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.

First examples

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 vectors-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.

Separability versus second countability

Any second-countable space gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf.

To further compare these two properties:

• An arbitrary subspace of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).

• Any continuous image of a separable space is separable ; even a quotient of a second-countable space need not be second countable.

• A product of at most continuum many separable spaces is separable . A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not even be first countable.

is open. Therefore, the space is separable but there cannot have a countable base.

Cardinality

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 the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases

if it is first countable.

The product of at most continuum many separable spaces is a separable space (Hewitt–Marczewski–Pondiczery theorem).

Constructive mathematics

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.

Further examples

Separable spaces

• Every compact metric space (or metrizable space) is separable.

• -dimensional Euclidean space is separable.

• of all continuous functions from a compact is separable.

• The Lebesgue spaces.

• 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. The Banach–Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace.

• A Hilbert space is separable if and only if it has a countable orthonormal basis 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 that is a separable space when considered as a metric space with metric and a given finite measure being the symmetric difference operator).

Non-separable spaces

• 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.

• The Banach space of functions of bounded variation is not separable.

Properties

• 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.

• In fact, every topological space is a subspace of a separable space of the same cardinality. Only countably many points need to be added: the points of the countable dense subset of the separable space. A construction is given in ; this construction additionally embeds every Hausdorff space into a separable Hausdorff space.

• , the cardinality of the continuum. This follows since such functions are determined by their values on dense subsets.

• From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal.

• For a compact Hausdorff space X, the following are equivalent:

Embedding separable metric spaces

• Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proof of the Urysohn metrization theorem.

• Every separable metric space is isometric to a subset of the (non-separable) Banach space l∞ of all bounded real sequences with the supremum norm; this is known as the Fréchet embedding.

• Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuous functions [0,1]→R, with the supremum norm. This is due to Stefan Banach.

• Every separable metric space is isometric to a subset of the Urysohn universal space.

For nonseparable spaces:

• A metric space of density equal to an infinite cardinal is isometric to a subspace of , the space of real continuous functions on the product of copies of the unit interval.

References