Hausdorff space T2Space 2002-02-08 18:23:40 2008-08-21 06:57:20 Definition Hausdorff Hausdorff topology T2 T2 topology T2 axiom \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\emptyset{\varnothing} A topological space $(X,\tau)$ is said to be $T_2$ (or said to satisfy the $T_2$ axiom) if given distinct $x,y\in X$, there exist disjoint open sets $U,V\in\tau$ (that is, $U\cap V=\emptyset$) such that $x\in U$ and $y\in V$. A $T_2$ space is also known as a \emph{Hausdorff space}. A \emph{Hausdorff topology} for a set $X$ is a topology $\tau$ such that $(X,\tau)$ is a Hausdorff space. \subsubsection*{Properties} The following properties are equivalent: \begin{enumerate} \item $X$ is a Hausdorff space. \item The set $$ \Delta=\{(x,y)\in X\times X:x=y\} $$ is closed in the product topology of $X\times X$. \item For all $x\in X$, we have $$ \{x\} = \bigcap \{A : A\subseteq X\ \mbox{closed}, \mbox{$\exists$ open set}\ U\ \mbox{such that}\ x\in U\subseteq A\}. $$ \end{enumerate} Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.