Prove closure of (A-union-B) = (closure A)-union-(closure B)

[felvt]

I am supposed to prove that the closure of A Union B = the closure of A Union the closure of B.

I think I supposed to start with the fact that the closure of the union contains all of its limit points, and somehow relate that to how if the closure of the union contains all of its limit points then it must contain the limit points of A and all of the limit points of B, hence the union of the closures. I am not sure how to state that as a proper proof though.

Also, does this result extend to infinite unions of sets?

 

[daon]

Re: Closure of sets

If p belongs to A or B, it is clearly true. Otherwise,

Let p be in the LHS. Let X be an open set containing p. X must intersect AUB. Thus X must intersect A or X must intersect B (intersections distribute over finite unions). Since X is an arbitrary open set in the topology, p is either a limit point of A or a limit point of B, so it is in the RHS.

Going the other way, assume p belongs to the closure of A. Any open set Y which contains p intersects A. Thus Y also intersects AUB. So p is a limit point of AUB. Therefore it is in the LHS. Same for B.

Now for the infinite unions question think of sets like Xn = [0, (n-1)/n]. Hint: these are closed in the standard topology.