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单词 DenseSet
释义

dense set


A subset D of a topological spaceMathworldPlanetmath Xis said to be dense (or everywhere dense) in Xif the closureMathworldPlanetmathPlanetmath of D is equal to X.Equivalently, D is dense if and only ifD intersects every nonempty open set.

In the special case that X is a metric space with metric d,then this can be rephrased as:for all ε>0 and all xXthere is yD such that d(x,y)<ε.

For example, both the rationals and the irrationals are dense in the reals .

The least cardinality of a dense set of a topological spaceis called the density of the space.It is conventional to take the density to be 0if it would otherwise be finite;with this convention,the spaces of density 0 are precisely the separable spacesMathworldPlanetmath.The density of a topological space X is denoted d(X).If X is a Hausdorff space,it can be shown that |X|22d(X).

Titledense set
Canonical nameDenseSet
Date of creation2013-03-22 12:05:42
Last modified on2013-03-22 12:05:42
Owneryark (2760)
Last modified byyark (2760)
Numerical id12
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54A99
Synonymdense subset
Synonymeverywhere dense set
Synonymeverywhere dense subset
Synonymeverywhere-dense set
Synonymeverywhere-dense subset
Related topicNowhereDense
Related topicDenseInAPoset
Definesdense
Defineseverywhere dense
Defineseverywhere-dense
Definesdensity
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