(locally finite cover)
Let be a topological space.
An open cover of is called locally finite if for all points , there exists a neighbourhood such that it intersects only finitely many elements of the cover, hence such that for only a finite number of .
(locally finite refinement induces locally finite cover with original index set)
Let be a topological space, let be an open cover, and let , be a refinement to a locally finite cover.
is still a refinement of to a locally finite cover.
It is clear by construction that , hence that we have a refinement. We need to show local finiteness.
Hence consider . By the assumption that is locally finite, it follows that there exists an open neighbourhood and a finitee subset such that
Hence by construction
Since the image is still a finite set, this shows that is locally finite.
Let be a normal topological space, and let be a locally finite open cover. Then there exists a shrinking to a locally finite open cover whose closures are still contained in the original cover: