Separable (idea)

Definition A topological space X is said to be separable if X has a countable dense subset. In other words, there is a countable subset D of X such that closure(D) = X. Equivalently, each nonempty open set in X intersects D.

Recall that a space is second countable if there is a countable basis for the space.

Theorem Let X be a topological space. If X is second countable then X is separable. If X is separable and metrizable, then X is second countable (and hence Lindelof).

A countable product of separable spaces is separable, but a subspace of a separable space is not necessarily separable; a counterexample can be found in the Sorgenfrey plane. However, a metrizable separable space is second countable, and so any subspace is second countable, and therefore separable.

We also have:

Theorem A compact metrizable space is separable.

One use of separability is this: suppose X,Y are topological spaces, f:X->Y is continuous and D is dense in X. Then f is uniquely determined by it's behaviour on D. If D is countable, then this lets us restrict our attention to a countable subset of X, which is generally much easier to handle than the whole space, and allows use of induction.

Some examples of separable spaces in topology and analysis:

• A countable space is separable
• Euclidean space Rⁿ in the usual topology is separable, with countable dense subset Qⁿ
• The space beta(N), the Stone-Cech compactification of the natural numbers, is separable
• The Banach space C([0,1]) of continuous functions on [0,1] is separable, with countable dense subset the set of polynomials on [0,1] with rational coefficients (this is the Weierstrass Approximation Theorem)
• The Hilbert space L²([0,2pi]) of square integrable functions f:[0,2pi]->C with the 2-norm is separable. This is of interest in analysis, since it can be shown that any two separable Hilbert spaces are isometrically isomorphic. See Fourier series

Spaces that aren't separable include:

• The first uncountable ordinal omega₁ in the order topology
• The space X^J in the product (or box) topology, where X is a topological space with at least two points and J is uncountable
• The Banach space l^infinity of bounded complex sequences with the supremum norm