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, $U\cap V=\mathrm{\varnothing}$) 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.
$X$ is a Hausdorff space.

2.
The set
$$\mathrm{\Delta}=\{(x,y)\in X\times X:x=y\}$$ is closed in the product topology of $X\times X$.

3.
For all $x\in X$, we have
$$\{x\}=\bigcap \{A:A\subseteq X\text{closed},\exists \text{open set}U\text{such that}x\in U\subseteq A\}.$$
Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces^{}.
Title  Hausdorff space 
Canonical name  HausdorffSpace 
Date of creation  20130322 12:18:18 
Last modified on  20130322 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 