Metric Space is Weakly Countably Compact iff Countably Compact

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Theorem

Let $M = \struct {A, d}$ be a metric space.

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


Proof

From Metric Space fulfils all Separation Axioms, a metric space is a $T_1$ (Fréchet) space.

The result follows from $T_1$ Space is Weakly Countably Compact iff Countably Compact.

$\blacksquare$


Sources