A topological space is said to be a 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 a separable space if there is a subset of such that (1) is countable (2) ( is dense in).
Let be a non-empty set and be a topology defined on . Suppose a subset . The closed sets are . Now we have . Since is finite and dense in , then is a separable space.
Consider that the set of rational numbers a subset of (with usual topology), then the only closed set containing is , which shows that . Since is dense in , then is also separable in . However, the set of irrational numbers is dense in but not countable.
• Every second countable space is a separable space.
• Every separable space is not a second countable space.
• Every separable metric space is a second countable space.
• The continuous image of a separable space is separable.