Topology, Locally Finite

Locally Finite

A collection of sets is locally finite in the space S if every point x in S is contained in an open set that intersects a finite number of these sets. In other words, sets do not pile up at any one point.

The arbitrary union of closed sets need not be closed, but if those sets are locally finite, the union is indeed closed. Let's prove this by covering the complement with open sets.

Let U be the union of a locally finite collection of closed sets. Let x be any point not in U and let Q be an open set that intersects T₁ T₂T₃ … T_n, a finite subcollection of closed sets. Let W_i be the intersection of Q with the complement of T_i. Now each W_i is open. Their intersection is also open, contained in Q, disjoint from U, and containing x. Thus x is in an open set apart from U. This holds for all x outside of U, so U is closed.