Let  be a topological space, then for every non-empty closed set  and a point x which does not belongs to , there exist open sets  and , such that  and .
            In other words, a topological space X is said to be a regular space if for any  and any closed set A of X, there exist an open sets  and  such that  and .

Example:
            Show that a regular space need not be a Hausdorff space.
            For this, let X be an indiscrete topological space, then the only non-empty closed set X, so for any , there does not exist and closed set A which does not contain x. so X is trivially a regular space. Since for any , there is only one open set X itself containing these points, so X is not a Hausdorff space.

T3-Space:
            A regular T1-space is called a T3-space.

Theorems:

• Every subspace of a regular space is a regular space.
• Every T3-space is a Hausdorff space.
•  Let X be a topological space, then the following statements are equivalent. (1) X is a regular space. (2) For every open set U in X and a point  there exist an open set V such that . (3) Every point of X has a local neighbourhood basis consisting of closed sets.