Difference between revisions of "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|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|metric space]] can be refined to a locally finite open covering. Spaces that have this property are called paracompact (cf. [[Paracompact space|Paracompact space]]). | ||
====Comments==== | ====Comments==== | ||
See also [[Locally finite covering|Locally finite covering]]. | See also [[Locally finite covering|Locally finite covering]]. | ||
Latest revision as of 22:17, 5 June 2020
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
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