Locally finite family

Summary

A family of subsets of a topological space is locally finite if each point has a neighbourhood meeting only finitely many members. The family may be infinite, but near each point it looks finite. Local finiteness lets one treat infinite unions and sums as if they were finite. For example, the closure of the union of a locally finite family is the union of the closures. It is the key notion in the definition of paracompact spaces and of partitions of unity.