Separable space

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In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. This condition is typical of spaces that are met in classical parts of mathematical analysis and geometry. In the same way that any real number can be approximated to any specified accuracy by rational numbers, a separable space has some countable subset with which all its elements can be approached, in the sense of a mathematical limit.

Separable spaces are topological spaces with a certain limitation on their size. The separability property is often listed as one of the axioms of countability. From an axiomatic point of view separability was rather frowned upon in the period 1940 to 1960 — where previously it had been basic to descriptive set theory. Subsequently the pendulum swung back, and textbooks would more often choose to admit separability, proving less general theorems (this attitude was adopted, for example, by Jean Dieudonné). For example taking Hilbert space to mean a complex Hilbert space of infinite dimension and separable, there is one such space up to isomorphism (there is a categorical theory, at least if our theory of the real numbers is categorical). This is a useful convention for discussion, at least. The possible use of non-separable Hilbert spaces in theoretical physics has provoked some inconclusive debate.

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the Hahn-Banach theorem.

[edit] Examples

A discrete space is separable if and only if it is countable. Given any uncountable discrete space D, one may form a new space X by adding a point p and using the particular point topology; the open sets in X are U∪{p}, with U open in D. Then X is separable, but has a subspace which is not separable.

The real numbers are separable; they have the rational numbers as a countable dense subset. More generally, Euclidean space Rⁿ is separable, as the set of all points with rational coordinates is dense.

The space of continuous functions on the unit interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (this is the Weierstrass approximation theorem). It can be shown that any separable Banach space is isometrically isomorphic to a closed linear subspace of this space, by the Banach-Mazur theorem.

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

An example of a separable space that is not second-countable is R_llt, the set of real numbers equipped with the lower limit topology.

The product topology on the set of all functions (not necessarily continuous) from the real line to itself is a separable Hausdorff space. This space has cardinality 2^c, showing that separable spaces can still be rather "large". However, for separable Hausdorff spaces this is the largest possible cardinality. Note that this space is not first-countable.

The trivial topology on any set is separable since any singleton is dense. This shows that by removing the Hausdorff requirement in the previous example we can get separable spaces with arbitrarily large cardinality.

The uncountable ordinal ω₁ in its order topology is not separable.

[edit] Properties

The continuous image of a separable space is separable. It follows that separability is a topological property preserved by homeomorphisms. It also follows that every quotient of a separable space is separable.

A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable. Also every subspace of a separable metric space is separable.

Every countable product of separable spaces is separable. Arbitrary products of separable spaces need not be separable.

A separable, Hausdorff space X has cardinality less than or equal to 2^c (where c is the cardinality of the real numbers). If X is also first-countable then its cardinality is less than or equal to c.

The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c. This follows since such functions are determined by their values on dense subsets.