# Hausdorff space

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, $U\cap V=\varnothing$) such that $x\in U$ and $y\in V$.

A $T_{2}$ space is also known as a Hausdorff space. A Hausdorff topology for a set $X$ is a topology $\tau$ such that $(X,\tau)$ is a Hausdorff space.

## Properties

The following properties are equivalent:

1. 1.

$X$ is a Hausdorff space.

2. 2.

The set

 $\Delta=\{(x,y)\in X\times X:x=y\}$

is closed in the product topology of $X\times X$.

3. 3.

For all $x\in X$, we have

 $\{x\}=\bigcap\{A:A\subseteq X\ \mbox{closed},\mbox{\exists open set}\ U\ % \mbox{such that}\ x\in U\subseteq A\}.$

Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.

 Title Hausdorff space Canonical name HausdorffSpace Date of creation 2013-03-22 12:18:18 Last modified on 2013-03-22 12:18:18 Owner yark (2760) Last modified by yark (2760) Numerical id 23 Author yark (2760) Entry type Definition Classification msc 54D10 Synonym Hausdorff topological space Synonym T2 space Related topic SeparationAxioms Related topic T1Space Related topic T0Space Related topic T3Space Related topic RegularSpace Related topic MetricSpace Related topic NormalTopologicalSpace Related topic ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed Related topic SierpinskiSpace Related topic HausdorffSpaceNotCompletelyHausdorff Related topic Tychonoff Related topic PropertyThatCompactSetsInASpaceAreClosedLies Defines Hausdorff Defines Hausdorff topology Defines T2 Defines T2 topology Defines T2 axiom