Separable Space

A topological space \left( {X,\tau } \right), is said to be separable space, if it has a countable dense subset in X. i.e., A \subseteq X, \overline A  = X, or A \cup U \ne \phi , where U is an open set.

In other words, a space X is said to be separable space if there is a subset A of X such that (1) A is countable (2) \overline A  = X (A is dense inX).


Let X = \left\{ {1,2,3,4,5} \right\} be a non-empty set and \tau  = \left\{ {\phi ,X,\left\{ 3 \right\},\left\{ {3,4} \right\},\left\{ {2,3} \right\},\left\{ {2,3,4} \right\}} \right\} is a topology defined on X. Suppose a subset A = \left\{ {1,3,5} \right\} \subseteq X. The closed set are X,\phi ,\left\{ {1,2,4,5} \right\},\left\{ {1,2,5} \right\},\left\{ {1,4,5} \right\},\left\{ {1,5} \right\}. Now we have \overline A  = X. Since A is finite and dense in X. So, X is a separable space.


Consider the set of rational number \mathbb{Q} a subset of \mathbb{R} (with usual topology), then the only closed set containing \mathbb{Q} is \mathbb{R} which shows that \overline {\Bbb Q}  = \mathbb{R}. Since \mathbb{Q} is dense in \mathbb{R}, so \mathbb{R} is also separable in \mathbb{R}. But set of irrational numbers is dense in \mathbb{R} 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.