Definition

A topological space is said to be separable if it has a countable dense subset.

Properties

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

A continuous image of a separable space is separable.

An open subset of a separable space is separable (in the subspace topology).

A product of $2^{\aleph_0}$ or fewer separable spaces is separable. This is a special case of the Hewitt-Marczewski-Pondiczery Theorem.

A Hilbert space is separable if and only if it has a countable orthonormal basis.