any families ${\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}},}$^[35]

Moreover, the following relations always hold for any family of sets ${\displaystyle {\mathcal {C}}}$:^[35]

where equality will hold if ${\displaystyle f}$ is surjective.^[35] Furthermore,

If ${\displaystyle {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {C}}\subseteq \wp (Y)}$ then^[10]

and ${\displaystyle g^{-1}(g({\mathcal {C}}))\leq {\mathcal {C}}}$^[35] where equality will hold if ${\displaystyle g}$ is injective.^[35]

Products of prefilters

Suppose ${\displaystyle X_{\bullet }=\left(X_{i}\right)_{i\in I}}$ is a family of one or more non–empty sets, whose product will be denoted by ${\displaystyle \prod X_{\bullet }:=\prod _{i\in I}X_{i},}$ and for every index ${\displaystyle i\in I,}$ let

denote the canonical projection. Let ${\displaystyle {\mathcal {B}}_{\bullet }:=\left({\mathcal {B}}_{i}\right)_{i\in I}}$ be non−empty families, also indexed by ${\displaystyle I,}$ such that ${\displaystyle {\mathcal {B}}_{i}\subseteq \wp \left(X_{i}\right)}$ for each ${\displaystyle i\in I.}$ The product of the families ${\displaystyle {\mathcal {B}}_{\bullet }}$^[11] is defined identically to how the basic open subsets of the product topology are defined (had all of these ${\displaystyle {\mathcal {B}}_{i}}$ been topologies). That is, both the notations denote the family of all subsets ${\displaystyle \prod _{i\in I}S_{i}\subseteq \prod _{}X_{\bullet }}$ such that ${\displaystyle S_{i}=X_{i}}$ for all but finitely many ${\displaystyle i\in I}$ and where ${\displaystyle S_{i}\in {\mathcal {B}}_{i}}$ for any one of these finitely many exceptions (that is, for any ${\displaystyle i}$ such that ${\displaystyle S_{i}\neq X_{i},}$ necessarily ${\displaystyle S_{i}\in {\mathcal {B}}_{i}}$). When every ${\displaystyle {\mathcal {B}}_{i}}$ is a filter subbase then the family ${\displaystyle \bigcup _{i\in I}\Pr {}_{X_{i}}^{-1}\left({\mathcal {B}}_{i}\right)}$ is a filter subbase for the filter on ${\displaystyle \prod X_{\bullet }}$ generated by ${\displaystyle {\mathcal {B}}_{\bullet }.}$^[11] If ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ is a filter subbase then the filter on ${\displaystyle \prod X_{\bullet }}$ that it generates is called the filter generated by ${\displaystyle {\mathcal {B}}_{\bullet }}$.^[11] If every ${\displaystyle {\mathcal {B}}_{i}}$ is a prefilter on ${\displaystyle X_{i}}$ then ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ will be a prefilter on ${\displaystyle \prod X_{\bullet }}$ and moreover, this prefilter is equal to the coarsest prefilter ${\displaystyle {\mathcal {F}}{\text{ on }}\prod X_{\bullet }}$ such that ${\displaystyle \Pr {}_{X_{i}}({\mathcal {F}})={\mathcal {B}}_{i}}$ for every ${\displaystyle i\in I.}$^[11] However, ${\displaystyle \prod {\mathcal {B}}_{\bullet }}$ may fail to be a filter on ${\displaystyle \prod X_{\bullet }}$ even if every ${\displaystyle {\mathcal {B}}_{i}}$ is a filter on ${\displaystyle X_{i}.}$^[11]

Set subtraction and some examples

Set subtracting away a subset of the kernel

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X,S\subseteq \ker {\mathcal {B}},{\text{ and }}S\not \in {\mathcal {B}}}$ then ${\displaystyle \{B\setminus S~:~B\in {\mathcal {B}}\}}$ is a prefilter, where this latter set is a filter if and only if ${\displaystyle {\mathcal {B}}}$ is a filter and ${\displaystyle S=\varnothing .}$ In particular, if ${\displaystyle {\mathcal {B}}}$ is a neighborhood basis at a point ${\displaystyle x}$ in a topological space ${\displaystyle X}$ having at least 2 points, then ${\displaystyle \{B\setminus \{x\}~:~B\in {\mathcal {B}}\}}$ is a prefilter on ${\displaystyle X.}$ This construction is used to define ${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$ in terms of prefilter convergence.

Using duality between ideals and dual ideals

There is a dual relation ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ or ${\displaystyle {\mathcal {C}}\vartriangleright {\mathcal {B}},}$ which is defined to mean that every ${\displaystyle B\in {\mathcal {B}}}$ is contained in some ${\displaystyle C\in {\mathcal {C}}.}$ Explicitly, this means that for every ${\displaystyle B\in {\mathcal {B}}}$ , there is some ${\displaystyle C\in {\mathcal {C}}}$ such that ${\displaystyle B\subseteq C.}$ This relation is dual to ${\displaystyle \,\leq \,}$ in sense that ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ if and only if ${\displaystyle (X\setminus {\mathcal {B}})\leq (X\setminus {\mathcal {C}}).}$^[6] The relation ${\displaystyle {\mathcal {B}}\vartriangleleft {\mathcal {C}}}$ is closely related to the downward closure of a family in a manner similar to how ${\displaystyle \,\leq \,}$ is related to the upward closure family.

For an example that uses this duality, suppose ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle \Xi \subseteq \wp (Y).}$ Define

which contains the empty set if and only if ${\displaystyle \Xi }$ does. It is possible for ${\displaystyle \Xi }$ to be an ultrafilter and for ${\displaystyle \Xi _{f}}$ to be empty or not closed under finite intersections (see footnote for example).^{[note 12]} Although ${\displaystyle \Xi _{f}}$ does not preserve properties of filters very well, if ${\displaystyle \Xi }$ is downward closed (resp. closed under finite unions, an ideal) then this will also be true for ${\displaystyle \Xi _{f}.}$ Using the duality between ideals and dual ideals allows for a construction of the following filter.

Suppose ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle Y}$ and let ${\displaystyle \Xi :=Y\setminus {\mathcal {B}}}$ be its dual in ${\displaystyle Y.}$ If ${\displaystyle X\not \in \Xi _{f}}$ then ${\displaystyle \Xi _{f}}$'s dual ${\displaystyle X\setminus \Xi _{f}}$ will be a filter.

Other topology related examples

Example: The set ${\displaystyle {\mathcal {B}}}$ of all dense open subsets of a topological space is a proper π–system and a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than ${\displaystyle {\mathcal {B}}.}$

Example: The family ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ of all dense open sets of ${\displaystyle X=\mathbb {R} }$ having finite Lebesgue measure is a proper π–system and a free prefilter. The prefilter ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of ${\displaystyle \mathbb {R} .}$ Since ${\displaystyle X}$ is a Baire space, every countable intersection of sets in ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ is dense in ${\displaystyle X}$ (and also comeagre and non–meager) so the set of all countable intersections of elements of ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }}$ is a prefilter and π–system; it is also finer than, and not equivalent to, ${\displaystyle {\mathcal {B}}_{\operatorname {Open} }.}$

Convergence, limits, and cluster points

Throughout, ${\displaystyle (X,\tau )}$ is a topological space.

Prefilters vs. filters

With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If ${\displaystyle S\subseteq X}$ is a proper subset then any filter on ${\displaystyle S}$ will not be a filter on ${\displaystyle X,}$ although it will be a prefilter.

One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ${\displaystyle \,\leq }$), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

A note on intuition

Suppose that ${\displaystyle {\mathcal {F}}}$ is a non–principal filter on an infinite set ${\displaystyle X.}$ ${\displaystyle {\mathcal {F}}}$ has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any ${\displaystyle F_{0}\in {\mathcal {F}},}$ there always exists some ${\displaystyle F_{1}\in {\mathcal {F}}}$ that is a proper subset of ${\displaystyle F_{0}}$; this may be continued ad infinitum to get a sequence ${\displaystyle F_{0}\supset F_{1}\supset \cdots }$ of sets in ${\displaystyle {\mathcal {F}}}$ with each ${\displaystyle F_{i+1}}$ being a proper subset of ${\displaystyle F_{i}.}$ The same is not true going "upward", for if ${\displaystyle F_{0}=X\in {\mathcal {F}}}$ then there is no set in ${\displaystyle {\mathcal {F}}}$ that contains ${\displaystyle X}$ as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to ${\displaystyle \,\subseteq ,}$ every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

Limits and convergence

The following well known definition will be generalized to prefilters. A point ${\displaystyle x\in X}$ is called a limit point, cluster point, or accumulation point of a subset ${\displaystyle S\subseteq X}$ if every neighborhood of ${\displaystyle x{\text{ in }}(X,\tau )}$ contains a point of ${\displaystyle S}$ different from ${\displaystyle x,}$ or equivalently, if ${\displaystyle x\in \operatorname {cl} _{(X,\tau )}(S\setminus \{x\}).}$ The set of all limit points of ${\displaystyle S}$ is called the derived set of ${\displaystyle S{\text{ in }}(X,\tau ).}$ The closure of a set ${\displaystyle S\subseteq X}$ is equal to the union of ${\displaystyle S}$ together with the set of all limit points of ${\displaystyle S.}$

A family ${\displaystyle {\mathcal {B}}}$ is said to converge in ${\displaystyle (X,\tau )}$ to a point or subset ${\displaystyle x}$ of ${\displaystyle X}$^[8] written ${\displaystyle {\mathcal {B}}\to x{\text{ or }}\lim {\mathcal {B}}\to x{\text{ in }}(X,\tau ),}$^[29] if ${\displaystyle {\mathcal {B}}\geq {\mathcal {N}}(x),}$ in which case ${\displaystyle x}$ is said to be a limit (or if ${\displaystyle x}$ is a point, also limit point)^[36] of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau ).}$ Denote^[8] the set of all these limit points by ${\displaystyle \lim {}_{(X,\tau )}{\mathcal {B}}{\text{ and }}\lim {\mathcal {B}}.}$ As usual, ${\displaystyle \lim {\mathcal {B}}=x}$ is defined to mean that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau )}$ and ${\displaystyle x\in X}$ is the only limit point of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau );}$ that is, if also ${\displaystyle {\mathcal {B}}\to z{\text{ in }}(X,\tau ){\text{ then }}z=x.}$^[29] (If the notation "${\displaystyle \lim {\mathcal {B}}=x}$" did not also require that the limit point ${\displaystyle x}$ be unique then the equals sign = would no longer be guaranteed to be transitive).

In words, ${\displaystyle {\mathcal {B}}}$ converges to a point if and only if ${\displaystyle {\mathcal {B}}}$ is finer than the neighborhood filter at that point. Explicitly, ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}}$ means that every neighborhood ${\displaystyle N{\text{ of }}x}$ contains some ${\displaystyle B\in {\mathcal {B}}}$ as a subset (that is, ${\displaystyle B\subseteq N}$); thus the following then holds: ${\displaystyle {\mathcal {N}}\ni N\supseteq B\in {\mathcal {B}}.}$

In the above definitions, it suffices to check that ${\displaystyle {\mathcal {B}}}$ is finer than some (or equivalently, finer than every) neighborhood base in ${\displaystyle (X,\tau )}$ of the point or set (for example, such as ${\displaystyle \tau (x)=\{U\in \tau :x\in U\}}$ or ${\displaystyle \tau (S)=\bigcap _{s\in S}\tau (s)}$). If ${\displaystyle \varnothing \neq S\subseteq X}$ then because ${\displaystyle {\mathcal {N}}(S)=\bigcap _{s\in S}{\mathcal {N}}(s),}$ if ${\displaystyle {\mathcal {B}}\to s{\text{ for all }}s\in S}$ then ${\displaystyle {\mathcal {B}}\to S.}$ Because ${\displaystyle \varnothing }$ is an open set, a family ${\displaystyle {\mathcal {B}}}$ converges to ${\displaystyle \varnothing }$ if and only if ${\displaystyle \varnothing \in {\mathcal {B}};}$ so in particular, no filter or other non-degenerate family can converge to the empty set, which is why when dealing with convergent prefilters (or filter subbases), it is typically assumed (often without mention) that ${\displaystyle S\neq \varnothing .}$

Given ${\displaystyle x\in X,}$ the following are equivalent for a prefilter ${\displaystyle {\mathcal {B}}:}$

1. ${\displaystyle {\mathcal {B}}}$ converges to ${\displaystyle x.}$
2. ${\displaystyle {\mathcal {B}}}$ converges to the set ${\displaystyle \{x\}.}$
3. ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ converges to ${\displaystyle x.}$
4. There exists a family equivalent to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter and ${\displaystyle B\in {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}}$ converges to a point (or subset) of ${\displaystyle X}$ if and only if this is true of the trace ${\displaystyle {\mathcal {B}}{\big \vert }_{B}.}$^[37] If ${\displaystyle {\mathcal {B}}}$ is a filter subbase that converges to ${\displaystyle x{\text{ or }}S}$ then this is also true of the filter that it generates (and also of any prefilter equivalent to this filter, such as the π-system generated by ${\displaystyle {\mathcal {B}}}$).

Because subordination is transitive, if ${\displaystyle {\mathcal {B}}\leq {\mathcal {C}}{\text{ then }}\lim {}_{(X,\tau )}{\mathcal {B}}\subseteq \lim {}_{(X,\tau )}{\mathcal {C}}}$ and moreover, for every ${\displaystyle x\in X,}$ both ${\displaystyle \{x\}}$ and the maximal/ultrafilter ${\displaystyle \{x\}^{\uparrow X}}$ converge to ${\displaystyle x{\text{ in }}(X,\tau ).}$ Thus every topological space ${\displaystyle (X,\tau )}$ induces a canonical convergence ${\displaystyle \xi \subseteq X\times \operatorname {Filters} (X)}$ defined by ${\displaystyle (x,{\mathcal {B}})\in \xi {\text{ if and only if }}x\in \lim {}_{(X,\tau )}{\mathcal {B}}.}$ At the other extreme, the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ is the smallest (that is, coarsest) filter on ${\displaystyle X}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau );}$ that is, any filter converging to ${\displaystyle x}$ must contain ${\displaystyle {\mathcal {N}}(x)}$ as a subset. Said differently, the family of filters that converge to ${\displaystyle x}$ consists exactly of those filter on ${\displaystyle X}$ that contain ${\displaystyle {\mathcal {N}}(x)}$ as a subset. Consequently, the finer the topology on ${\displaystyle X}$ then the fewer prefilters exist that have any limit points in ${\displaystyle X.}$

Cluster points

Say that ${\displaystyle x\in X}$ is a cluster point or an accumulation point of a family ${\displaystyle {\mathcal {B}}}$^[8] if ${\displaystyle {\mathcal {B}}}$ meshes with the neighborhood filter at ${\displaystyle x}$; that is, if ${\displaystyle {\mathcal {B}}\#{\mathcal {N}}(x).}$ The set of all cluster points of ${\displaystyle {\mathcal {B}}}$ is denoted by ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}{\text{ or }}\operatorname {cl} {\mathcal {B}}.}$

Explicitly, this means that ${\displaystyle B\cap N\neq \varnothing {\text{ for every }}B\in {\mathcal {B}}}$ and every neighborhood ${\displaystyle N}$ of ${\displaystyle x.}$ When ${\displaystyle {\mathcal {B}}}$ is a prefilter then the definition of "${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {N}}}$ mesh" can be characterized entirely in terms of the preorder ${\displaystyle \,\leq \,.}$

More generally, given ${\displaystyle S\subseteq X,}$ say that

${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle S}$ if ${\displaystyle {\mathcal {B}}}$ meshes with the neighborhood filter of ${\displaystyle S}$; that is, if ${\displaystyle {\mathcal {B}}\#{\mathcal {N}}(S).}$

In the above definitions, it suffices to check that ${\displaystyle {\mathcal {B}}}$ meshes with some (or equivalently, meshes with every) neighborhood base in ${\displaystyle (X,\tau )}$ of ${\displaystyle x{\text{ or }}S.}$ Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every ${\displaystyle x\in X,}$ both ${\displaystyle \{x\}}$ and the principal ultrafilter ${\displaystyle \{x\}^{\uparrow X}}$ cluster at ${\displaystyle x.}$ For any ${\displaystyle S\subseteq X,}$ if ${\displaystyle {\mathcal {B}}}$ clusters at some ${\displaystyle s\in S}$ then ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle S.}$ No family clusters at ${\displaystyle S:=\varnothing }$ and if ${\displaystyle \varnothing \in {\mathcal {B}}{\text{ then }}\varnothing =\operatorname {cl} {\mathcal {B}}.}$

Given ${\displaystyle x\in X,}$ the following are equivalent for a prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$:

1. ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle x.}$
2. ${\displaystyle {\mathcal {B}}}$ clusters at the set ${\displaystyle \{x\}.}$
3. The family ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ generated by ${\displaystyle {\mathcal {B}}}$ clusters at ${\displaystyle x.}$
4. There exists a family equivalent to ${\displaystyle {\mathcal {B}}}$ that clusters at ${\displaystyle x.}$
5. ${\displaystyle x\in \bigcap _{F\in {\mathcal {B}}}\operatorname {cl} _{X}F.}$
6. ${\displaystyle X\setminus N\not \in {\mathcal {B}}^{\uparrow X}}$ for every neighborhood ${\displaystyle N}$ of ${\displaystyle x.}$
   • If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}{\text{ if and only if }}X\setminus N\not \in {\mathcal {B}}}$ for every neighborhood ${\displaystyle N{\text{ of }}x.}$
7. There exists a prefilter ${\displaystyle {\mathcal {F}}}$ subordinate to ${\displaystyle {\mathcal {B}}}$ (that is, ${\displaystyle {\mathcal {F}}\geq {\mathcal {B}}}$) such that ${\displaystyle {\mathcal {F}}\to x.}$
   • This is the filter equivalent of "${\displaystyle x}$ is a cluster point of a sequence if and only if there exists a subsequence converging to ${\displaystyle x.}$
   • In particular, if ${\displaystyle x}$ is a cluster point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)}$ is a prefilter subordinate to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau ).}$

If ${\displaystyle {\mathcal {B}}}$ is an ultra prefilter on ${\displaystyle X{\text{ and }}x\in X,}$ then ${\displaystyle x}$ is a cluster point of ${\displaystyle {\mathcal {B}}{\text{ if and only if }}{\mathcal {B}}\to x{\text{ in }}(X,\tau ).}$^[30]

The set ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ of all cluster points of a prefilter ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau )}$ satisfies

which in particular shows that the set ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ of all cluster points of any prefilter ${\displaystyle {\mathcal {B}}}$ is a closed subset of ${\displaystyle X.}$^[38][8] This also justifies the notation ${\displaystyle \operatorname {cl} _{X}{\mathcal {B}}}$ for the set of cluster points.^[8]

Properties and relationships

Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have any cluster points or limit points.^[38]

If ${\displaystyle x}$ is a limit point of ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle x}$ is necessarily a limit point of any family ${\displaystyle {\mathcal {C}}}$ finer than ${\displaystyle {\mathcal {B}}}$ (that is, if ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {B}}{\text{ and }}{\mathcal {B}}\leq {\mathcal {C}}}$ then ${\displaystyle {\mathcal {N}}(x)\leq {\mathcal {C}}}$).^[38] In contrast, if ${\displaystyle x}$ is a cluster point of ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle x}$ is necessarily a cluster point of any family ${\displaystyle {\mathcal {C}}}$ coarser than ${\displaystyle {\mathcal {B}}}$ (that is, if ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}}$ mesh and ${\displaystyle {\mathcal {C}}\leq {\mathcal {B}}}$ then ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {C}}}$ mesh).

Equivalent families and subordination

Any two equivalent families ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}}$ can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that ${\displaystyle {\mathcal {N}}\leq {\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}\leq {\mathcal {C}},}$ and also that ${\displaystyle {\mathcal {N}}\#{\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}\#{\mathcal {C}}.}$ In essence, the preorder ${\displaystyle \,\leq \,}$ is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder ${\displaystyle \,\leq \,}$ is of such great importance in applying (pre)filters to Topology.

Limit and cluster point relationships and sufficient conditions

Every limit point of a prefilter ${\displaystyle {\mathcal {B}}}$ is also a cluster point of ${\displaystyle {\mathcal {B}},}$ since if ${\displaystyle x}$ is a limit point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {N}}(x){\text{ and }}{\mathcal {B}}}$ mesh,^[19][38] which makes ${\displaystyle x}$ a cluster point of ${\displaystyle {\mathcal {B}}.}$^[8] Every accumulation point of an ultrafilter is also a limit point.

If ${\displaystyle \varnothing \neq {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {S}}\geq {\mathcal {B}}}$ is a filter subbase such that ${\displaystyle {\mathcal {S}}\to x{\text{ in }}(X,\tau )}$ then ${\displaystyle x\in \operatorname {cl} _{X}{\mathcal {B}}.}$ In particular, any limit point of a filter subbase subordinate to ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is necessarily also a cluster point of ${\displaystyle {\mathcal {B}}.}$ If ${\displaystyle x}$ is a cluster point of a prefilter ${\displaystyle {\mathcal {B}}}$ then ${\displaystyle {\mathcal {B}}(\cap ){\mathcal {N}}(x)}$ is a prefilter subordinate to ${\displaystyle {\mathcal {B}}}$ that converges to ${\displaystyle x{\text{ in }}(X,\tau ).}$

If ${\displaystyle S\subseteq X}$ and if ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle S}$ then every cluster point of ${\displaystyle {\mathcal {B}}{\text{ in }}X}$ belongs to ${\displaystyle \operatorname {cl} _{X}S}$ and any point in ${\displaystyle \operatorname {cl} _{X}S}$ is a limit point of a filter on ${\displaystyle S.}$^[38]

Primitive sets

A subset ${\displaystyle P\subseteq X}$ is called primitive^[39] if it is the set of limit points of some ultrafilter on ${\displaystyle X.}$ That is, if there exists an ultrafilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ such that ${\displaystyle P}$ is equal to ${\displaystyle \operatorname {lim} _{X}{\mathcal {B}},}$ which recall denotes the set of limit points of ${\displaystyle {\mathcal {B}}{\text{ in }}(X,\tau ).}$

Any closed singleton subset of ${\displaystyle X}$ is a primitive subset of ${\displaystyle X.}$^[39] The image of a primitive subset of ${\displaystyle X}$ under a continuous map ${\displaystyle f:X\to Y}$ is contained in a primitive subset of ${\displaystyle Y.}$^[39]

Assume that ${\displaystyle P,Q\subseteq X}$ are two primitive subset of ${\displaystyle X.}$ If ${\displaystyle U}$ is an open subset of ${\displaystyle X}$ such that ${\displaystyle P\cap Q\neq \varnothing ,}$ then ${\displaystyle U\in {\mathcal {B}}}$ for any ultrafilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ such that ${\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}.}$^[39] In addition, if ${\displaystyle P{\text{ and }}Q}$ are distinct then there exists some ${\displaystyle S\subseteq X}$ and some ultrafilters ${\displaystyle {\mathcal {B}}_{P}{\text{ and }}{\mathcal {B}}_{Q}{\text{ on }}X}$ such that ${\displaystyle P=\operatorname {lim} _{X}{\mathcal {B}}_{P},Q=\operatorname {lim} _{X}{\mathcal {B}}_{Q},S\in {\mathcal {B}}_{P},}$ and ${\displaystyle X\setminus S\in {\mathcal {B}}_{Q}.}$^[39]

Other results

If ${\displaystyle X}$ is a complete lattice then:^{[citation needed]}

• The limit inferior of ${\displaystyle B}$ is the infimum of the set of all cluster points of ${\displaystyle B.}$
• The limit superior of ${\displaystyle B}$ is the supremum of the set of all cluster points of ${\displaystyle B.}$
• ${\displaystyle B}$ is a convergent prefilter if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the prefilter.

Limits of functions defined as limits of prefilters

If ${\displaystyle f:X\to Y}$ is a map from a set into a topological space ${\displaystyle Y{\text{ and }}{\mathcal {B}}\subseteq \wp (X)}$ then ${\displaystyle y\in Y}$ is called a limit point or limit (respectively, a cluster point) of ${\displaystyle f}$ with respect to ${\displaystyle {\mathcal {B}}}$^[38] if ${\displaystyle y}$ is a limit point (resp. a cluster point) of ${\displaystyle f({\mathcal {B}}){\text{ in }}Y,}$ in which case this may be expressed by writing ${\displaystyle f({\mathcal {B}})\to y,{\text{ or }}\lim f({\mathcal {B}})\to y{\text{ in }}Y.}$ If the limit ${\displaystyle y}$ is unique then the arrow ${\displaystyle \to }$ may be replaced with an equals sign ${\displaystyle =.}$^[29]

Explicitly, ${\displaystyle y}$ is a limit of ${\displaystyle f}$ with respect to ${\displaystyle {\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {N}}(y)\leq f({\mathcal {B}}).}$

The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if ${\displaystyle x\in X{\text{ and }}\chi :(I,\leq )\to X}$ is a net then

where the left hand side states that ${\displaystyle x}$ is a limit of the net${\displaystyle \chi }$ while the right hand side states that ${\displaystyle x}$ is a limit of the function${\displaystyle \chi }$ with respect to ${\displaystyle {\mathcal {B}}:=\operatorname {Tails} (I,\leq )}$ (as defined above).

The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under ${\displaystyle f}$) of particular prefilters on the domain ${\displaystyle X.}$ This shows that prefilters provide a general framework into which many of the various definitions of limits fit.^[37] The limits in the left–most column are defined in their usual way with their obvious definitions.

Throughout, let ${\displaystyle f:X\to Y}$ be a map between topological spaces, ${\displaystyle x_{0}\in X,{\text{ and }}y\in Y.}$ If ${\displaystyle Y}$ is Hausdorff then all arrows "${\displaystyle \to y}$" in the table may be replaced with equal signs "${\displaystyle =y}$" and "${\displaystyle \lim f({\mathcal {B}})\to y}$" may be replaced with "${\displaystyle \lim f({\mathcal {B}})=y}$".^[29]

Type of limit	if and only if	Definition in terms of prefilters[37]	Assumptions
${\displaystyle \lim _{x\to x_{0}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,{\mathcal {N}}\left(x_{0}\right)}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{N\setminus \left\{x_{0}\right\}:N\in {\mathcal {N}}\left(x_{0}\right)\right\}}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\in S}}f(x)\to y}$ or ${\displaystyle \lim _{x\to x_{0}}f{\big \vert }_{S}(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{N\cap S:N\in {\mathcal {N}}\left(x_{0}\right)\right\}}$	${\displaystyle S\subseteq X{\text{ and }}x_{0}\in \operatorname {cl} _{X}S}$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\neq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x_{0}-r,x_{0}\right)\cup \left(x_{0},x_{0}+r\right):0<r\in \mathbb {R} \right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x<x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x,x_{0}\right):x<x_{0}\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\leq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x,x_{0}\right]:x<x_{0}\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x>x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left(x_{0},x\right):x_{0}<x\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{\stackrel {x\to x_{0}}{x\geq x_{0}}}f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\left\{\left[x_{0},x\right):x_{0}\leq x\right\}}$	${\displaystyle x_{0}\in X=\mathbb {R} }$
${\displaystyle \lim _{n\to \infty }f(n)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{\{n,n+1,\ldots \}~:~n\in \mathbb {N} \}\}}$	${\displaystyle X=\mathbb {N} {\text{ so }}f:\mathbb {N} \to Y}$ is a sequence in ${\displaystyle Y}$
${\displaystyle \lim _{x\to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,(\mathbb {R} ,\infty )\,:=\,\{(x,\infty ):x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} }$
${\displaystyle \lim _{x\to -\infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,(-\infty ,\mathbb {R} )\,:=\,\{(-\infty ,x):x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} }$
${\displaystyle \lim _{ x \to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{X\cap [(-\infty ,x)\cup (x,\infty )]:x\in \mathbb {R} \}}$	${\displaystyle X=\mathbb {R} {\text{ or }}X=\mathbb {Z} }$ for a double-ended sequence
${\displaystyle \lim _{\ x\ \to \infty }f(x)\to y}$	⇔	${\displaystyle f({\mathcal {B}})\to y{\text{ where }}{\mathcal {B}}\,:=\,\{\{x\in X:\ x\ >r\}~:~0<r\in \mathbb {R} \}}$	${\displaystyle (X,\ \cdot \ ){\text{ is}}}$ a seminormed space; for example, a Banach space ${\displaystyle {\text{like }}X=\mathbb {C} }$

By defining different prefilters, many other notions of limits can be defined; for example, ${\displaystyle \lim _{\stackrel { x \to x_{0} }{ x \neq x_{0} }}f(x)\to y.}$

Limits of ${\displaystyle f:X\to Y}$ that diverge to infinity can be defined by using the prefilters

For example, if ${\displaystyle {\mathcal {B}}\,:=\,{\mathcal {N}}\left(x_{0}\right)}$ and ${\displaystyle Y=\mathbb {R} }$ then ${\displaystyle \lim _{x\to x_{0}}f(x)\to \infty }$ can be defined to mean that ${\displaystyle f({\mathcal {B}})\leq (\mathbb {R} ,\infty )}$ holds and similarly, ${\displaystyle \lim _{x\to x_{0}}f(x)\to -\infty }$ can be defined to mean ${\displaystyle f({\mathcal {B}})\leq (-\infty ,\mathbb {R} ).}$ If ${\displaystyle f}$ is valued in ${\displaystyle Y=\mathbb {R} ^{n}{\text{ or }}Y=\mathbb {C} ^{n}}$ (or some other seminormed vector space) then ${\displaystyle \lim _{x\to x_{0}} f(x) \to \infty }$ if and only if ${\displaystyle f({\mathcal {B}})\leq Y\setminus B_{\bullet }}$ holds.

Filters and nets

This article will describe the relationships between prefilters and nets in great detail so as to make it easier to understand later why subnets (with their most commonly used definitions) are not generally equivalent with "sub–prefilters".

Nets to prefilters

In the definitions below, the first statement is the standard definition of a limit point of a net (resp. a cluster point of a net) and it is gradually reworded until the corresponding filter concept is reached.

A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}{\text{ in }}X}$ is said to converge in${\displaystyle (X,\tau )}$ to a point ${\displaystyle x\in X,}$ written ${\displaystyle x_{\bullet }\to x{\text{ in }}(X,\tau ),}$ and ${\displaystyle x}$ is called a limit or limit point of ${\displaystyle x_{\bullet },}$^[40] if any of the following equivalent conditions hold:

1. Definition: For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle i\in I}$ such that if ${\displaystyle i\leq j\in I{\text{ then }}x_{j}\in N.}$
2. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle i\in I}$ such that the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ is contained in ${\displaystyle N}$ (that is, such that ${\displaystyle x_{\geq i}\subseteq N}$).
3. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x),}$ there exists some ${\displaystyle B\in \operatorname {Tails} \left(x_{\bullet }\right)}$ such that ${\displaystyle B\subseteq N.}$
4. ${\displaystyle {\mathcal {N}}_{\tau }(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).}$
5. ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\to x{\text{ in }}(X,\tau );}$ that is, the prefilter ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ converges to ${\displaystyle x.}$

As usual, ${\displaystyle \lim x_{\bullet }=x}$ is defined to mean that ${\displaystyle x_{\bullet }\to x}$ and ${\displaystyle x}$ is the only limit point of ${\displaystyle x_{\bullet };}$ that is, if also ${\displaystyle x_{\bullet }\to z{\text{ then }}z=x.}$^[40]

A point ${\displaystyle x\in X}$ is called a cluster point or an accumulation point of a net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}{\text{ in }}(X,\tau )}$ if any of the following equivalent conditions hold:

1. Definition: For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle i\in I,}$ there exists some ${\displaystyle i\leq j\in I}$ such that ${\displaystyle x_{j}\in N.}$
2. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle i\in I,}$ the tail of ${\displaystyle x_{\bullet }}$ starting at ${\displaystyle i}$ intersects ${\displaystyle N.}$
3. For every ${\displaystyle N\in {\mathcal {N}}_{\tau }(x)}$ and every ${\displaystyle B\in \operatorname {Tails} \left(x_{\bullet }\right),B\cap N\neq \varnothing .}$
4. ${\displaystyle {\mathcal {N}}_{\tau }(x){\text{ and }}\operatorname {Tails} \left(x_{\bullet }\right)}$ mesh (by definition of "mesh").
5. ${\displaystyle x}$ is a cluster point of ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right).}$
6. For every neighborhood ${\displaystyle N}$ of ${\displaystyle x}$ there exists a subnet of ${\displaystyle x_{\bullet }}$ that converges to ${\displaystyle x.}$^{[citation needed]}

If ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle x_{\bullet }}$ is a net in ${\displaystyle X}$ then ${\displaystyle \operatorname {Tails} \left(f\left(x_{\bullet }\right)\right)=f\left(\operatorname {Tails} \left(x_{\bullet }\right)\right).}$^[4]

Prefilters to nets

A pointed set is a pair ${\displaystyle (S,s)}$ consisting of a non–empty set ${\displaystyle S}$ and an element ${\displaystyle s\in S.}$ For any family ${\displaystyle {\mathcal {B}},}$ let

Define a canonical preorder ${\displaystyle \,\leq \,}$ on pointed sets by declaring

If ${\displaystyle s_{0},s_{1}\in S{\text{ then }}\left(S,s_{0}\right)\leq \left(S,s_{1}\right){\text{ and }}\left(S,s_{1}\right)\leq \left(S,s_{0}\right)}$ even if ${\displaystyle s_{0}\neq s_{1},}$ so this preorder is not antisymmetric and given any family of sets ${\displaystyle {\mathcal {B}},}$ ${\displaystyle (\operatorname {PointedSets} ({\mathcal {B}}),\leq )}$ is partially ordered if and only if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ consists entirely of singleton sets. If ${\displaystyle \{x\}\in {\mathcal {B}}{\text{ then }}(\{x\},x)}$ is a maximal element of ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}})}$; moreover, all maximal elements are of this form. If ${\displaystyle \left(B,b_{0}\right)\in \operatorname {PointedSets} ({\mathcal {B}}){\text{ then }}\left(B,b_{0}\right)}$ is a greatest element if and only if ${\displaystyle B=\ker {\mathcal {B}},}$ in which case ${\displaystyle \{(B,b)~:~b\in B\}}$ is the set of all greatest elements. However, a greatest element ${\displaystyle (B,b)}$ is a maximal element if and only if ${\displaystyle B=\{b\}=\ker {\mathcal {B}},}$ so there is at most one element that is both maximal and greatest. There is a canonical map ${\displaystyle \operatorname {Point} _{\mathcal {B}}~:~\operatorname {PointedSets} ({\mathcal {B}})\to X}$ defined by ${\displaystyle (B,b)\mapsto b.}$ If ${\displaystyle i_{0}=\left(B_{0},b_{0}\right)\in \operatorname {PointedSets} ({\mathcal {B}})}$ then the tail of the assignment ${\displaystyle \operatorname {Point} _{\mathcal {B}}}$ starting at ${\displaystyle i_{0}}$ is ${\displaystyle \left\{c~:~(C,c)\in \operatorname {PointedSets} ({\mathcal {B}}){\text{ and }}\left(B_{0},b_{0}\right)\leq (C,c)\right\}=B_{0}.}$

Although ${\displaystyle (\operatorname {PointedSets} ({\mathcal {B}}),\leq )}$ is not, in general, a partially ordered set, it is a directed set if (and only if) ${\displaystyle {\mathcal {B}}}$ is a prefilter. So the most immediate choice for the definition of "the net in ${\displaystyle X}$ induced by a prefilter ${\displaystyle {\mathcal {B}}}$" is the assignment ${\displaystyle (B,b)\mapsto b}$ from ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}})}$ into ${\displaystyle X.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then the net associated with ${\displaystyle {\mathcal {B}}}$ is the map

$${\displaystyle {\begin{alignedat}{4}\operatorname {Net} _{\mathcal {B}}:\;&&(\operatorname {PointedSets} ({\mathcal {B}}),\leq )&&\,\to \;&X\\&&(B,b)&&\,\mapsto \;&b\\\end{alignedat}}}$$

that is, ${\displaystyle \operatorname {Net} _{\mathcal {B}}(B,b):=b.}$

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X{\text{ then }}\operatorname {Net} _{\mathcal {B}}}$ is a net in ${\displaystyle X}$ and the prefilter associated with ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}}$; that is:^{[note 13]}

This would not necessarily be true had ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ been defined on a proper subset of ${\displaystyle \operatorname {PointedSets} ({\mathcal {B}}).}$ For example, suppose ${\displaystyle X}$ has at least two distinct elements, ${\displaystyle {\mathcal {B}}:=\{X\}}$ is the indiscrete filter, and ${\displaystyle x\in X}$ is arbitrary. Had ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ instead been defined on the singleton set ${\displaystyle D:=\{(X,x)\},}$ where the restriction of ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ to ${\displaystyle D}$ will temporarily be denote by ${\displaystyle \operatorname {Net} _{D}:D\to X,}$ then the prefilter of tails associated with ${\displaystyle \operatorname {Net} _{D}:D\to X}$ would be the principal prefilter ${\displaystyle \{\,\{x\}\,\}}$ rather than the original filter ${\displaystyle {\mathcal {B}}=\{X\}}$; this means that the equality ${\displaystyle \operatorname {Tails} \left(\operatorname {Net} _{D}\right)={\mathcal {B}}}$ is false, so unlike ${\displaystyle \operatorname {Net} _{\mathcal {B}},}$ the prefilter ${\displaystyle {\mathcal {B}}}$ can not be recovered from ${\displaystyle \operatorname {Net} _{D}.}$ Worse still, while ${\displaystyle {\mathcal {B}}}$ is the unique minimal filter on ${\displaystyle X,}$ the prefilter ${\displaystyle \operatorname {Tails} \left(\operatorname {Net} _{D}\right)=\{\{x\}\}}$ instead generates a maximal filter (that is, an ultrafilter) on ${\displaystyle X.}$

However, if ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ then it is not in general true that ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}}$ is equal to ${\displaystyle x_{\bullet }}$ because, for example, the domain of ${\displaystyle x_{\bullet }}$ may be of a completely different cardinality than that of ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)}}$ (since unlike the domain of ${\displaystyle \operatorname {Net} _{\operatorname {Tails} \left(x_{\bullet }\right)},}$ the domain of an arbitrary net in ${\displaystyle X}$ could have any cardinality).

Ultranets and ultra prefilters

A net ${\displaystyle x_{\bullet }{\text{ in }}X}$ is called an ultranet or universal net in ${\displaystyle X}$ if for every subset ${\displaystyle S\subseteq X,x_{\bullet }}$ is eventually in ${\displaystyle S}$ or it is eventually in ${\displaystyle X\setminus S}$; this happens if and only if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is an ultra prefilter. A prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ is an ultra prefilter if and only if ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is an ultranet in ${\displaystyle X.}$

Partially ordered net

The domain of the canonical net ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered^[41] a construction that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.^[4] It begins with the construction of a strict partial order (meaning a transitive and irreflexive relation) ${\displaystyle \,<\,}$ on a subset of ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ that is similar to the lexicographical order on ${\displaystyle {\mathcal {B}}\times \mathbb {N} }$ of the strict partial orders ${\displaystyle ({\mathcal {B}},\supsetneq ){\text{ and }}(\mathbb {N} ,<).}$ For any ${\displaystyle i=(B,m,b){\text{ and }}j=(C,n,c)}$ in ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X,}$ declare that ${\displaystyle i<j}$ if and only if

or equivalently, if and only if ${\displaystyle {\text{(1) }}B\supseteq C,{\text{ and (2) if }}B=C{\text{ then }}m<n.}$

The non−strict partial order associated with ${\displaystyle \,<,}$ denoted by ${\displaystyle \,\leq ,}$ is defined by declaring that ${\displaystyle i\leq j\,{\text{ if and only if }}i<j{\text{ or }}i=j.}$ Unwinding these definitions gives the following characterization:

${\displaystyle i\leq j}$ if and only if ${\displaystyle {\text{(1) }}B\supseteq C,{\text{ and (2) if }}B=C{\text{ then }}m\leq n,}$ and also ${\displaystyle {\text{(3) if }}B=C{\text{ and }}m=n{\text{ then }}b=c,}$

which shows that ${\displaystyle \,\leq \,}$ is just the lexicographical order on ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ induced by ${\displaystyle ({\mathcal {B}},\supseteq ),\,(\mathbb {N} ,\leq ),{\text{ and }}(X,=),}$ where ${\displaystyle X}$ is partially ordered by equality ${\displaystyle \,=.\,}$^{[note 14]} Both ${\displaystyle \,<{\text{ and }}\leq \,}$ are serial and neither possesses a greatest element or a maximal element; this remains true if they are each restricted to the subset of ${\displaystyle {\mathcal {B}}\times \mathbb {N} \times X}$ defined by

where it will henceforth be assumed that they are. Denote the assignment ${\displaystyle i=(B,m,b)\mapsto b}$ from this subset by: If ${\displaystyle i_{0}=\left(B_{0},m_{0},b_{0}\right)\in \operatorname {Poset} _{\mathcal {B}}}$ then just as with ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ before, the tail of the ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}}$ starting at ${\displaystyle i_{0}}$ is equal to ${\displaystyle B_{0}.}$ If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ then ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}}$ is a net in ${\displaystyle X}$ whose domain ${\displaystyle \operatorname {Poset} _{\mathcal {B}}}$ is a partially ordered set and moreover, ${\displaystyle \operatorname {Tails} \left(\operatorname {PosetNet} _{\mathcal {B}}\right)={\mathcal {B}}.}$^[4] Because the tails of ${\displaystyle \operatorname {PosetNet} _{\mathcal {B}}{\text{ and }}\operatorname {Net} _{\mathcal {B}}}$ are identical (since both are equal to the prefilter ${\displaystyle {\mathcal {B}}}$), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.^[4] If the set ${\displaystyle \mathbb {N} }$ is replaced with the positive rational numbers then the strict partial order ${\displaystyle <}$ will also be a dense order.

Subordinate filters and subnets

The notion of "${\displaystyle {\mathcal {B}}}$ is subordinate to ${\displaystyle {\mathcal {C}}}$" (written ${\displaystyle {\mathcal {B}}\vdash {\mathcal {C}}}$) is for filters and prefilters what "${\displaystyle x_{n_{\bullet }}=\left(x_{n_{i}}\right)_{i=1}^{\infty }}$ is a subsequence of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$" is for sequences.^[24] For example, if ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}:i\in \mathbb {N} \right\}}$ denotes the set of tails of ${\displaystyle x_{\bullet }}$ and if ${\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)=\left\{x_{n_{\geq i}}:i\in \mathbb {N} \right\}}$ denotes the set of tails of the subsequence ${\displaystyle x_{n_{\bullet }}}$ (where ${\displaystyle x_{n_{\geq i}}:=\left\{x_{n_{i}}~:~i\in \mathbb {N} \right\}}$) then ${\displaystyle \operatorname {Tails} \left(x_{n_{\bullet }}\right)~\vdash ~\operatorname {Tails} \left(x_{\bullet }\right)}$ (that is, ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(x_{n_{\bullet }}\right)}$) is true but ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)~\vdash ~\operatorname {Tails} \left(x_{n_{\bullet }}\right)}$ is in general false. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in a topological space ${\displaystyle X}$ and if ${\displaystyle {\mathcal {N}}(x)}$ is the neighborhood filter at a point ${\displaystyle x\in X,}$ then ${\displaystyle x_{\bullet }\to x{\text{ in }}X{\text{ if and only if }}{\mathcal {N}}(x)\leq \operatorname {Tails} \left(x_{\bullet }\right).}$

Subordination analogs of results involving subsequences

The following results are the prefilter analogs of statements involving subsequences.^[42] The condition "${\displaystyle {\mathcal {C}}\geq {\mathcal {B}},}$" which is also written ${\displaystyle {\mathcal {C}}\vdash {\mathcal {B}},}$ is the analog of "${\displaystyle {\mathcal {C}}}$ is a subsequence of ${\displaystyle {\mathcal {B}}.}$" So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."

Non–equivalence of subnets and subordinate filters

A subset ${\displaystyle R\subseteq I}$ of a preordered space ${\displaystyle (I,\leq )}$ is frequent or cofinal in ${\displaystyle I}$ if for every ${\displaystyle i\in I}$ there exists some ${\displaystyle r\in R{\text{ such that }}i\leq r.}$ If ${\displaystyle R\subseteq I}$ contains a tail of ${\displaystyle I}$ then ${\displaystyle R}$ is said to be eventual or eventually in ${\displaystyle I}$; explicitly, this means that there exists some ${\displaystyle i\in I{\text{ such that }}I_{\geq i}\subseteq R}$ (that is, ${\displaystyle j\in R{\text{ for all }}j\in I{\text{ satisfying }}i\leq j}$). A subset is eventual if and only if its complement is not frequent (which is termed infrequent).^[43] A map ${\displaystyle h:A\to I}$ between two preordered sets is order–preserving if whenever ${\displaystyle a,b\in A{\text{ satisfy }}a\leq b,{\text{ then }}h(a)\leq h(b).}$

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."^[43] The first definition of a subnet was introduced by John L. Kelley in 1955.^[43] Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.^[43] AA–subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA–subnets were studied in great detail by Aarnes and Andenaes but they are not often used.^[43]

Let ${\displaystyle S=S_{\bullet }~:~(A,\leq )\to X{\text{ and }}N=N_{\bullet }~:~(I,\leq )\to X}$ be nets. Then^[43]

1. ${\displaystyle S_{\bullet }}$ is a Willard–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Willard if there exists an order–preserving map ${\displaystyle h:A\to I}$ such that ${\displaystyle S=N\circ h{\text{ and }}h(A)}$ is cofinal in ${\displaystyle I.}$
2. ${\displaystyle S_{\bullet }}$ is a Kelley–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Kelley if there exists a map ${\displaystyle h~:~A\to I{\text{ such that }}S=N\circ h}$ and whenever ${\displaystyle E\subseteq I}$ is eventually in ${\displaystyle I}$ then ${\displaystyle h^{-1}(E)}$ is eventually in ${\displaystyle A.}$
3. ${\displaystyle S_{\bullet }}$ is an AA–subnet of ${\displaystyle N_{\bullet }}$ or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
   1. ${\displaystyle \operatorname {Tails} \left(N_{\bullet }\right)\leq \operatorname {Tails} \left(S_{\bullet }\right).}$
   2. ${\displaystyle \operatorname {TailsFilter} \left(N_{\bullet }\right)\subseteq \operatorname {TailsFilter} \left(S_{\bullet }\right).}$
   3. If ${\displaystyle J}$ is eventually in ${\displaystyle I{\text{ then }}S^{-1}(N(J))}$ is eventually in ${\displaystyle A.}$
   4. For any subset ${\displaystyle R\subseteq X,{\text{ if }}\operatorname {Tails} \left(S_{\bullet }\right){\text{ and }}\{R\}}$ mesh, then so do ${\displaystyle \operatorname {Tails} \left(N_{\bullet }\right){\text{ and }}\{R\}.}$
   5. For any subset ${\displaystyle R\subseteq X,{\text{ if }}\operatorname {Tails} \left(S_{\bullet }\right)\leq \{R\}{\text{ then }}\operatorname {Tails} \left(N_{\bullet }\right)\leq \{R\}.}$

Kelley did not require the map ${\displaystyle h}$ to be order preserving while the definition of an AA–subnet does away entirely with any map between the two nets' domains and instead focuses entirely on ${\displaystyle X}$ − the nets' common codomain. Every Willard–subnet is a Kelley–subnet and both are AA–subnets.^[43] In particular, if ${\displaystyle y_{\bullet }=\left(y_{a}\right)_{a\in A}}$ is a Willard–subnet or a Kelley–subnet of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ then ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)\leq \operatorname {Tails} \left(y_{\bullet }\right).}$

AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.^[43][44] Explicitly, what is meant is that the following statement is true for AA–subnets:

If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters then ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ if and only if }}\operatorname {Net} _{\mathcal {F}}}$ is an AA–subnet of ${\displaystyle \;\operatorname {Net} _{\mathcal {B}}.}$

If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, the problem is that the following statement is in general false:

False statement: If ${\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}}$ are prefilters such that ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}{\text{ then }}\operatorname {Net} _{\mathcal {F}}}$ is a Kelley–subnet of ${\displaystyle \;\operatorname {Net} _{\mathcal {B}}.}$

Since every Willard–subnet is a Kelley–subnet, this statement remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".

• Counter example: For all ${\displaystyle n\in \mathbb {N} ,}$ let ${\displaystyle B_{n}=\{1\}\cup \mathbb {N} _{\geq n}.}$ Let ${\displaystyle {\mathcal {B}}=\{B_{n}~:~n\in \mathbb {N} \},}$ which is a proper π–system, and let ${\displaystyle {\mathcal {F}}=\{\{1\}\}\cup {\mathcal {B}},}$ where both families are prefilters on the natural numbers ${\displaystyle X:=\mathbb {N} =\{1,2,\ldots \}.}$ Because ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}},{\mathcal {F}}}$ is to ${\displaystyle {\mathcal {B}}}$ as a subsequence is to a sequence. So ideally, ${\displaystyle S=\operatorname {Net} _{\mathcal {F}}}$ should be a subnet of ${\displaystyle B=\operatorname {Net} _{\mathcal {B}}.}$ Let ${\displaystyle I:=\operatorname {PointedSets} ({\mathcal {B}})}$ be the domain of ${\displaystyle \operatorname {Net} _{\mathcal {B}},}$ so ${\displaystyle I}$ contains a cofinal subset that is order isomorphic to ${\displaystyle \mathbb {N} }$ and consequently contains neither a maximal nor greatest element. Let ${\displaystyle A:=\operatorname {PointedSets} ({\mathcal {F}})=\{M\}\cup I,{\text{ where }}M:=(1,\{1\})}$ is both a maximal and greatest element of ${\displaystyle A.}$ The directed set ${\displaystyle A}$ also contains a subset that is order isomorphic to ${\displaystyle \mathbb {N} }$ (because it contains ${\displaystyle I,}$ which contains such a subset) but no such subset can be cofinal in ${\displaystyle A}$ because of the maximal element ${\displaystyle M.}$ Consequently, any order–preserving map ${\displaystyle h:A\to I}$ must be eventually constant (with value ${\displaystyle h(M)}$) where ${\displaystyle h(M)}$ is then a greatest element of the range ${\displaystyle \operatorname {image} h.}$ Because of this, there can be no order preserving map ${\displaystyle h:A\to I}$ that satisfies the conditions required for ${\displaystyle \operatorname {Net} _{\mathcal {F}}}$ to be a Willard–subnet of ${\displaystyle \operatorname {Net} _{\mathcal {B}}}$ (because the range of such a map ${\displaystyle h}$ cannot be cofinal in ${\displaystyle I}$). Suppose for the sake of contradiction that there exists a map ${\displaystyle h:A\to I}$ such that ${\displaystyle h^{-1}\left(I_{\geq i}\right)}$ is eventually in ${\displaystyle A}$ for all ${\displaystyle i\in I.}$ Because ${\displaystyle h(M)\in I,}$ there exist ${\displaystyle n,n_{0}\in \mathbb {N} }$ such that ${\displaystyle h(M)=\left(n_{0},B_{n}\right){\text{ with }}n_{0}\in B_{n}.}$ For every ${\displaystyle i\in I,}$ because ${\displaystyle h^{-1}\left(I_{\geq i}\right)}$ is eventually in ${\displaystyle A,}$ it is necessary that ${\displaystyle h(M)\in I_{\geq i}.}$ In particular, if ${\displaystyle i:=\left(n+2,B_{n+2}\right)}$ then ${\displaystyle h(M)\geq i=\left(n+2,B_{n+2}\right),}$ which by definition is equivalent to ${\displaystyle B_{n}\subseteq B_{n+2},}$ which is false. Consequently, ${\displaystyle \operatorname {Net} _{\mathcal {F}}}$ is not a Kelley–subnet of ${\displaystyle \operatorname {Net} _{\mathcal {B}}.}$^[44]

If "subnet" is defined to mean Willard–subnet or Kelley–subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley–subnets and Willard–subnets are not fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.^[43][44]

Topologies and prefilters

Throughout, ${\displaystyle (X,\tau )}$ is a topological space.

Examples of relationships between filters and topologies

Bases and prefilters

Let ${\displaystyle {\mathcal {B}}\neq \varnothing }$ be a family of sets that covers ${\displaystyle X}$ and define ${\displaystyle {\mathcal {B}}_{x}=\{B\in {\mathcal {B}}~:~x\in B\}}$ for every ${\displaystyle x\in X.}$ The definition of a base for some topology can be immediately reworded as: ${\displaystyle {\mathcal {B}}}$ is a base for some topology on ${\displaystyle X}$ if and only if ${\displaystyle {\mathcal {B}}_{x}}$ is a filter base for every ${\displaystyle x\in X.}$ If ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ and ${\displaystyle {\mathcal {B}}\subseteq \tau }$ then the definitions of ${\displaystyle {\mathcal {B}}}$ is a basis (resp. subbase) for ${\displaystyle \tau }$ can be reworded as:

${\displaystyle {\mathcal {B}}}$ is a base (resp. subbase) for ${\displaystyle \tau }$ if and only if for every ${\displaystyle x\in X,{\mathcal {B}}_{x}}$ is a filter base (resp. filter subbase) that generates the neighborhood filter of ${\displaystyle (X,\tau )}$ at ${\displaystyle x.}$

Neighborhood filters

The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter."

Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If ${\displaystyle X=\mathbb {R} }$ has its usual topology and if ${\displaystyle x\in X,}$ then any neighborhood filter base ${\displaystyle {\mathcal {B}}}$ of ${\displaystyle x}$ is fixed by ${\displaystyle x}$ (in fact, it is even true that ${\displaystyle \ker {\mathcal {B}}=\{x\}}$) but ${\displaystyle {\mathcal {B}}}$ is not principal since ${\displaystyle \{x\}\not \in {\mathcal {B}}.}$ In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non–principal filter on an infinite set is not necessarily free.

The neighborhood filter of every point ${\displaystyle x}$ in topological space ${\displaystyle X}$ is fixed since its kernel contains ${\displaystyle x}$ (and possibly other points if, for instance, ${\displaystyle X}$ is not a T₁ space). This is also true of any neighborhood basis at ${\displaystyle x.}$ For any point ${\displaystyle x}$ in a T₁ space (for example, a Hausdorff space), the kernel of the neighborhood filter of ${\displaystyle x}$ is equal to the singleton set ${\displaystyle \{x\}.}$

However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis ${\displaystyle {\mathcal {B}}}$ of a point ${\displaystyle x}$ in a topological space is principal if and only if the kernel of ${\displaystyle {\mathcal {B}}}$ is an open set. If in addition the space is T₁ then ${\displaystyle \ker {\mathcal {B}}=\{x\}}$ so that this basis ${\displaystyle {\mathcal {B}}}$ is principal if and only if ${\displaystyle \{x\}}$ is an open set.

Generating topologies from filters and prefilters

Suppose ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is not empty (and ${\displaystyle X\neq \varnothing }$). If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ then ${\displaystyle \{\varnothing \}\cup {\mathcal {B}}}$ is a topology on ${\displaystyle X}$ but the converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form ${\displaystyle \{\varnothing \}\cup {\mathcal {B}}}$ where ${\displaystyle {\mathcal {B}}}$ is an ultrafilter on ${\displaystyle X}$ are an even more specialized subclass of such topologies; they have the property that every proper subset ${\displaystyle \varnothing \neq S\subseteq X}$ is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces.

If ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. filter subbase, π–system, proper) on ${\displaystyle X}$ then the same is true of both ${\displaystyle \{X\}\cup {\mathcal {B}}}$ and the set ${\displaystyle {\mathcal {B}}_{\cup }}$ of all possible unions of one or more elements of ${\displaystyle {\mathcal {B}}.}$ If ${\displaystyle {\mathcal {B}}}$ is closed under finite intersections then the set ${\displaystyle \tau _{\mathcal {B}}=\{\varnothing ,X\}\cup {\mathcal {B}}_{\cup }}$ is a topology on ${\displaystyle X}$ with both ${\displaystyle \{X\}\cup {\mathcal {B}}_{\cup }{\text{ and }}\{X\}\cup {\mathcal {B}}}$ being bases for it. If the π–system ${\displaystyle {\mathcal {B}}}$ covers ${\displaystyle X}$ then both ${\displaystyle {\mathcal {B}}_{\cup }{\text{ and }}{\mathcal {B}}}$ are also bases for ${\displaystyle \tau _{\mathcal {B}}.}$ If ${\displaystyle \tau }$ is a topology on ${\displaystyle X}$ then ${\displaystyle \tau \setminus \{\varnothing \}}$ is a prefilter (or equivalently, a π–system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset ${\displaystyle {\mathcal {B}}\subseteq \tau }$ will be a basis for ${\displaystyle \tau }$ if and only if ${\displaystyle {\mathcal {B}}\setminus \{\varnothing \}}$ is equivalent to ${\displaystyle \tau \setminus \{\varnothing \},}$ in which case ${\displaystyle {\mathcal {B}}\setminus \{\varnothing \}}$ will be a prefilter.

Topologies on directed sets and net convergence

Let ${\displaystyle (I,\leq )}$ be a non–empty directed set and let ${\displaystyle \operatorname {Tails} (I)=\left\{I_{\geq i}~:~i\in I\right\},}$ where ${\displaystyle I_{\geq i}=\{j\in I~:~i\leq j\}.}$ Then ${\displaystyle \operatorname {Tails} (I)}$ is a prefilter that covers ${\displaystyle I}$ and if ${\displaystyle I}$ is totally ordered then ${\displaystyle \operatorname {Tails} (I)}$ is also closed under finite intersections. This particular prefilter ${\displaystyle \operatorname {Tails} (I)}$ forms a base for a topology on ${\displaystyle I}$ in which all sets of the form ${\displaystyle I_{>i}=\{j\in I~:~i<j\}}$ are also open. The same is true of the topology ${\displaystyle \tau _{I}:=\{\varnothing \}\cup \operatorname {FilterTails} (I){\text{ on }}I,}$ where ${\displaystyle \operatorname {FilterTails} (I)}$ is the filter on ${\displaystyle I}$ generated by ${\displaystyle \operatorname {Tails} (I).}$ With this topology, convergent nets can be viewed as continuous functions in the following way. Let ${\displaystyle (X,\tau )}$ be a topological space, let ${\displaystyle x\in X,}$ let ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}~:~I\to X}$ be a net in ${\displaystyle X,}$ and let ${\displaystyle \tau (x)\subseteq \tau }$ denote the set of all open neighborhoods of ${\displaystyle x.}$ If the net ${\displaystyle x_{\bullet }}$ converges to ${\displaystyle x{\text{ in }}(X,\tau )}$ then ${\displaystyle x_{\bullet }~:~\left(I,\tau _{I}\right)\to \left(X,\{\varnothing \}\cup \tau (x)\right)}$ is necessarily continuous although in general, the converse is false (for example, consider if ${\displaystyle x_{\bullet }}$ is constant and not equal to ${\displaystyle x}$). But if in addition to continuity, the preimage under ${\displaystyle x_{\bullet }}$ of every ${\displaystyle N\in \tau (x)}$ is not empty, then the net ${\displaystyle x_{\bullet }}$ will necessarily converge to ${\displaystyle x{\text{ in }}(X,\tau ).}$ In this way, the empty set is all that separates net convergence and continuity.

Another way in which a convergent nets can be viewed as continuous functions is, for any given ${\displaystyle x\in X}$ and net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}~:~I\to X,}$ to first extend the net to a new net ${\displaystyle {\hat {x}}_{\bullet }:=\left({\hat {x}}_{i}\right)_{i\in I}:I\cup \{\infty \}\to X,}$ where ${\displaystyle \infty \not \in I}$ is a new symbol, by defining ${\displaystyle {\hat {x}}_{\infty }:=x{\text{ and }}{\hat {x}}_{i}:=x_{i}}$ for every ${\displaystyle i\in I.}$ If ${\displaystyle I\cup \{\infty \}}$ is endowed with the topology

then ${\displaystyle x_{\bullet }\to x{\text{ in }}X}$ (that is, the net ${\displaystyle x_{\bullet }}$converges to ${\displaystyle x}$) if and only if ${\displaystyle {\hat {x}}_{\bullet }:\left(I\cup \{\infty \},\tau _{I\cup \{\infty \}}\right)\to (X,\tau )}$ is a continuous function. Moreover, ${\displaystyle I}$ is always a dense subset of ${\displaystyle I\cup \{\infty \}.}$

Topological properties and prefilters

Throughout ${\displaystyle (X,\tau )}$ will be a topological space with ${\displaystyle X\neq \varnothing .}$

Neighborhoods and topologies

The neighborhood filter of a non–empty subset ${\displaystyle S\subseteq X}$ in a topological space ${\displaystyle X}$ is equal to the intersection of all neighborhood filters of all points in ${\displaystyle S.}$^[28] If ${\displaystyle S\subseteq X}$ then ${\displaystyle S}$ is open in ${\displaystyle X}$ if and only if whenever ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ and ${\displaystyle s\in S,}$ then ${\displaystyle {\mathcal {F}}\to s{\text{ in }}(X,\tau ){\text{ implies }}S\in {\mathcal {F}}.}$

Suppose ${\displaystyle \sigma {\text{ and }}\tau }$ are topologies on ${\displaystyle X.}$ Then ${\displaystyle \tau }$ is finer than ${\displaystyle \sigma }$ (that is, ${\displaystyle \sigma \subseteq \tau }$) if and only if whenever ${\displaystyle x\in X{\text{ and }}{\mathcal {B}}}$ is a filter on ${\displaystyle X,}$ if ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau )}$ then ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\sigma ).}$^[39] Consequently, ${\displaystyle \sigma =\tau }$ if and only if for every filter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ and every ${\displaystyle x\in X,{\mathcal {B}}\to x{\text{ in }}(X,\sigma )}$ if and only if ${\displaystyle {\mathcal {B}}\to x{\text{ in }}(X,\tau ).}$^[29] However, it is possible that ${\displaystyle \sigma \neq \tau }$ while also for every filter ${\displaystyle {\mathcal {B}}{\text{ on }}X,{\mathcal {B}}}$ converges to some point of ${\displaystyle X{\text{ in }}(X,\sigma )}$ if and only if ${\displaystyle {\mathcal {B}}}$ converges to some point of ${\displaystyle X{\text{ in }}(X,\tau ).}$^[29]

Closure

If ${\displaystyle x\in X{\text{ and }}S\subseteq X{\text{ with }}S\neq \varnothing }$ then the following are equivalent:

1. ${\displaystyle x\in \operatorname {ck} _{X}S}$
2. ${\displaystyle x}$ is a limit point of the prefilter ${\displaystyle \{S\}}$ (that is, ${\displaystyle \{S\}\to x{\text{ in }}X}$).
3. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (X){\text{ on }}X}$ such that ${\displaystyle S\in {\mathcal {F}}{\text{ and }}{\mathcal {F}}\to x{\text{ in }}X.}$
4. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}S}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X.}$^[42]
5. ${\displaystyle x}$ is a cluster point of the prefilter ${\displaystyle \{S\}.}$
6. The prefilter ${\displaystyle \{S\}}$ meshes with the neighborhood filter ${\displaystyle {\mathcal {N}}(x).}$
7. The prefilter ${\displaystyle \{S\}}$ meshes with some (or equivalently, with every) prefilter of ${\displaystyle {\mathcal {N}}(x).}$

The following are equivalent:

1. ${\displaystyle x}$ is a limit points of ${\displaystyle S{\text{ in }}X.}$
2. There exists a prefilter ${\displaystyle {\mathcal {F}}\subseteq \wp (S){\text{ on }}\{S\}\setminus \{x\}}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X.}$^[42]

Closed sets

If ${\displaystyle S\subseteq X}$ is not empty then the following are equivalent:

1. ${\displaystyle S}$ is a closed subset of ${\displaystyle X.}$
2. If ${\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)}$ is a prefilter on ${\displaystyle S}$ such that ${\displaystyle {\mathcal {F}}\to x{\text{ in }}X,}$ then ${\displaystyle x\in S.}$
3. If ${\displaystyle x\in X{\text{ and }}{\mathcal {F}}\subseteq \wp (S)}$ is a prefilter on ${\displaystyle S}$ such that ${\displaystyle x}$ is an accumulation points of ${\displaystyle {\mathcal {F}}{\text{ in }}X,}$ then ${\displaystyle x\in S.}$^[42]
4. If ${\displaystyle x\in X}$ is such that the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ meshes with ${\displaystyle \{S\}}$ then ${\displaystyle x\in S.}$
   • The proof of this characterization depends the ultrafilter lemma, which depends on the axiom of choice.

Hausdorffness

The following are equivalent:

1. ${\displaystyle X}$ is Hausdorff.
2. Every prefilter on ${\displaystyle X}$ converges to at most one point in ${\displaystyle X.}$^[8]
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.^[8]

Compactness

As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.

The following are equivalent:

1. ${\displaystyle (X,\tau )}$ is a compact space.
2. Every ultrafilter on ${\displaystyle X}$ converges to at least one point in ${\displaystyle X.}$
   • That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma (or even the axiom of choice).
3. The above statement but with the words "prefilter" replaced by any one of the following: filter, ultrafilter.^[45]
4. For every filter ${\displaystyle {\mathcal {C}}{\text{ on }}X}$ there exists a filter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ converges to some point of ${\displaystyle X.}$
5. For every prefilter ${\displaystyle {\mathcal {C}}{\text{ on }}X}$ there exists a prefilter ${\displaystyle {\mathcal {F}}{\text{ on }}X}$ such that ${\displaystyle {\mathcal {C}}\leq {\mathcal {F}}}$ and ${\displaystyle {\mathcal {F}}}$ converges to some point of ${\displaystyle X.}$
6. Every maximal (i.e. ultra) prefilter on ${\displaystyle X}$ converges to at least one point in ${\displaystyle X.}$^[8]
7. The above statement but with the words "maximal prefilter" replaced by any one of the following: prefilter, filter, ultra prefilter, ultrafilter.
8. Every prefilter on ${\displaystyle X}$ has at least one cluster point in ${\displaystyle X.}$^[8]
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
9. Alexander subbase theorem: There exists a subbase${\displaystyle {\mathcal {S}}{\text{ for }}\tau }$ such that every cover of ${\displaystyle X}$ by sets in ${\displaystyle {\mathcal {S}}}$ has a finite subcover.
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

If ${\displaystyle (X,\tau )}$ is topological space and ${\displaystyle {\mathcal {F}}}$ is the set of all complements of compact subsets of ${\displaystyle (X,\tau ),}$ then ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ if and only if ${\displaystyle (X,\tau )}$ is not compact.

Continuity

Let ${\displaystyle f:X\to Y}$ is a map between topological spaces ${\displaystyle (X,\tau ){\text{ and }}(Y,\upsilon ).}$

Given ${\displaystyle x\in X,}$ the following are equivalent:

1. ${\displaystyle f:X\to Y}$ is continuous at ${\displaystyle x.}$
2. Definition: For every neighborhood ${\displaystyle V}$ of ${\displaystyle f(x){\text{ in }}Y}$ there exists some neighborhood ${\displaystyle N}$ of ${\displaystyle x{\text{ in }}X}$ such that ${\displaystyle f(N)\subseteq V.}$
3. ${\displaystyle f({\mathcal {N}}(x))}$ is a filter base for ${\displaystyle {\mathcal {N}}(f(x))}$; that is, the upward closure of ${\displaystyle f({\mathcal {N}}(x)){\text{ in }}Y}$ is equal to ${\displaystyle {\mathcal {N}}(f(x)).}$^[42]
4. ${\displaystyle f({\mathcal {N}}(x))\to f(x){\text{ in }}Y.}$^[42]
5. If ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ such that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}X}$ then ${\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.}$
6. The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

1. ${\displaystyle f:X\to Y}$ is continuous.
2. If ${\displaystyle x\in X{\text{ and }}{\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ such that ${\displaystyle {\mathcal {B}}\to x{\text{ in }}X}$ then ${\displaystyle f({\mathcal {B}})\to f(x){\text{ in }}Y.}$
3. If ${\displaystyle x\in X}$ is a limit point of a prefilter ${\displaystyle {\mathcal {B}}{\text{ on }}X}$ then ${\displaystyle f(x)}$ is a limit point of ${\displaystyle f({\mathcal {B}}){\text{ in }}Y.}$
4. Any one of the above two statements but with the word "prefilter" replaced by any one of the following: filter.

If ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X,x\in X}$ is a cluster point of ${\displaystyle {\mathcal {B}},{\text{ and }}f:X\to Y}$ is continuous, then ${\displaystyle f(x)}$ is a cluster point in ${\displaystyle Y}$ of the prefilter ${\displaystyle f({\mathcal {B}}).}$^[39]

Products

Suppose ${\displaystyle X_{\bullet }:=\left(X_{i}\right)_{i\in I}}$ is a non–empty family of non–empty topological spaces and that is a family of prefilters where each ${\displaystyle {\mathcal {B}}_{i}}$ is a prefilter on ${\displaystyle X_{i}.}$ Then the product ${\displaystyle {\mathcal {B}}_{\bullet }}$ of these prefilters (defined above) is a prefilter on the product space ${\displaystyle \prod X_{\bullet },}$ which as usual, is endowed with the product topology.

If ${\displaystyle x_{\bullet }:=\left(x_{i}\right)_{i\in I}\in \prod X_{\bullet },}$ then ${\displaystyle {\mathcal {B}}_{\bullet }\to x_{\bullet }{\text{ in }}\prod X_{\bullet }}$ if and only if ${\displaystyle {\mathcal {B}}_{i}\to x_{i}{\text{ in }}X_{i}{\text{ for every }}i\in I.}$

Suppose ${\displaystyle X{\text{ and }}Y}$ are topological spaces, ${\displaystyle {\mathcal {B}}}$ is a prefilter on ${\displaystyle X}$ having ${\displaystyle x\in X}$ as a cluster point, and ${\displaystyle {\mathcal {C}}}$ is a prefilter on ${\displaystyle Y}$ having ${\displaystyle y\in Y}$ as a cluster point. Then ${\displaystyle (x,y)}$ is a cluster point of ${\displaystyle {\mathcal {B}}\times {\mathcal {C}}}$ in the product space ${\displaystyle X\times Y.}$^[39] However, if ${\displaystyle X=Y=\mathbb {Q} }$ then there exist sequences ${\displaystyle x_{\bullet }:=\left(x_{i}\right)_{i=1}^{\infty }\subseteq X{\text{ and }}y_{\bullet }:=\left(y_{i}\right)_{i=1}^{\infty }\subseteq Y}$ such that both of these sequences have a cluster point in ${\displaystyle \mathbb {Q} }$ but the sequence ${\displaystyle \left(x_{i},y_{i}\right)_{i=1}^{\infty }\subseteq X\times Y}$ does not have a cluster point in ${\displaystyle X\times Y.}$^[39]

Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:

Examples of applications of prefilters

Uniformities and Cauchy prefilters

A uniform space is a set ${\displaystyle X}$ equipped with a filter on ${\displaystyle X\times X}$ that has certain properties. A base or fundamental system of entourages is a prefilter on ${\displaystyle X\times X}$ whose upward closure is a uniform space. A prefilter ${\displaystyle {\mathcal {B}}}$ on a uniform space ${\displaystyle X}$ with uniformity ${\displaystyle {\mathcal {F}}}$ is called a Cauchy prefilter if for every entourage ${\displaystyle N\in {\mathcal {F}},}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ that is ${\displaystyle N}$–small, which means that ${\displaystyle B\times B\subseteq N.}$ A uniform space ${\displaystyle (X,{\mathcal {F}})}$ is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$ A minimal Cauchy filter is a minimal element (with respect to ${\displaystyle \,\leq \,}$ or equivalently, to ${\displaystyle \,\subseteq }$) of the set of all Cauchy filters on ${\displaystyle X.}$ Examples of minimal Cauchy filters include the neighborhood filter ${\displaystyle {\mathcal {N}}_{X}(x)}$ of any point ${\displaystyle x\in X.}$

Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first–countable, or even sequential. The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS) ${\displaystyle X}$ can made into a vector space and topologized in such a way that it becomes a completion of ${\displaystyle X}$ (with the assignment ${\displaystyle x\mapsto {\mathcal {N}}_{X}(x)}$ becoming a linear topological embedding that identifies ${\displaystyle X}$ as a dense vector subspace of this completion).

Convergence of nets of sets

There is often a personal preference of nets over filters or filters over nets. This example shows that the choice between nets and filters is not a dichotomy by combining them together.

A net of sets in ${\displaystyle X}$ or a net of subsets of ${\displaystyle X}$ refers to a net in the power set ${\displaystyle \wp (X)}$ of ${\displaystyle X;}$ that is, a net of sets in ${\displaystyle X}$ is a function from a non–empty directed set into ${\displaystyle \wp (X).}$ However, a "net in ${\displaystyle X}$" will always refer to a net valued in ${\displaystyle X}$ and never to a net valued in ${\displaystyle \wp (X)}$ although for emphasis or contrast, a net in ${\displaystyle X}$ may also be referred to as a net of points in ${\displaystyle X}$. A net ${\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}}$ of sets in ${\displaystyle X}$ is called a net of singleton (resp. non–empty, finite, compact, etc.) sets in ${\displaystyle X}$ if every ${\displaystyle S_{i}}$ has this property. Similarly, ${\displaystyle S_{\bullet }}$ is called eventually empty (resp. non–empty, finite, compact, etc.) if there is some index ${\displaystyle i}$ such that this is true of ${\displaystyle S_{j}}$ for every index ${\displaystyle j\geq i.}$

The following definition generalizes the notion of the set of tails of a net of points in ${\displaystyle X}$ to nets of subsets of ${\displaystyle X.}$

Suppose ${\displaystyle S_{\bullet }=\left(S_{i}\right)_{i\in I}}$ is a net of sets in ${\displaystyle X.}$ Define for every index ${\displaystyle i}$ the tail of ${\displaystyle S_{\bullet }}$ starting at ${\displaystyle i}$ to be the set

and define the set or family of tails generated by ${\displaystyle S_{\bullet }}$ to be the family The family ${\displaystyle \operatorname {Tails} \left(S_{\bullet }\right)}$ is a prefilter if and only if it does not contain the empty set, which is equivalent to ${\displaystyle S_{\bullet }}$ not being eventually empty; in this case the upward closure in ${\displaystyle X}$ of this prefilter of tails is called the filter of tails or eventuality filter in ${\displaystyle X}$ generated by ${\displaystyle S_{\bullet }.}$ A net ${\displaystyle y_{\bullet }}$ (of sets or points) is eventually contained in a set ${\displaystyle C}$ if and only if ${\displaystyle \{C\}\leq \operatorname {Tails} \left(y_{\bullet }\right);}$ so ${\displaystyle S_{\bullet }}$ is eventually empty if and only if ${\displaystyle \{\varnothing \}\leq \operatorname {Tails} \left(S_{\bullet }\right).}$

Nets of sets arise naturally when pulling back nets in a function's codomain. If ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle y_{\bullet }=\left(y_{i}\right)_{i\in I}}$ is a net of sets (or of points) then let ${\displaystyle f^{-1}\left(y_{\bullet }\right):=\left(f^{-1}\left(y_{i}\right)\right)_{i\in I}}$ and ${\displaystyle f\left(y_{\bullet }\right):=\left(f\left(y_{i}\right)\right)_{i\in I};}$ that is, ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ denotes the net of sets ${\displaystyle I\to \wp (X)}$ defined by ${\displaystyle i\mapsto f^{-1}\left(y_{i}\right).}$ The tail of ${\displaystyle f\left(y_{\bullet }\right)}$ starting at an index ${\displaystyle i}$ is equal to ${\displaystyle f\left(y_{\geq i}\right)}$ and similarly, the tail of ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ starting at ${\displaystyle i}$ is ${\displaystyle f^{-1}\left(y_{\geq i}\right).}$ Consequently, ${\displaystyle \operatorname {Tails} \left(f\left(y_{\bullet }\right)\right)=f\left(\operatorname {Tails} \left(y_{\bullet }\right)\right)}$ where this family if a prefilter if and only if ${\displaystyle \operatorname {Tails} \left(S_{\bullet }\right)}$ is a prefilter; similarly, ${\displaystyle \operatorname {Tails} \left(f^{-1}\left(y_{\bullet }\right)\right)=f^{-1}\left(\operatorname {Tails} \left(y_{\bullet }\right)\right).}$ One useful consequence of this definition is that ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ is a prefilter if and only if ${\displaystyle y_{\bullet }}$ cofinally intersects (or for points, is cofinally in) ${\displaystyle f(X),}$ meaning that for every index ${\displaystyle i,}$ there is some ${\displaystyle j\geq i}$ such that ${\displaystyle y_{j}\cap f(X)\neq \varnothing }$ (where this intersection means ${\displaystyle y_{j}\in f(X)}$ if ${\displaystyle y_{j}}$ is a point instead of a set). In particular, ${\displaystyle y_{\bullet }}$ eventually being contained in ${\displaystyle f(X)}$ (meaning that ${\displaystyle y_{\geq i}\subseteq f(X)}$ for some ${\displaystyle i}$) is not a necessary condition for ${\displaystyle \operatorname {Tails} \left(f^{-1}\left(y_{\bullet }\right)\right)}$ to be a prefilter. So even if a net ${\displaystyle \left(y_{i}\right)_{i\in I}}$ of points in ${\displaystyle Y}$ cannot be pulled back by ${\displaystyle f}$ to a net ${\displaystyle \left(x_{i}\right)_{i\in I}}$ of points in ${\displaystyle X}$ (say because it is not entirely/eventually in the image of ${\displaystyle f}$), it is nevertheless still possible to talk about the net of sets ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ and its properties (such as convergence or clustering).

Convergence and clustering

Consideration of the following bijective correspondence leads naturally to the following definition, which is completely analogous to the previously given definition of the tails of a net (of points) in ${\displaystyle X.}$

(Nets of points ${\displaystyle \leftrightarrow }$ Nets of singleton sets): Every net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ of points in ${\displaystyle X}$ can be uniquely associated with the canonical net of singleton sets ${\displaystyle \left(\left\{x_{i}\right\}\right)_{i\in I}}$ and conversely, every net of singleton sets in ${\displaystyle X}$ is uniquely associated with a canonical net of points (defined in the obvious way). The tail of ${\displaystyle \left(\left\{x_{i}\right\}\right)_{i\in I}}$ starting at an index ${\displaystyle i}$ is equal to that of ${\displaystyle x_{\bullet }}$ (that is, to ${\displaystyle x_{\geq i}}$); consequently, ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\operatorname {Tails} \left(\left(\left\{x_{i}\right\}\right)_{i\in I}\right).}$ This makes it apparent that the following definition of "convergence of a net of sets" in ${\displaystyle X}$ is indeed a generalization of the original definition of "convergence of a net of points" in ${\displaystyle X}$ (because ${\displaystyle x_{\bullet }\to R}$ if and only if ${\displaystyle \left(\left\{x_{i}\right\}\right)_{i\in I}\to R}$); similarly, a net of points clusters at a given point or subset ${\displaystyle x}$ (according to the original definition) if and only if its associated net of singleton sets clusters at ${\displaystyle x}$ (according to the definition below).

A net of sets ${\displaystyle S_{\bullet }}$ is said to converge in ${\displaystyle (X,\tau )}$ to a given point or subset ${\displaystyle x}$ of ${\displaystyle X,}$ written ${\displaystyle S_{\bullet }\to x{\text{ in }}(X,\tau ),}$ if ${\displaystyle \,\operatorname {Tails} \left(S_{\bullet }\right)\to x{\text{ in }}(X,\tau ),}$ which recall was defined to mean that ${\displaystyle {\mathcal {N}}_{\tau }(x)\leq \operatorname {Tails} \left(S_{\bullet }\right).}$ Explicitly, this happens if and only if for every neighborhood ${\displaystyle U}$ of ${\displaystyle x,}$ there exists some index ${\displaystyle i}$ such that ${\displaystyle S_{\geq i}\subseteq U.}$ Similarly, ${\displaystyle S_{\bullet }}$ is said to cluster at a given point or subset ${\displaystyle x}$ of ${\displaystyle X}$ if ${\displaystyle \operatorname {Tails} \left(S_{\bullet }\right)}$ meshes with ${\displaystyle {\mathcal {N}}_{\tau }(x)}$ (written ${\displaystyle {\mathcal {N}}_{\tau }(x)\;\#\;\operatorname {Tails} \left(S_{\bullet }\right)}$); explicitly, this means that ${\displaystyle \varnothing \neq N\cap S_{\geq i}}$ for every index ${\displaystyle i\in I}$ and neighborhood ${\displaystyle N}$ of ${\displaystyle x.}$

Every net of sets that is eventually empty converges to every point/subset. However, a net of sets converges to ${\displaystyle \varnothing }$ if and only if it is eventually empty. No net of sets clusters at ${\displaystyle \varnothing .}$ If a net of sets converges to ${\displaystyle x}$ then it will cluster at ${\displaystyle x}$ if and only if it is not eventually empty (which implies ${\displaystyle x\neq \varnothing }$). If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ then ${\displaystyle \left(x_{\geq i}\right)_{i\in I}}$ is a net of sets in ${\displaystyle X}$ and for any point or subset ${\displaystyle x}$ of ${\displaystyle X,}$ ${\displaystyle x_{\bullet }}$ converges to (respectively, clusters at) ${\displaystyle x}$ in ${\displaystyle X}$ if and only if this is true of ${\displaystyle \left(x_{\geq i}\right)_{i\in I}.}$ This statement remains true if ${\displaystyle x_{\bullet }}$ is instead a net of sets. If ${\displaystyle f:X\to Y}$ is a map and ${\displaystyle y_{\bullet }=\left(y_{i}\right)_{i\in I}}$ is a net (of points or of sets) then ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ converges to (respectively, clusters at) some given point or subset of ${\displaystyle X}$ if and only if every neighborhood of it contains (respectively, intersects) some set of the form ${\displaystyle f^{-1}\left(y_{\geq i}\right).}$ Moreover, the net ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ converges in ${\displaystyle X}$ to some given point or subset if and only if this is true of ${\displaystyle f^{-1}\left(\operatorname {Tails} \left(y_{\bullet }\right)\right).}$

Applications

Some applications are now given showing how nets of sets can be used to characterize various properties. In the statements below, the map ${\displaystyle f:X\to Y}$ is not necessarily surjective and ${\displaystyle y}$ is a point and ${\displaystyle y_{\bullet }}$ is nets of points in ${\displaystyle Y.}$

• A map ${\displaystyle f:X\to Y}$ is closed (meaning it sends closed sets to closed subset of ${\displaystyle Y}$) if and only if whenever ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ then ${\displaystyle f^{-1}\left(y_{\bullet }\right)\to f^{-1}(y){\text{ in }}X.}$ This characterization remains true if ${\displaystyle y_{\bullet }{\text{ and }}y}$ are allowed to be sets (instead of restricted to being points) such that ${\displaystyle y_{\bullet }\to y{\text{ in }}Y.}$

  Proof

  Assume ${\displaystyle f:X\to Y}$ is closed and ${\displaystyle y_{\bullet }\to y{\text{ in }}Y.}$ If ${\displaystyle y\not \in f(X)}$ then ${\displaystyle y}$ is in the open set ${\displaystyle Y\setminus f(X)}$ so that ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ implies that ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ is eventually empty and thus that ${\displaystyle f^{-1}\left(y_{\bullet }\right)\to \varnothing =f^{-1}(y)}$ in ${\displaystyle X.}$ So assume ${\displaystyle f^{-1}(y)\neq \varnothing }$ and let ${\displaystyle U}$ be an open neighborhood of ${\displaystyle f^{-1}(y)}$ in ${\displaystyle X.}$ It remains to show that ${\displaystyle f^{-1}\left(y_{\geq i}\right)\subseteq U}$ for some index ${\displaystyle i.}$ Since ${\displaystyle f}$ is closed, ${\displaystyle Y\setminus f(X\setminus U)}$ is an open neighborhood of ${\displaystyle y}$ in ${\displaystyle Y}$ so there must exists some index ${\displaystyle i}$ such that ${\displaystyle y_{\geq i}\subseteq Y\setminus f(X\setminus U).}$ This implies ${\displaystyle f^{-1}\left(y_{\geq i}\right)\subseteq f^{-1}(Y\setminus f(X\setminus U))}$ where the right hand side is a subset of ${\displaystyle U,}$ as desired. For the converse, assume that ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ implies ${\displaystyle f^{-1}\left(y_{\bullet }\right)\to f^{-1}(y){\text{ in }}X.}$ Let ${\displaystyle C\subseteq X}$ be closed and assume it is not empty. Let ${\displaystyle y_{\bullet }=\left(y_{i}\right)_{i\in I}}$ be a net in ${\displaystyle f(C)}$ (meaning ${\displaystyle y_{i}\in f(C)}$ for all ${\displaystyle i}$) and let ${\displaystyle y\in Y}$ be such that ${\displaystyle y_{\bullet }\to y{\text{ in }}Y.}$ It remains to show that ${\displaystyle y\in f(C).}$ The hypotheses guarantee that ${\displaystyle f^{-1}\left(y_{\bullet }\right)\to f^{-1}(y){\text{ in }}X.}$ The fact that every fiber ${\displaystyle f^{-1}\left(y_{i}\right)}$ is not empty and that these fibers converge to ${\displaystyle f^{-1}(y)}$ imply that ${\displaystyle f^{-1}(y)\neq \varnothing .}$ Since ${\displaystyle X\setminus C}$ is open, were it true that ${\displaystyle f^{-1}(y)\subseteq X\setminus C}$ then there would exist some index ${\displaystyle i}$ such that ${\displaystyle f^{-1}\left(y_{\geq i}\right)\subseteq X\setminus C,}$ which is impossible since ${\displaystyle y_{j}\in f(C)}$ for every index ${\displaystyle j.}$ Thus ${\displaystyle f^{-1}(y)\not \subseteq X\setminus C}$ so there is some ${\displaystyle c\in C\cap f^{-1}(y),}$ which proves that ${\displaystyle y=f(c)\in f(C).}$ ${\displaystyle \blacksquare }$

• A map ${\displaystyle f:X\to Y}$ is open (meaning it sends open sets to open subset of ${\displaystyle Y}$) if and only if whenever ${\displaystyle x}$ is a point of ${\displaystyle X}$ and ${\displaystyle y_{\bullet }}$ is a net that clusters at ${\displaystyle f(x)}$ in ${\displaystyle Y,}$ then ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ clusters at ${\displaystyle x}$ in ${\displaystyle X.}$ This characterization remains true if ${\displaystyle y_{\bullet }{\text{ and }}x}$ are allowed to be sets.

  Proof

  For the non-trivial direction, suppose that ${\displaystyle f:X\to Y}$ is not an open map. Pick an open subset ${\displaystyle U\subseteq X}$ such that ${\displaystyle f(U)}$ is not open in ${\displaystyle Y,}$ where non-openness means that there is some point ${\displaystyle y\in f(U)}$ such that ${\displaystyle f(U)}$ is not a neighborhood of ${\displaystyle y}$ in ${\displaystyle Y.}$ Explicitly, this means that ${\displaystyle N\not \subseteq f(U)}$ for every neighborhood ${\displaystyle N}$ of ${\displaystyle y}$ in ${\displaystyle Y,}$ which guarantees the existence of some ${\displaystyle y_{N}\in N\setminus f(U).}$ Let ${\displaystyle I:={\mathcal {N}}_{Y}(y)}$ denote the neighborhood filter of ${\displaystyle y}$ in ${\displaystyle Y}$ and direct it by ${\displaystyle \,\supseteq \,}$ to make ${\displaystyle y_{\bullet }:=\left(y_{N}\right)_{N\in I}}$ into a net that converges to ${\displaystyle y}$ in ${\displaystyle Y,}$ which implies that ${\displaystyle y_{\bullet }}$ clusters at ${\displaystyle y}$ in ${\displaystyle Y.}$ Because ${\displaystyle y\in f(U),}$ there is some ${\displaystyle x\in U\cap f^{-1}(y).}$ But ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ does not clusters at ${\displaystyle x}$ since ${\displaystyle f^{-1}\left(y_{N}\right)\cap U=\varnothing }$ for every ${\displaystyle N\in I.}$ ${\displaystyle \blacksquare }$

• A map ${\displaystyle f:X\to Y}$ is open if and only if whenever ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ then any closed subset of ${\displaystyle X}$ that contains ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ will necessarily also contain ${\displaystyle f^{-1}(y).}$ (A set ${\displaystyle C}$contains${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ means that ${\displaystyle f^{-1}\left(y_{i}\right)\subseteq C}$ for every index ${\displaystyle i.}$) This characterization remains true if ${\displaystyle y_{\bullet }}$ is allowed to be a net of sets that is not eventually empty (instead of being a net of points) and ${\displaystyle y\in Y}$ remains a point such that ${\displaystyle y_{\bullet }\to y{\text{ in }}Y.}$ The same is true of the quotient map characterization below.
  • In comparison, by the closure characterization of continuity, ${\displaystyle f:X\to Y}$ is continuous if and only if whenever ${\displaystyle x_{\bullet }\to x{\text{ in }}X}$ then any closed subset of ${\displaystyle Y}$ that contains ${\displaystyle f\left(x_{\bullet }\right)}$ will necessarily also contain ${\displaystyle f(x).}$

  Proof

  If ${\displaystyle R\subseteq X}$ is any subset then it is readily verified that ${\displaystyle f(X\setminus R)=Y\setminus \left\{y\in Y:f^{-1}(y)\subseteq R\right\}.}$ This implies that a map ${\displaystyle f:X\to Y}$ is open if and only if whenever ${\displaystyle C\subseteq X}$ is closed in ${\displaystyle X}$ then ${\displaystyle \left\{y\in Y~:~f^{-1}(y)\subseteq C\right\}}$ is closed in ${\displaystyle Y.}$ This characterization of "open map" combined with the convergent net characterization of closed sets produces the desired conclusion: ${\displaystyle f:X\to Y}$ is open if and only if whenever ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ and ${\displaystyle C\subseteq X}$ is a closed subset of ${\displaystyle X}$ that contains ${\displaystyle f^{-1}\left(y_{\bullet }\right),}$ then necessarily ${\displaystyle f^{-1}(y)\subseteq C.}$ ${\displaystyle \blacksquare }$

• A continuous surjection ${\displaystyle f:X\to Y}$ is a quotient map if and only if whenever ${\displaystyle y_{\bullet }\to y{\text{ in }}Y}$ then any saturated closed subset of ${\displaystyle X}$ that contains ${\displaystyle f^{-1}\left(y_{\bullet }\right)}$ will necessarily also contain ${\displaystyle f^{-1}(y).}$ (A set ${\displaystyle C\subseteq X}$ is saturated if ${\displaystyle f^{-1}(f(C))=C.}$)
• A subset ${\displaystyle C\subseteq X}$ is closed in ${\displaystyle X}$ if and only if for every point ${\displaystyle x\in X}$ and every net ${\displaystyle c_{\bullet }}$ of subsets of ${\displaystyle C}$ that is not eventually empty, if ${\displaystyle c_{\bullet }\to x{\text{ in }}X}$ then ${\displaystyle x\in C.}$
• A map ${\displaystyle f:X\to Y}$ is continuous if and only if whenever a net ${\displaystyle S_{\bullet }}$ of points or sets in ${\displaystyle X}$ converges to a point or set ${\displaystyle S{\text{ in }}X,}$ then ${\displaystyle f\left(S_{\bullet }\right)\to f(S){\text{ in }}Y.}$

  Proof

  The proof is essentially identical to the usual proof involving only nets of points. One direction (that whose conclusion is that ${\displaystyle f}$ is continuous) only requires consideration of nets of points and so it is omitted. So suppose that the map is continuous and that ${\displaystyle S_{\bullet }\to S{\text{ in }}X.}$ Let ${\displaystyle V\subseteq Y}$ be an open neighborhood of ${\displaystyle f(S)}$ in ${\displaystyle Y.}$ Then ${\displaystyle f^{-1}(V)}$ is an open neighborhood of ${\displaystyle S}$ in ${\displaystyle X}$ so there exists some index ${\displaystyle i}$ such that ${\displaystyle S_{\geq i}\subseteq f^{-1}(V).}$ Thus ${\displaystyle f\left(S_{\geq i}\right)\subseteq V,}$ as desired. ${\displaystyle \blacksquare }$

• If ${\displaystyle f:X\to Y}$ is continuous and ${\displaystyle S_{\bullet }}$ is a net of sets (or points) in ${\displaystyle X}$ that clusters at (respectively, converges to) some given point or subset ${\displaystyle S}$ of ${\displaystyle X,}$ then ${\displaystyle f\left(S_{\bullet }\right):=\left(f\left(S_{i}\right)\right)_{i\in I}}$ clusters at (respectively, converges to) ${\displaystyle f(S)}$ in ${\displaystyle Y.}$

Topologizing the set of prefilters

Starting with nothing more than a set ${\displaystyle X,}$ it is possible to topologize the set

of all filter bases on ${\displaystyle X}$ with the Stone topology, which is named after Marshall Harvey Stone.

To reduce confusion, this article will adhere to the following notational conventions:

• Lower case letters for elements ${\displaystyle x\in X.}$
• Upper case letters for subsets ${\displaystyle S\subseteq X.}$
• Upper case calligraphy letters for subsets ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ (or equivalently, for elements ${\displaystyle {\mathcal {B}}\in \wp (\wp (X)),}$ such as prefilters).
• Upper case double–struck letters for subsets ${\displaystyle \mathbb {P} \subseteq \wp (\wp (X)).}$

For every ${\displaystyle S\subseteq X,}$ let

where ${\displaystyle \mathbb {O} (X)=\mathbb {P} {\text{ and }}\mathbb {O} (\varnothing )=\varnothing .}$^{[note 15]} These sets will be the basic open subsets of the Stone topology. If ${\displaystyle R\subseteq S\subseteq X}$ then

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of ${\displaystyle \mathbb {O} (R\cap S)~\supseteq ~\mathbb {O} (R)\cap \mathbb {O} (S).}$^{[note 16]} For all ${\displaystyle R\subseteq S\subseteq X,}$

where in particular, the equality ${\displaystyle \mathbb {O} (R\cap S)=\mathbb {O} (R)\cap \mathbb {O} (S)}$ shows that the family ${\displaystyle \{\mathbb {O} (S)~:~S\subseteq X\}}$ is a ${\displaystyle \pi }$–system that forms a basis for a topology on ${\displaystyle \mathbb {P} }$ called the Stone topology. It is henceforth assumed that ${\displaystyle \mathbb {P} }$ carries this topology and that any subset of ${\displaystyle \mathbb {P} }$ carries the induced subspace topology.

In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on ${\displaystyle \mathbb {P} }$ was defined without using anything other than the set ${\displaystyle X;}$ there were no preexisting structures or assumptions on ${\displaystyle X}$ so this topology is completely independent of everything other than ${\displaystyle X}$ (and its subsets).

The following criteria can be used for checking for points of closure and neighborhoods. If ${\displaystyle \mathbb {B} \subseteq \mathbb {P} {\text{ and }}{\mathcal {F}}\in \mathbb {P} }$ then:

• Closure in ${\displaystyle \mathbb {P} }$: ${\displaystyle \ {\mathcal {F}}}$ belongs to the closure of ${\displaystyle \mathbb {B} {\text{ in }}\mathbb {P} }$ if and only if ${\displaystyle {\mathcal {F}}\subseteq \bigcup _{{\mathcal {B}}\in \mathbb {B} }{\mathcal {B}}^{\uparrow X}.}$
• Neighborhoods in ${\displaystyle \mathbb {P} }$: ${\displaystyle \ \mathbb {B} }$ is a neighborhood of ${\displaystyle {\mathcal {F}}{\text{ in }}\mathbb {P} }$ if and only if there exists some ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle \mathbb {O} (F)=\left\{{\mathcal {B}}\in \mathbb {P} ~:~F\in {\mathcal {B}}^{\uparrow X}\right\}\subseteq \mathbb {B} }$ (that is, such that for all ${\displaystyle {\mathcal {B}}\in \mathbb {P} ,{\text{ if }}F\in {\mathcal {B}}^{\uparrow X}{\text{ then }}{\mathcal {B}}\in \mathbb {B} }$).

It will be henceforth assumed that ${\displaystyle X\neq \varnothing }$ because otherwise ${\displaystyle \mathbb {P} =\varnothing }$ and the topology is ${\displaystyle \{\varnothing \},}$ which is uninteresting.

Subspace of ultrafilters

The set of ultrafilters on ${\displaystyle X}$ (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If ${\displaystyle X}$ has the discrete topology then the map ${\displaystyle \beta :X\to \operatorname {UltraFilters} (X),}$ defined by sending ${\displaystyle x\in X}$ to the principal ultrafilter at ${\displaystyle x,}$ is a topological embedding whose image is a dense subset of ${\displaystyle \operatorname {UltraFilters} (X)}$ (see the article Stone–Čech compactification for more details).

Relationships between topologies on ${\displaystyle X}$ and the Stone topology on ${\displaystyle \mathbb {P} }$

Every ${\displaystyle \tau \in \operatorname {Top} (X)}$ induces a canonical map ${\displaystyle {\mathcal {N}}_{\tau }:X\to \operatorname {Filters} (X)}$ defined by ${\displaystyle x\mapsto {\mathcal {N}}_{\tau }(x),}$ which sends ${\displaystyle x\in X}$ to the neighborhood filter of ${\displaystyle x{\text{ in }}(X,\tau ).}$ The map ${\displaystyle {\mathcal {N}}_{\tau }:X\to \operatorname {Filters} (X)}$ is injective if and only if ${\displaystyle \tau {\text{ is }}T_{0}}$ (that is, a Kolmogorov space) and moreover, if ${\displaystyle \tau ,\sigma \in \operatorname {Top} (X)}$ then ${\displaystyle \tau =\sigma {\text{ if and only if }}{\mathcal {N}}_{\tau }={\mathcal {N}}_{\sigma }.}$ Thus every ${\displaystyle \tau \in \operatorname {Top} (X)}$ can be identified with the canonical map ${\displaystyle {\mathcal {N}}_{\tau },}$ which allows ${\displaystyle \operatorname {Top} (X)}$ to be canonically identified as a subset of ${\displaystyle \operatorname {Func} (X;\mathbb {P} )}$ (as a side note, it is now possible to place on ${\displaystyle \operatorname {Func} (X;\mathbb {P} ),}$ and thus also on ${\displaystyle \operatorname {Top} (X),}$ the topology of pointwise convergence on ${\displaystyle X}$ so that it now makes sense to talk about things such as topologies converging pointwise on ${\displaystyle X}$). For every ${\displaystyle \tau \in \operatorname {Top} (X),}$ the surjection ${\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \operatorname {image} {\mathcal {N}}_{\tau }}$ is continuous, closed, and open. In particular, for every ${\displaystyle T_{0}}$ topology ${\displaystyle \tau {\text{ on }}X,}$ the map ${\displaystyle {\mathcal {N}}_{\tau }:(X,\tau )\to \mathbb {P} }$ is a topological embedding.

In addition, if ${\displaystyle {\mathfrak {F}}:X\to \operatorname {Filters} (X)}$ is a map such that ${\displaystyle x\in \ker {\mathfrak {F}}(x):=\bigcap _{F\in {\mathfrak {F}}(x)}F{\text{ for every }}x\in X}$ (which is true of ${\displaystyle {\mathfrak {F}}:={\mathcal {N}}_{\tau },}$ for instance), then for every ${\displaystyle x\in X{\text{ and }}F\in {\mathfrak {F}}(x),}$ the set ${\displaystyle {\mathfrak {F}}(F):=\{{\mathfrak {F}}(f):f\in F\}}$ is a neighborhood of ${\displaystyle {\mathfrak {F}}(x){\text{ in }}\operatorname {image} {\mathfrak {F}}}$ (where ${\displaystyle \operatorname {image} {\mathfrak {F}}}$ has the subspace topology inherited from ${\displaystyle \mathbb {P} }$).

See also

• Characterizations of the category of topological spaces
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Filtration (mathematics)
• Filtration (probability theory) – Model of information available at a given point of a random process
• Filtration (abstract algebra)
• Fréchet filter
• Generic filter
• Ideal (set theory) – Non-empty family of sets that is closed under finite unions and subsets
• Stone–Čech compactification#Construction using ultrafilters
• The fundamental theorem of ultraproducts

Notes

Proofs

Citations

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