# T3 space

A regular space is a topological space in 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 $\mathrm{T}_{3}$ 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 spaces in which points and closed sets can be separated by open sets, and then define regular spaces to be topological spaces that are both $\mathrm{T}_{3}$ and $\mathrm{T}_{0}$. (With these definitions, $\mathrm{T}_{3}$ does not imply $\mathrm{T}_{2}$.)

 Title T3 space Canonical name T3Space Date of creation 2013-03-22 12:18:24 Last modified on 2013-03-22 12:18:24 Owner yark (2760) Last modified by yark (2760) Numerical id 14 Author yark (2760) Entry type Definition Classification msc 54D10 Related topic Tychonoff Related topic T2Space Related topic T1Space Related topic T0Space Defines T3 Defines regular Defines regular space