Difference between revisions of "Separable space"
From Encyclopedia of Mathematics
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A topological space containing a countable everywhere-dense set. | A topological space containing a countable everywhere-dense set. | ||
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| − | Thus, a space | + | Thus, a space $X$ is separable if and only if its density $d(X)\leq\aleph_0$; cf. [[Cardinal characteristic|Cardinal characteristic]]. |
A metrizable space is separable if and only if it satisfies the [[Second axiom of countability|second axiom of countability]]. | A metrizable space is separable if and only if it satisfies the [[Second axiom of countability|second axiom of countability]]. | ||
Revision as of 19:07, 12 April 2014
A topological space containing a countable everywhere-dense set.
Comments
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.
References
| [1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 43ff (Translated from Russian) |
How to Cite This Entry:
Separable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_space&oldid=31649
Separable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_space&oldid=31649
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article