1. ## Compact Metric Spaces

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2. 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.
Hi EricaMae,

To prove A is disconnected, you will need two proper subsets in A being relatively open and closed other than the empty set. So you have the right idea there.
Now let $P'=P \cap A$ and $Q'=Q \cap A$. See if you can prove them to be relatively open and closed in A.

3. Okay, I have P and Q open sets in A such that A is contained in (P intersect Q), P intersect A is not empty, Q intersect A not empty, and P intersect Q is empty. I know that P' and Q' are relatively open and closed, (because P' is A intersect P with A closed and P open?) same for Q'. which would imply that A is disconnected because more than the whole set and empty set are relatively open and closed, but does it matter that the question says in a compact metric space A is a subset?

4. Originally Posted by EricaMae
Okay, I have P and Q open sets in A such that A is contained in (P intersect Q), P intersect A is not empty, Q intersect A not empty, and P intersect Q is empty. I know that P' and Q' are relatively open and closed, (because P' is A intersect P with A closed and P open?) same for Q'. which would imply that A is disconnected because more than the whole set and empty set are relatively open and closed, but does it matter that the question says in a compact metric space A is a subset?
P' is relatively open since A is relatively open and closed in A when you considered A as a subspace of Y, So a finite intersection of open sets is open. Similarly, Q' is relatively open.
Though you will have to showed that they are also relatively closed. (Hint: P' and Q' are complement of each other in A).

As far as the part of A being compact, I am not so sure. I suspect that it is to make sure the set is not "too big" as you may have problems in some cases. It will have to leave it to someone who is more knowledgeable to answer. Sorry.