Difference between revisions of "Separable space"
From Encyclopedia of Mathematics
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| + | A [[topological space]] containing a [[Countable set|countable]] [[everywhere-dense set]]. | ||
====Comments==== | ====Comments==== | ||
| − | Thus, a space $X$ is separable if and only if its density $d(X)\leq\aleph_0$; cf. [[ | + | Thus, a space $X$ is separable if and only if its [[Density (of a topological space)|density]] $d(X)\leq\aleph_0$; cf. [[Cardinal characteristic]]. |
| − | A metrizable space is separable if and only if it satisfies the [[Second | + | A [[metrizable space]] is separable if and only if it satisfies the [[Second axiom of countability]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 43ff (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 43ff (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 19:33, 12 December 2015
2020 Mathematics Subject Classification: Primary: 54D65 [MSN][ZBL]
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