Separable space

In topology and related branches of mathematics, separable spaces are topological spaces with a certain limit on their size.

To be specific, a topological space is separable if and only if it has a subset that is both countable and dense.

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.

Every second countable space is separable. As a partial converse, every separable metric space must be second countable. More generally, every separable uniform space whose uniformity has a countable basis must be second countable.

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. To avoid violating the previous paragraph, it follows that R_llt must not be metrisable -- it can't be made into a metric space. On the other hand, because R_llt is completely regular, it is uniformisable -- it can be made into a uniform space. But again, to avoid violating the previous paragraph, none of its uniformities could possibly have a countable basis.