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
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces