# Metric Space is Separable iff Second-Countable

## Theorem

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

## Proof

Follows directly from:

$\blacksquare$

## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$