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	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