Separable space - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A topological space $X$ is separable if it contains a countable dense subset.

Equivalent formulation

This means that some countable subset meets every nonempty open subset of $X$ .

Metric spaces

For a metric space, separability says that the whole space can be approximated arbitrarily closely by countably many points. A Banach or Hilbert space is called separable when it is separable in the topology induced by its norm.