Locally finite family

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of sets in a topological space

A family $ F $ of sets such that every point of the space has a neighbourhood that intersects only finitely many elements of $ F $. Locally finite families of open sets and locally finite open coverings are important. Thus, a regular space is metrizable if and only if has a base that splits into countably many locally finite families. Any open covering of a metric space can be refined to a locally finite open covering. Spaces that have this property are called paracompact (cf. Paracompact space).


See also Locally finite covering.

How to Cite This Entry:
Locally finite family. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article