# Locally finite family

From Encyclopedia of Mathematics

*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).

#### Comments

See also Locally finite covering.

**How to Cite This Entry:**

Locally finite family.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_finite_family&oldid=47695

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