Compact Space satisfies Finite Intersection Axiom

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Theorem

The following definitions of the concept of Compact Space in the context of Topology are equivalent:

Definition by Open Covers

A topological space $T = \left({S, \tau}\right)$ is compact if and only if every open cover for $S$ has a finite subcover.

Definition by Finite Intersection Axiom

A topological space $T = \left({S, \tau}\right)$ is compact if and only if it satisfies the Finite Intersection Axiom.


Proof

Let every open cover of $S$ have a finite subcover.

Let $\mathcal A$ be any set of closed subsets of $S$ satisfying $\displaystyle \bigcap \mathcal A = \varnothing$.

We define the set:

$\mathcal V := \left\{{S \setminus A : A \in \mathcal A}\right\}$

which is clearly an open cover of $S$.


From De Morgan's Laws: Difference with Union:

$\displaystyle S \setminus \bigcup \mathcal V = \bigcap \left\{{S \setminus V : V \in \mathcal V}\right\} = \bigcap \left\{{A : A \in \mathcal A}\right\} = \varnothing$

and therefore:

$S = \displaystyle \bigcup \mathcal V$


By definition, there exists a finite subcover $\tilde{\mathcal V} \subseteq \mathcal V$.

We define:

$\tilde{\mathcal A} := \left\{{S \setminus V : V \in \tilde{\mathcal V}}\right\}$

then $\tilde{\mathcal A} \subseteq \mathcal A$ by definition of $\mathcal V$.


Because $\tilde{\mathcal V}$ covers $S$, it follows directly that:

$\displaystyle \bigcap \tilde{\mathcal A} = \bigcap \left\{{S \setminus V : V \in \tilde{\mathcal V}}\right\} = S \setminus \bigcup \tilde{\mathcal V} = \varnothing$


Thus, in every set $\mathcal A$ of closed subsets of $S$ satisfying $\displaystyle \bigcap \mathcal A = \varnothing$ exists a finite subset $\tilde{\mathcal A}$ such that $\displaystyle \bigcap \tilde{\mathcal A} = \varnothing$.

That is, $S$ satisfies the Finite Intersection Axiom.

$\Box$


The converse works exactly as the previous, but with the roles of the open cover and $\mathcal A$ reversed.

$\blacksquare$


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