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Topological space with a dense countable subset

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.

Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset. Similarly the set of all length vectors of rational numbers, , is a countable dense subset of the set of all length vectors of real numbers, ; so for every , dimensional Euclidean space is separable. A simple example of a space that is not separable is a discrete space of uncountable cardinality. Further examples are given below.

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi compact, and connected. The "trouble" with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one point space. A first countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality In such a space, closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of
A separable Hausdorff space has cardinality at most , where is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if and , then if and only if there exists a filter base consisting of subsets of that converges to The cardinality of the set of such filter bases is at most Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection when

The same arguments establish a more general result: suppose that a Hausdorff topological space contains a dense subset of cardinality Then has cardinality at most and cardinality at most if it is first countable. The product of at most continuum many separable spaces is a separable space In particular the space of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality More generally, if is any infinite cardinal, then a product of at most spaces with dense subsets of size at most has itself a dense subset of size at most (Hewitt–Marczewski–Pondiczery theorem).

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.

Every compact metric space (or metrizable space) is separable. Any topological space that is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that dimensional Euclidean space is separable. The space of all continuous functions from a compact subset to the real line is separable. The Lebesgue spaces , over a measure space whose σ algebra is countably generated and whose measure is σ finite, are separable for any The space of continuous real valued functions on the unit interval with the metric of uniform convergence is a separable space, since it follows from the Weierstrass approximation theorem that the set of polynomials in one variable with rational coefficients is a countable dense subset of The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of A Hilbert space is separable if and only if it has a countable orthonormal basis. It follows that any separable, infinite dimensional Hilbert space is isometric to the space of square summable sequences. An example of a separable space that is not second countable is the Sorgenfrey line , the set of real numbers equipped with the lower limit topology. A separable σ algebra is a σ algebra that is a separable space when considered as a metric space with metric for and a given finite measure (and with being the symmetric difference operator).

The first uncountable ordinal , equipped with its natural order topology, is not separable. The Banach space of all bounded real sequences, with the supremum norm, is not separable. The same holds for The Banach space of functions of bounded variation is not separable; note however that this space has very important applications in mathematics, physics and engineering.