T1 Space is Weakly Countably Compact iff Countably Compact

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Theorem

Let $T = \left({S, \tau}\right)$ be a $T_1$ (Fréchet) topological space.


Then $T$ is weakly countably compact if and only if $T$ is countably compact.


Proof

Sufficient Condition

We have that a Countably Compact Space is Weakly Countably Compact whatever the separation properties.

$\Box$


Necessary Condition

We need to show that a $T_1$ space which is weakly countably compact is also necessarily countably compact. ‎

So, let $T = \left({S, \tau}\right)$ be a $T_1$ space which is weakly countably compact.

Let $A \subseteq S$.

Suppose $x$ to be a limit point but not an $\omega$-accumulation point of $A$.

Then there exists some open set $U_x \in \tau$ such that $x \in U_x$ which contains only a finite number of points of $A$, say:

$U_x = \left\{{a_1, a_2, \ldots, a_n}\right\}$

But this, we have specified, is a $T_1$ space.

So this would imply that $x$ has an open neighborhood which contains no points of $A$.

That is, $x$ is not a limit point of $A$ after all.

Hence if $x$ is a limit point of $A$, it is an $\omega$-accumulation point of $A$.

Thus $S$ is countably compact.

$\blacksquare$


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