Countably Compact Metric Space is Compact

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Theorem

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

Let $M$ be countably compact.


Then $M$ is compact.


Proof 1

This follows directly from:

Metric Space is First-Countable
Countably Compact First-Countable Space is Sequentially Compact
Sequentially Compact Metric Space is Second-Countable
Second-Countable Space is Compact iff Countably Compact

$\blacksquare$


Proof 2

We have that a Metric Space is Countably Compact iff Sequentially Compact.

Then we have that a Sequentially Compact Metric Space is Separable.

For each $n$, a metric space which is countably compact can be covered by finitely many open $\left({1/n}\right)$-balls: $B_{1/n} \left({x_i}\right)$.


So $\left\{{x_i}\right\}$ is a dense subset of $A$ which is countable.

So if a metric space is countably compact it is by definition second-countable.


The result follows from Second-Countable Space is Compact iff Countably Compact.

$\blacksquare$


Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice.

Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.


Also see