# Difference between revisions of "Lindelöf space"

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− | A topological space | + | 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=22755