Separable space

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2020 Mathematics Subject Classification: Primary: 54D65 [MSN][ZBL]

A topological space containing a countable everywhere-dense set.


Thus, a space $X$ is separable if and only if its density $d(X)\leq\aleph_0$; cf. Cardinal characteristic.

A metrizable space is separable if and only if it satisfies the Second axiom of countability.


[1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 43ff (Translated from Russian)
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Separable space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article