# Separable Space

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.