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Difference between revisions of "Lindelöf space"

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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058980/l0589801.png" /> such that every open covering (cf. [[Covering (of a set)|Covering (of a set)]]) of it contains a countable subcovering. For example, a space with a countable base is a Lindelöf space; every [[Quasi-compact space|quasi-compact space]] is a Lindelöf space. Every closed subspace of a Lindelöf space is a Lindelöf space. For every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058980/l0589802.png" /> of a Lindelöf space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058980/l0589803.png" /> into a topological space, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058980/l0589804.png" /> of the latter is a Lindelöf space. Every Hausdorff space that is the union of a countable family of compact (Hausdorff) sets is a Lindelöf space. Every regular Lindelöf space is paracompact (cf. [[Paracompact space|Paracompact space]]). The product of a Lindelöf space and a compact (Hausdorff) space is a Lindelöf space.
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A topological space $X$ such that every open covering (cf. [[Covering (of a set)|Covering (of a set)]]) of it contains a countable subcovering. For example, a space with a countable base is a Lindelöf space; every [[Quasi-compact space|quasi-compact space]] is a Lindelöf space. Every closed subspace of a Lindelöf space is a Lindelöf space. For every continuous mapping $f$ of a Lindelöf space $X$ into a topological space, the subspace $f(X)$ of the latter is a Lindelöf space. Every Hausdorff space that is the union of a countable family of compact (Hausdorff) sets is a Lindelöf space. Every regular Lindelöf space is paracompact (cf. [[Paracompact space|Paracompact space]]). The product of a Lindelöf space and a compact (Hausdorff) space is a Lindelöf space.

Latest revision as of 09:07, 27 April 2014

finally-compact space

A topological space $X$ such that every open covering (cf. Covering (of a set)) of it contains a countable subcovering. For example, a space with a countable base is a Lindelöf space; every quasi-compact space is a Lindelöf space. Every closed subspace of a Lindelöf space is a Lindelöf space. For every continuous mapping $f$ of a Lindelöf space $X$ into a topological space, the subspace $f(X)$ of the latter is a Lindelöf space. Every Hausdorff space that is the union of a countable family of compact (Hausdorff) sets is a Lindelöf space. Every regular Lindelöf space is paracompact (cf. Paracompact space). The product of a Lindelöf space and a compact (Hausdorff) space is a Lindelöf space.

How to Cite This Entry:
Lindelöf space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_space&oldid=31939
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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