Hausdorff space

A topological spaceMathworldPlanetmath (X,τ) is said to be T2 (or said to satisfy the T2 axiom) if given distinct x,yX, there exist disjoint open sets U,Vτ (that is, UV=) such that xU and yV.

A T2 space is also known as a Hausdorff space. A Hausdorff topology for a set X is a topology τ such that (X,τ) is a Hausdorff space.


The following properties are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    X is a Hausdorff space.

  2. 2.

    The set


    is closed in the product topology of X×X.

  3. 3.

    For all xX, we have

    {x}={A:AXclosed, open setUsuch thatxUA}.

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

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 TychonoffPlanetmathPlanetmath
Related topic PropertyThatCompactSetsInASpaceAreClosedLies
Defines Hausdorff
Defines Hausdorff topology
Defines T2
Defines T2 topology
Defines T2 axiom