1. The problem statement, all variables and given/known data Let X be a topological space, and A a locally finite family of sets in X (i.e. such a family of sets that every point in X has a neighborhood which intersects a finite number of sets in A). One needs to show that Cl(U A) = U (Cl(A)) (i.e. the closure of the union of sets in A equals the union of the closures of sets in A). 3. The attempt at a solution Inclusion [tex]\subseteq[/tex]. Let x be in Cl(U A). Then every neighborhood of x intersects U A. Since A is locally finite, there exists some neighborhood N of x which intersects A in a finite number of sets. This is where I'm stuck, right at the beginning. Somehow, I need to show that this very x is contained in some set of the family A, since then it's contained in U (Cl(A)), too. Any push in the right direction is highly appreciated.