A topological space , is said to be separable space, if it has a countable dense subset in . i.e., , , or , where  is an open set.
            In other words, a space  is said to be separable space if there is a subset  of  such that (1)  is countable (2)  ( is dense in).

Example:
            Let  be a non-empty set and  is a topology defined on . Suppose a subset . The closed set are . Now we have . Since  is finite and dense in . So, is a separable space.

Example:
            Consider the set of rational number  a subset of  (with usual topology), then the only closed set containing  is  which shows that . Since  is dense in , so  is also separable in . But set of irrational numbers is dense in  but not countable.

Theorems:

• Every second countable space is a separable space.
• Every separable space is not second countable space.
• Every separable metric space is second countable.
• The continuous image of a separable space is separable.