Separable space

Symbol-free definition
A topological space is said to be separable if it has a countable dense subset.

Metaproperties
A finite product of separable spaces is separable. The countable dense subset that we take for the product is the Cartesian product of countable dense subsets for each.

Any open subset of a separable space is separable. Indeed, given a countable dense subset for the whole space, we intersect it with the open subset to get a countable dense subset of the open subset.

It is not true in general that a subspace of a separable space is separable. A counterexample is the antidiagonal in the Sorgenfrey plane.

Textbook references

 * , Page 192 (definition in paragraph)