Let be a topological space, then for every non-empty closed set and a point which does not belong 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 open sets and such that and .
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 is , so for any , there does not exist a 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.
A regular space is called a space.
• 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 exists an open set such that ; (3) every point of has a local neighborhood basis consisting of closed sets.