Originally Posted by
EricaMae I have this proof I've been working on, but I seem to be stuck. I have to show that in a compact metric space (Y,d) a closed subset A is disconnected if and only if there are open sets P and Q such that A contained in (P union Q), (P intersect A) is not empty and (Q intersect A) is also not empty and (P intersect Q) is empty. I think I'm mostly stuck on the fact that this is a compact metric space.
I started of by saying assume the second part is true. I can see this space is Hausdorff, which means there is a p in P and q in Q such that P intersect Q is empty, but that was given. So I'm not sure if it helps. I know that to show it is disconnected there is a space other than A and other than the nulset that is relatively open and closed at the same time. I can see what this picture looks like geometrically, but I'm going in circles.
Any hint will help.