Let be a topological space, then for every non-empty closed set and a point which does not belongs to , there exist open sets and , such that and .

In other words, a topological space is said to be a regular space if for any and any closed set of , there exist an open sets and such that and .

**Example:**

Show that a regular space need not be a Hausdorff space.

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

**T3-Space:**

A regular space is called a space.

**Theorems:**

• Every subspace of a regular space is a regular space.

• Every space is a Hausdorff space.

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