Compactness

Subjects: Topology
Links: Topological Spaces, Topological Covers

Def: A topological space $X$ is compact if every open cover has a finite subcover.

Obs: A topological space $X$ is compact if every open cover has finite refinement.

Def: We say that a family $\mathcal{F}\subseteq \mathcal{P}(X)$ has the finite intersection property if $\mathcal{F}\ne \varnothing$ and for every $\mathcal{G}\in [\mathcal{F}{]}^{<\omega }$ satisfies $\bigcap \mathcal{G}\ne \varnothing$.

Th: Let $X$ be a topological space. $X$ is compact iff every family $\mathcal{F}$ of closed subsets of $X$ that has the finite intersection property has nonempty intersection.

Prop: Every closed subspace of a compact space is compact.

Prop: If a subspace $A$ of a topological space $X$ is compact, then every family $\mathcal{U}\subseteq {\tau }_X$ such that $A\subseteq \bigcup \mathcal{U}$, then there exists $\mathcal{V}\in [\mathcal{U}{]}^{<\omega }$ such that $A\subseteq \bigcup \mathcal{V}$.

Cor: Let $X$ be a topological space, $n\in \omega \setminus 1$ and $\{F_m\mid n<\omega \}$ be a family of closed subsets of $X$. The subspace $F=\bigcup _{m<n}{F}_m$ of $X$ is compact iff $F_m$ is compact for every $m<n$.

Cor: Let $U$ be an open set of a topological space $X$. If a family $\mathcal{F}$ of closed subsets of $X$ contains at least one compact set (in particular, if $X$ is compact) and $\bigcap \mathcal{F}\subseteq U$, then there's $\mathcal{G}\in [\mathcal{F}{]}^{<\omega }$ such that $\bigcap \mathcal{G}\subseteq U$.

Th: If $A$ is a compact subspace of a regular space $X$, then for every closed set $B$ disjoint from $A$ there exist $U,V\in {\tau }_X$ such that $A\subseteq U$, $B\subseteq V$ and $U\cap V=\varnothing$. If, moreover, $B$ is a compact subspace of $X$, then we only need that $X$ is a $T_2$ space.

Th: Let $X$ be a completely regular space. If $A$ is a compact subspace of $X$, and $B$ is a closed subset of $X$ with $A\cap B=\varnothing$, then there's a continuous function $f:X\to I$ such that $f(x)=0$ for all $x\in A$ and $f(x)=1$ for all $x\in B$.

Prop: Every compact subspace of a $T_2$ space is closed.

Cor: Every $T_2$ compact space is normal.

Cor: Every $T_2$ compact space is collectionwise normal.

Prop: If $f:X\to Y$ is a continuous and surjective function, and $X$ is a compact space, then $Y$ is compact, meaning that compactness is preserved by continuous surjective functions.

Prop: If $f:X\to Y$ is a continuous function, and $X$ is compact, then $f[X]$ is compact.

Cor: If $f:X\to Y$ is a continuous function, $X$ and $Y$ are Hausdorff spaces, and $X$ is compact, then any $A\in \mathcal{P}(X)$ satisfies ${\text{cl}}_Y(f[A])=f[{\text{cl}}_X(A)]$.

Cor: Every continuous function from a $T_2$ compact space to a $T_2$ space is closed.

Cor: Every continuous bijective function from a $T_2$ compact space to a $T_2$ space is a homeomorphism.

Cor: Let ${\tau }_1$ and ${\tau }_2$ be topologies defined on a set $X$, and let ${\tau }_1$ be finer than ${\tau }_2$, both of them $T_2$ topologies. If the space $(X,{\tau }_1)$ is a compact space, then ${\tau }_1={\tau }_2$. In other words, among all Hausdorff topologies, compact topologies are minimal.

Lemma: If $A$ is a $T_2$ compact subspace of a space $X$ and $y\in Y$, then for every open set $W\subseteq X\times Y$ containing $A\times \{y\}$ there exist open sets $U\in {\tau }_X$ and $V\in {\tau }_Y$ such that $A\times \{y\}\subseteq U\times V\subseteq W$.

Kuratowski's Theorem: The following are equivalent for a $T_2$ space $X$.

• $X$ is a compact space.
• For every topological space $Y$, the projection ${\pi }_Y:X\times Y\to Y$ is closed.
• For every $T_4$ space $Y$, the projection ${\pi }_Y:X\times Y\to Y$ is closed.

Cor: Let $Y$ be a $T_2$ compact space. The function $f:X\to Y$ is continuous iff the set $f=\{(x,f(x))\mid x\in X\}$ is closed in $X\times Y$.

Prop: If $X$ is a $T_2$ compact space, then $nw(X)=w(X)$.

Cor: If $X$ is a $T_2$ compact space and a has a cover $\{A_{\alpha }\mid \alpha <\kappa \}$ such that $nw(A_{\alpha })\le \lambda \ge \omega$ for $\alpha <\kappa$ and $\kappa \le \lambda$, then $nw(X)\le \lambda$.

Th: For every $T_2$ compact space $X$ we have $w(X)\le |X|$

Th: Let $X$ and $Y$ be $T_2$ spaces. If there's a continuous surjective function $f:X\to Y$, and $Y$ is compact, then $w(Y)\le nw(X)$.

Th: A topological space $X$ is compact iff every net in $X$ has a cluster point.

Th: A topological space $X$ is compact iff every filter in $X$ has a cluster point.

Obs: Every finite space is compact.

Th: Every infinite $T_2$ compact space $X$ satisfies $|X|\le \exp (\chi (X))$

Cor: very infinite first countable $T_2$ compact space $X$ satisfies $|X|\le \mathfrak{c}$.

Tube Lemma: Let $X$ be any space and $Y$ be a compact space. If $x\in X$ and $U\subseteq X\times Y$ is an open subset containing $\{x\}\times Y$, then there is a neighbourhood $V$ of $X$ such that $V\times Y\subseteq U$.

Prop: Let $X$ be a metrizable space, if $A\subseteq X$ is compact, then $A$ is totally bounded.