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Separable space

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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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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=36898

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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Separable spaceencyclopedia_of_mathematics

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Second axiom of countabilityoutgoing citation Countable setoutgoing citation Metrizable spaceoutgoing citation Cardinal characteristicoutgoing citation Topological spaceoutgoing citation Density (of a topological space)outgoing citation Dense setoutgoing citation Alternative set theoryincoming citation Lévy-Prokhorov metricincoming citation Wijsman convergenceincoming citation Separate and joint continuityincoming citation Weak convergence of probability measuresincoming citation Measurable spaceincoming citation Polish spaceincoming citation Sierpiński gameincoming citation Distributions, convergence ofincoming citation Density (of a topological space)incoming citation

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