적절한 지도

Let ${\displaystyle f\colon X\to Y}$ be a closed map, such that ${\displaystyle f^{-1}(y)}$ is compact (in X) for all ${\displaystyle y\in Y.}$ Let ${\displaystyle K}$ be a compact subset of ${\displaystyle Y.}$ It remains to show that ${\displaystyle f^{-1}($$K)}$은(는) 콤팩트하다.

Let {${\displaystyle U{}_{}}$ ${\displaystyle a_{}}$ :${\displaystyle {}_{}\in }$ ${\displaystyle A}$ ${\displaystyle \}}$ ${\$$U_{a}~\colon ~a\in A\right\}}$은(는) ${\displaystyle {\mathrm{f}}^{}}$- 1${\displaystyle {}^{}}$( K ${\displaystyle )}$의 오픈 커버${\displaystyle .}$ ${\displaystyle$ f$^{-1}(K$)이다$.$${\displaystyle \}}$ ${\displaystyle k\in K}$에 대한 이 $역시$ ${\displaystyle {\mathrm{f}}^{}}$- 1${\displaystyle {}^{}}$( ${\displaystyle k}$)${\displaystyle }$. ${\displaystyle f^{-1}(k$)의 개방형 커버임$.$$}}$ 후자는 콤팩트하다고 가정하기 때문에$$ 유한 서브커버를 가지고 있다.In other words, for every ${\displaystyle k\in K,}$ there exists a finite subset ${\displaystyle \gamma _{k}\subseteq A}$ such that ${\displaystyle f^{-1}(k)\subseteq \cup _{a\in \gamma _{k}}$$U_{a}}$ 세트$$ $X{\displaystyle }$ ${\displaystyle \setminus }$ ${\displaystyle {}_{}}$ ${\displaystyle {\cup }_{}}$ ${\displaystyle {{}_{}}_{}}$ ${\displaystyle {k_{}}_{}}$ k u u ${\displaystyle {{}_{}}_{}{}_{}}$ u a ${\displaystyle a_{}}$ { { { { ${$_{k}}}}}}}$U\_{\displaystyle }$${a}$는 $X{\displaystyle }$ ${\displaystyle$ X$}$에서 $닫히고$, f ${\displaystyle f}$ $아래$의 이미지는 $Y{\displaystyle }$ ${\$$displaystyle Y}$에서 닫힌다. f$$ {\displaystyle $f}$는 닫힌 맵이기$$ 때문이다$$.그러므로 집합.

${\displaystyle V_{k}=Y\setminus f\왼쪽(X\setminus \cup _{a\in \gamma_{k})U_{a}\오른쪽)}$

is open in ${\displaystyle Y.}$ It follows that ${\displaystyle V_{k}}$ contains the point ${\displaystyle k.}$ Now ${\displaystyle K\subseteq \cup _{k\in K}V_{k}}$ and because ${\displaystyle K}$ is assumed to be compact, there are finitely many points ${\displaystyle k_1,\dots ,k_s}$${\displaystyle k_{1},\dots,k_{s}.$ $K{\displaystyle }$ ${\displaystyle \subseteq }$ ${\displaystyle {}_{}^{}}$ ${\displaystyle i_{}^{}}$= 1 ${\displaystyle {\mathrm{s}}_{}^{}}$ ${\displaystyle {}_{}^{}{}_{}}$ k ${\displaystyle {i_{}}_{}}$ ${\displaystyle K\subseteq \cup _{i=1}^{s$}s$$.$}V_{k_{i}}}.$ 게다가$$ $set{\displaystyle \mathrm{}}$= ${\displaystyle {}_{}^{}}$ i${\displaystyle {}_{}^{}}$= ${\displaystyle {}_{}^{}}$ ${\displaystyle {{\gamma }_{}}_{}}$ i ${\$1}^{$s}\gamma$$_{k_$${i}}}}}}$ 집합은$$ 유한 집합의 유한 $결합체$다$.$

이제 $f{\displaystyle {}^{}}$- ${\displaystyle 1^{}}$($$) ${\displaystyle \subseteq }$ ${\displaystyle f_{}^{}{}^{}}$- ${\displaystyle 1^{}}$ (${\displaystyle {}^{}\cup {}_{}^{}}$ i${\displaystyle {}_{}^{}}$= 1 ${\displaystyle {\mathrm{s}}_{}^{}}$ ${\displaystyle {}_{}^{}{}_{}}$ ${\displaystyle {k_{}}_{}}$ i${\displaystyle {{}_{}}_{}}$) ${\displaystyle \subseteq }$ ${\displaystyle {}_{}}$ ${\displaystyle {\cup }_{}}$ ∈ ∈ ${\displaystyle {\mathrm{a}}_{}}$ ${\displaystyle {}_{}{}_{}}$ a a a a a a ${\displaystyle a_{}}$ a a a a { { { { { { { { ${$ { { {{-1}($K)\subsetecq$ $f^{-$1}\{-1}\{-$1}\cup$_ $cup$ _ _ _ _ $_$_ _ _ _ _ _ _ _ _ _ _$}V_{k_{i}}\$$오른쪽$$)\subseteq \cup_{$a}컵$_{a\$$in$ $Gamma$$}$U_${\displaystyle {}^{}}$a${\displaystyle \}\}\}\}\}\}}$의 유한 서브커버를 발견해 ${\displaystyle {\mathrm{증거}}^{}}$를 완성했다$.$