separable space


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


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 ( of 20 or fewer separable spaces is 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.

Title separable space
Canonical name SeparableSpace
Date of creation 2013-03-22 12:05:45
Last modified on 2013-03-22 12:05:45
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Definition
Classification msc 54D65
Synonym separable topological space
Related topic SecondCountable
Related topic Lindelof
Related topic EverySecondCountableSpaceIsSeparable
Related topic HewittMarczewskiPondiczeryTheorem
Defines separable