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 ${\mathrm{T}}_{\mathrm{3}}$ 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 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