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).
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.
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.
• 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.