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单词 SeparableSpace
释义

separable space


Definition

A topological spaceMathworldPlanetmath is said to be separablePlanetmathPlanetmathif it has a countableMathworldPlanetmath dense subset.

Properties

All second-countable spaces are separable.A metric space is separable if and only if it is second-countable.

A continuousPlanetmathPlanetmath image of a separable space is separable.

An open subset of a separable space is separable (in the subspace topology).

A productPlanetmathPlanetmathPlanetmath (http://planetmath.org/ProductTopology) of 20 or fewer separable spacesis separable. This is a special case of the Hewitt-Marczewski-Pondiczery Theorem.

A Hilbert spaceMathworldPlanetmath is separable if and only if it has a countable orthonormal basisMathworldPlanetmath.

Titleseparable space
Canonical nameSeparableSpace
Date of creation2013-03-22 12:05:45
Last modified on2013-03-22 12:05:45
Owneryark (2760)
Last modified byyark (2760)
Numerical id13
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54D65
Synonymseparable topological space
Related topicSecondCountable
Related topicLindelof
Related topicEverySecondCountableSpaceIsSeparable
Related topicHewittMarczewskiPondiczeryTheorem
Definesseparable
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