Editing Separable space (section)

{{Short description|Topological space with a dense countable subset}}{{distinguish|Separated space|Separation axiom}}

In [[mathematics]], a [[topological space]] is called '''separable''' if it contains a [[countable set|countable]], [[dense (topology)|dense]] subset; that is, there exists a [[sequence]] <math>( x_n )_{n=1}^{\infty} </math> 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 space|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.