nLab
separable space

Definitions

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

To be explicit, $X$ is separable if there exists an infinite sequence $a:\mathbb{N}\to X$ such that, given any point $b$ in $X$ and any neighbourhood $U$ of $b$, we have $a_i\in U$ for some $i$.

Properties

A second-countable space is separable and first-countable, but the converse need not (see Steen Seebach Example 51 ).

Many results in analysis are easiest for separable spaces. This is particularly true if one wishes to avoid using strong forms of the axiom of choice or to be predicative over the natural numbers.

Related concepts

• first-countable topological space

• second-countable topological space

• separable space

• separable Hilbert space

• Hausdorff topological space