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单词 HausdorffSpace
释义

Hausdorff space


A topological spaceMathworldPlanetmath (X,τ) is said to be T2(or said to satisfy the T2 axiom) if givendistinct x,yX, there exist disjointopen 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 topologyτ such that (X,τ) is a Hausdorff space.

Properties

The following properties are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    X is a Hausdorff space.

  2. 2.

    The set

    Δ={(x,y)X×X:x=y}

    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.

TitleHausdorff space
Canonical nameHausdorffSpace
Date of creation2013-03-22 12:18:18
Last modified on2013-03-22 12:18:18
Owneryark (2760)
Last modified byyark (2760)
Numerical id23
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54D10
SynonymHausdorff topological space
SynonymT2 space
Related topicSeparationAxioms
Related topicT1Space
Related topicT0Space
Related topicT3Space
Related topicRegularSpace
Related topicMetricSpace
Related topicNormalTopologicalSpace
Related topicASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed
Related topicSierpinskiSpace
Related topicHausdorffSpaceNotCompletelyHausdorff
Related topicTychonoffPlanetmathPlanetmath
Related topicPropertyThatCompactSetsInASpaceAreClosedLies
DefinesHausdorff
DefinesHausdorff topology
DefinesT2
DefinesT2 topology
DefinesT2 axiom
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更新时间:2025/5/4 3:57:43