Locally Finite Covers

Definition¹

English

An open cover $\mathcal{O}$ of a set $S \subset \mathbb{R}^n$ is said to be locally finite if any compact set in $\mathbb{R}^n$ can intersect at most finitely many members of $\mathcal{O}$.

Explanation

Even an infinite cover can be locally finite. By definition, a locally finite cover is at most countable, and a finite set is naturally locally finite. Furthermore, if $S$ is closed, then any open cover of a uniformly bounded $S$ has a locally finite subcover.