# Locally finite family problem

1. Aug 17, 2010

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 $$\subseteq$$. 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.

2. Aug 19, 2010

### Eynstone

Let V be a neighbourhood of x intersecting A1,A2,...An. Assume the contrary, that x doesn't belong to the closure of any Ai.Then, we can choose a neighbourhood Vi of x which doesn't intersect Ai for each i.
The intersection W of V & Vi (1<=i<=n) is open & doesn't intersect any of A's & hence is not in U A. Therefore, x is not in Cl(U A), a contradiction.

3. Aug 19, 2010