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 whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.

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.


First examples

Any topological space which 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 vectors (r_1,\ldots,r_n) \in \mathbb{R}^n in which ri 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.

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

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