Separable space

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In mathematics a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence $\{ x_n \}_{n=1}^{\infty}$ 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 is determined by its values on the countable dense subset. If one has an explicit bijection between the countable dense subset and the natural numbers, it makes sense to talk about computable functions on the space.

In general, separability is a technical hypothesis on a space which is quite useful and -- among the classes of spaces studied in geometry and classical analysis -- generally considered to be quite mild. It is important to compare separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.

Here is a sample result showing the way separability naturally arises:

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

(i) X is second countable.

(ii) The space $\mathcal{C}(X,\mathbb{R})$ of continuous real-valued functions on X is separable.

(iii) X is metrizable.

1 First examples
2 Separability versus second countability
3 Cardinality
4 Constructive mathematics
5 Further examples
6 Properties
7 References

First examples

Evidently any topological space which is itself finite or countably infinite is separable. 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 vectors $(r_1,\ldots,r_n)$ in $\mathbb{R}^n$ in which $r i$ is rational for all i is a countable dense subset of $\mathbb{R}^n$ , so for every $n$ the $n$ -dimensional Euclidean space is separable. This can also be deduced from the facts that any compact metrizable space is separable and that any topological space which is the union of a countable number of separable subspaces is separable.

A simple example of a space which is not separable is a discrete space of uncountable cardinality.

Further examples are given below.

Separability versus second countability

Any second-countable space is separable: if ${U n}$ is a countable basis, choosing any $x_n \in U_n$ gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable 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 (Willard 1970, Th. 16.4a).; even a quotient of a second countable space need not be second countable.
A countable product of separable (resp. second countable) spaces is separable (resp. second countable). An uncountable product of second countable spaces need not even be separable. However, there is more to say about certain uncountable products of Hausdorff separable spaces (see below).

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 determined by limits of sequences and any sequence has at most one limit, so we get a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.

A separable Hausdorff space has cardinality at most $2 c$ , where c is the cardinality of the continuum. For this we use the characterization of closure in terms of limits of filter bases: if Y is a subset of X and z is a point of X, then z is in the closure of Y if and only if there exists a filter base B consisting of subsets of Y which converges to z. The cardinality of the set $S (Y)$ of such filter bases is at most $2^{2^{|Y|}}$ . Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection $S(Y) \rightarrow X$ when $\bar Y=X.$

The same arguments establish a more general result: suppose that a Hausdorff topological space X contains a dense subset of cardinality $κ$ . Then X has cardinality at most $2^{2^{\kappa}}$ and cardinality at most $2 κ$ if it is first countable.

The product of at most continuum many separable Hausdorff spaces is a separable space (Willard 1970, Th 16.4c). In particular the space $\mathbb{R}^{\mathbb{R}}$ of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality $2 c$ .

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

It follows easily from the Weierstrass approximation theorem that the set $\mathbb{Q}[t]$ of polynomials with rational coefficients is a countable dense subset of the space of continuous functions on the unit interval [0,1] with the metric of uniform convergence. The Banach-Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of this space.
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 first uncountable ordinal ω₁ in its order topology 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, (Willard 1970, Th 16.4b). 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. A construction adding at most countably many points is given in (Sierpinski 1952, p. 49).
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.

References

Kelley, John L. (1975), General Topology, Berlin, New York: Springer-Verlag, MR 0370454, ISBN 978-0-387-90125-1
Sierpinski, Waclaw (1952), General topology, Mathematical Expositions, No. 7, Toronto, Ont.: University of Toronto Press, MR 0050870
Steen, Lynn Arthur & Seebach, J. Arthur Jr. (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, MR 507446, ISBN 978-0-486-68735-3
Willard, Stephen (1970), General Topology, Addison-Wesley, MR 0264581, ISBN 978-0-201-08707-9