separable space

Definition

A topological space is said to be separable if it has a countable dense subset.

Properties

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

A continuous image of a separable space is separable.

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

A product (http://planetmath.org/ProductTopology) of $2^{{\mathrm{\aleph }}_0}$ or fewer separable spaces is separable. This is a special case of the Hewitt-Marczewski-Pondiczery Theorem.

A Hilbert space is separable if and only if it has a countable orthonormal basis.

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