# Regular Space

Let $\left( {X,\tau } \right)$ be a topological space, then for every non-empty closed set $F$ and a point $x$ which does not belong to $F$, there exist open sets $U$ and $V$, such that $x \in U,{\text{ }}F \subseteq U$ and $U \cap V = \phi$.

In other words, a topological space $X$ is said to be a regular space if for any $x \in X$ and any closed set $A$ of $X$, there exist open sets $U$ and $V$ such that $x \in U,{\text{ A}} \subseteq U$ and $U \cap V = \phi$.

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 is $X$, so for any $x \in X$, there does not exist a closed set $A$ which does not contain $x$. so $X$ is trivially a regular space. Since for any $x,y \in X,\;x \ne y$, there is only one open set $X$ itself containing these points, so $X$ is not a Hausdorff space.

T3-Space

A regular ${T_1}$ space is called a ${T_3}$ space.

Theorems
• Every subspace of a regular space is a regular space.
• Every ${T_3}$ 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 $a \in U$ there exists an open set $V$ such that $a \in V \subseteq \overline V \subseteq U$; (3) every point of $X$ has a local neighborhood basis consisting of closed sets.