Compact Space satisfies Finite Intersection Axiom

Theorem

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

Definition by Open Covers

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

Definition by Finite Intersection Axiom

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

Proof

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

Let $\AA$ be any set of closed subsets of $S$ satisfying $\ds \bigcap \AA = \O$.

We define the set:

$\VV := \set {S \setminus A : A \in \AA}$

which is trivially an open cover of $S$.

$\ds S \setminus \bigcup \VV = \bigcap \set {S \setminus V : V \in \VV} = \bigcap \set {A : A \in \AA} = \O$

and therefore:

$S = \ds \bigcup \VV$

By definition, there exists a finite subcover $\tilde \VV \subseteq \VV$.

We define:

$\tilde \AA := \set {S \setminus V : V \in \tilde \VV}$

then $\tilde \AA \subseteq \AA$ by definition of $\VV$.

Because $\tilde \VV$ covers $S$, it follows directly that:

$\ds \bigcap \tilde \AA = \bigcap \set {S \setminus V : V \in \tilde \VV} = S \setminus \bigcup \tilde \VV = \O$

Thus, in every set $\AA$ of closed subsets of $S$ satisfying $\ds \bigcap \AA = \O$, there exists a finite subset $\tilde \AA$ such that $\ds \bigcap \tilde \AA = \O$.

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 $\AA$ reversed.

$\blacksquare$