Countably Compact Space satisfies Countable Finite Intersection Axiom

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Theorem

The following definitions of the concept of Countably Compact Space are equivalent:

Definition by Finite Subcover

A topological space $T = \struct {S, \tau}$ is countably compact if and only if:

every countable open cover of $T$ has a finite subcover.

Definition by Countable Finite Intersection Axiom

A topological space $T = \struct {S, \tau}$ is countably compact if and only if:

every countable set of closed sets of $T$ whose intersection is empty has a finite subset whose intersection is empty.

That is, $T$ satisfies the countable finite intersection axiom.


Proof

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

Let $\AA$ be any set of closed sets of $T$ satisfying $\bigcap \AA = \O$.

We define the set:

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

which is clearly an open cover of $S$.


From De Morgan's Laws: Difference with Union:

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

and therefore $S = \bigcup \VV$.


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:

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


Thus, in every countable set $\AA$ of closed sets of $T$ satisfying $\displaystyle \bigcap \AA = \O$ exists a finite subset $\tilde \AA$ such that $\displaystyle \bigcap \tilde \AA = \O$.

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

$\Box$


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

$\blacksquare$


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