regular space

Definition 1.

A topological space is a regular space ifit is both a $T_{\mathrm{0}}$ space (http://planetmath.org/T0Space) and a$T_{\mathrm{3}}$ space (http://planetmath.org/T3Space).

Example.Consider the set $ℝ$ with the topology $\sigma$ generated by the basis

$$\beta =\{U=V-C:V\text{ is open with the standard topology and }C\text{ is (infinite) numerable}\}.$$

Since $\mathbb{Q}$ is numerable and $ℝ$ open, the set of irrational numbers $ℝ-\mathbb{Q}$ is open and therefore $\mathbb{Q}$ is closed.It can be shown that $ℝ-\mathbb{Q}$ is an open set with this topology and $\mathbb{Q}$ is closed.

Take any irrational number $x$. Any open set $V$ containing all $\mathbb{Q}$ must contain also $x$, so the regular space property cannot be satisfied. Therefore, $(ℝ,\sigma )$ is not a regular space.

Note

In topology, the terminology for separation axioms is notstandard. Therefore there are also other meanings of regular.In some references (e.g. [2])the meanings of regular and $T_3$ is exchanged. That is,$T_3$ is a stronger property than regular.