Locally finite family

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.