Weakly Countably Compact Metric Space is Countably Compact

Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is weakly countably compact iff $M$ is countably compact.

Proof

We have that a metric space is a $T_1$ space.

Then from Weakly Countably Compact T1 Space is Countably Compact, in a $T_1$ (Fréchet) space, weakly countable compactness is equivalent to countable compactness.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$