Metric Space is Separable iff Second-Countable

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Theorem

A metric space is separable if and only if it is second-countable.


Proof

Follows directly from:

Separable Metric Space is Second-Countable
Second-Countable Space is Separable

$\blacksquare$


Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Second-Countable 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.


Sources