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Difference between revisions of "Separable space"

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A topological space containing a countable everywhere-dense set.
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A [[topological space]] containing a [[Countable set|countable]] [[everywhere-dense set]].
  
 
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Thus, a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084510/s0845101.png" /> is separable if and only if its density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084510/s0845102.png" />; cf. [[Cardinal characteristic|Cardinal characteristic]].
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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]].
  
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A metrizable space is separable if and only if it satisfies the [[Second axiom of countability|second axiom of countability]].
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A [[metrizable space]] is separable if and only if it satisfies the [[Second axiom of countability]].
  
 
====References====
 
====References====
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<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>
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<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=11352
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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