Topological Spaces

Ok, So I need to show that clos( A union B ) = clos(A) union clos(B), where A,B are subsets of a topological space X and clos is the closure of the set.

What I have so far:
Take a element x in clos( A union B )
i.e x belongs to { U contained in X s.t A union B is contained in U}
iff x belongs to { U contained in X s.t A is contained in U} or x belongs to {U contained in X s.t B is contained in U}
iff x belongs to clos(A) or clos(B)
iff x belongs to clos(A) union clos(B)

So we have every element of clos( A union B ) belongs to clos(A) union clos(B) and vice versa. Thus we have equality.

Could someone comment on the validity.

the first two sets in the first two steps should have intersection as defined in closure. sorry.

outside the set that is.

hahaha you still can't solve this

Well I think I solved it, I am just not too sure about it.

Ok one more question.

Let A and B be open and dense subsets of a topological space X. Prove that A intersection B is dense in X.

1: Realise you are too stupid to win.
2: ???
3: Fail.