Totally Bounded Metric Space is Second-Countable/Proof 2

Theorem

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

Then $M$ is second-countable.

Proof

Follows directly from:

Totally Bounded Metric Space is Separable

Separable Metric Space is Second-Countable

$\blacksquare$

Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Totally Bounded Metric Space is Separable.

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.