Editing Separable space (section)

==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 space|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 [[cardinality of the continuum|continuum cardinality]] <math>\mathfrak{c}</math>. In such a space, [[Closure (topology)|closure]] is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of <math>X</math>.

A separable Hausdorff space has cardinality at most <math>2^\mathfrak{c}</math>, where <math>\mathfrak{c}</math> is the cardinality of the continuum. For this closure is characterized in terms of [[Filters in topology|limits of filter bases]]: if <math>Y\subseteq X</math> and <math>z\in X</math>, then <math>z\in\overline{Y}</math> if and only if there exists a filter base <math>\mathcal{B}</math> consisting of subsets of <math>Y</math> that converges to <math>z</math>. The cardinality of the set <math>S(Y)</math> of such filter bases is at most <math>2^{2^{|Y|}}</math>. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection <math>S(Y) \rightarrow X</math> when <math>\overline{Y}=X.</math>

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

The product of at most continuum many separable spaces is a separable space {{harv | Willard | 1970 | loc=Th 16.4c | p=109 }}. In particular the space <math>\mathbb{R}^{\mathbb{R}}</math> of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality <math>2^\mathfrak{c}</math>. More generally, if <math>\kappa</math> is any infinite cardinal, then a product of at most <math>2^\kappa</math> spaces with dense subsets of size at most <math>\kappa</math> has itself a dense subset of size at most <math>\kappa</math> ([[Hewitt–Marczewski–Pondiczery theorem]]).