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单词 SeparationAxioms
释义

separation axioms


The separation axiomsMathworldPlanetmathPlanetmath are additional conditions which may be required to a topological spaceMathworldPlanetmath in order to ensure that some particular types of sets can beseparated by open sets, thus avoiding certain pathological cases.

AxiomDefinition
T0given two distinct points, there is an open set containing exactly one of them;
T1 (http://planetmath.org/T1Space)given two distinct points, there is a neighborhoodMathworldPlanetmathPlanetmath of each of them which does not contain the other point;
T2 (http://planetmath.org/T2Space)given two distinct points, there are two disjoint open sets each of which contains one of the points;
T212given two distinct points, there are two open sets, each of which contains one of the points, whose closuresMathworldPlanetmathPlanetmath are disjoint;
T3 (http://planetmath.org/T3Space)given a closed setPlanetmathPlanetmath A and a point xA, there are two disjoint open sets U and V such that xU and AV;
T312given a closed set A and a point xA, there is an Urysohn function for A and {b};
T4given two disjoint closed sets A and B, there are two disjoint open sets U and V such that AU and BV;
T5given two separated sets A and B, there are two disjoint open sets U and V such that AU and BV.

If a topological space satisfies a Ti axiom, it is called a Ti-space.The following table shows other common names for topological spaces with these or other additional separation properties.

NameSeparation properties
Kolmogorov spaceT0
Fréchet spaceT1
Hausdorff spaceT2
Completely Hausdorff spaceT212
Regular spaceMathworldPlanetmathPlanetmathT3 and T0
TychonoffPlanetmathPlanetmath or completely regular spaceT312 and T0
Normal spaceT4 and T1
Perfectly T4 spaceT4 and every closed set is a Gδ (see here (http://planetmath.org/G_deltaSet))
Perfectly normal spaceT1 and perfectly T4
Completely normal spaceT5 and T1

The following implicationsMathworldPlanetmath hold strictly:

(T2 and T3)T212
(T3 and T4)T312
T312T3
T5T4
Completely normal normal  completely regular
 regular T212T2T1T0

Remark. Some authors define T3 spaces in the way we defined regular spaces, and T4 spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.

Bibliography: Counterexamples in Topology, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)

Titleseparation axioms
Canonical nameSeparationAxioms
Date of creation2013-03-22 13:28:47
Last modified on2013-03-22 13:28:47
OwnerKoro (127)
Last modified byKoro (127)
Numerical id26
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 54D10
Classificationmsc 54D15
Synonymseparation properties
Related topicNormalTopologicalSpace
Related topicHausdorffSpaceNotCompletelyHausdorff
Related topicSierpinskiSpace
Related topicMetricSpacesAreHausdorff
Related topicZeroDimensional
Related topicT2Space
Related topicRegularSpace
Related topicT4Space
DefinesHausdorff
Definescompletely Hausdorff
Definesnormal
Definescompletely normal
DefinesregularPlanetmathPlanetmath
DefinesTychonoff
Definescompletely regular
Definesperfectly normal
DefinesTychonov
Definesperfectly T4
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更新时间:2025/5/4 9:45:28