T1 Space is Weakly Countably Compact iff Countably Compact
Theorem
Let $T = \struct {S, \tau}$ 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 $T = \struct {S, \tau}$ 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:
- $U_x = \set {a_1, a_2, \ldots, a_n}$
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.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties