Locally finite family

A family $\{A_i\}_{i\in I}$ of subsets of a topological space $X$ is locally finite if every point $x\in X$ has a neighborhood that meets only finitely many of the sets $A_i$ .

Consequences

If $\{A_i\}$ is locally finite, then its subfamilies are locally finite and the union of the closures satisfies

\overline{\bigcup_i A_i}=\bigcup_i\overline{A_i}.

where the bars denote closures.

Local finiteness lets constructions indexed by $I$ reduce to finite ones near each point. In particular, a partition of unity is locally a finite sum because the family of supports of its functions is locally finite.

Distinction from point finiteness

Point finiteness requires each point to belong to only finitely many $A_i$ . Local finiteness is stronger: a whole neighborhood must meet only finitely many members.

References

James R. Munkres, Topology, 2nd ed., Prentice Hall, 2000. Relevant: locally finite families and partitions of unity.