Separable Space - Separability Versus Second Countability

Separability Versus Second Countability

Any second-countable space is separable: if is a countable base, choosing any from the non-empty gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.

To further compare these two properties:

• An arbitrary subspace of a second countable space is second countable; subspaces of separable spaces need not be separable (see below).
• Any continuous image of a separable space is separable (Willard 1970, Th. 16.4a).; even a quotient of a second countable space need not be second countable.
• A product of at most continuum many separable spaces is separable. A countable product of second countable spaces is second countable, but an uncountable product of second countable spaces need not even be first countable.