# 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 $2^{\aleph_{0}}$ 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