Editing Separable space (section)

==First examples==
Any topological space that is itself [[finite set|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 length-<math>n</math> [[Vector (mathematics and physics)|vectors]] of rational numbers, <math>\boldsymbol{r}=(r_1,\ldots,r_n) \in \mathbb{Q}^n</math>, is a countable dense subset of the set of all length-<math>n</math> vectors of real numbers, <math>\mathbb{R}^n</math>; so for every <math>n</math>, <math>n</math>-dimensional [[Euclidean space]] is separable.

A simple example of a space that is not separable is a [[discrete space]] of uncountable cardinality.

Further examples are given below.