separability

Separability is a property of a topological space describing the existence of a countable dense subset. A subset D of a space X is dense if its closure equals X, equivalently if every nonempty open set in X meets D. Thus X is separable if there exists such a D. In metric spaces, separability implies second countability, meaning the space has a countable base for its topology.

Examples and basic facts: The real line R with its usual topology is separable; the rationals Q are a countable dense subset. Euclidean space R^n is also separable, and in fact every separable metric space has a countable base. Subspaces of a separable space are separable as well: if D is dense in X, then D ∩ A is dense in any subspace A ⊆ X.

In analysis and functional analysis, several standard spaces are separable. For 1 ≤ p < ∞, L^p spaces on sigma-finite measure spaces are separable, and l^p (the space of p-summable sequences) is separable. C(K) is separable when K is a compact metric space. A separable Hilbert space has a countable dense subset, and hence a countable orthonormal basis; classic examples include l^2.

Non-separable spaces do occur, for example L^\infty on a nontrivial measure space or the space l^\infty of all bounded sequences.

Applications and significance: Separability allows the use of sequences to approximate and study objects, facilitates the construction of bases and bases-like tools, and underpins many results in analysis, probability, and applied mathematics where countable data suffice to capture the structure of the space.