T3 space

A regular space is a topological spacein which points and closed sets can be separated by open sets;in other words, given a closed set $A$ and a point $x\notin A$,there are disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subseteq V$.

A ${\mathrm{T}}_{\mathrm{3}}$ space is a regular ${\mathrm{T}}_0$-space (http://planetmath.org/T0Space).A ${\mathrm{T}}_3$ space is necessarily also ${\mathrm{T}}_2$, that is, Hausdorff.

Note that some authors make the opposite distinction between${\mathrm{T}}_3$ spaces and regular spaces,that is, they define ${\mathrm{T}}_3$ spaces to be topological spacesin which points and closed sets can be separated by open sets,and then define regular spaces to be topological spacesthat are both ${\mathrm{T}}_3$ and ${\mathrm{T}}_0$.(With these definitions, ${\mathrm{T}}_3$ does not imply ${\mathrm{T}}_2$.)