Analysis Quesion - Completeness?

Question

Let (X,d) be a connected metric space.

Suppose A and B are closed subsets of X such that X=AUB, and AB is connected (AB means A intersection B).

Prove tha A and B are connected.
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I'm not quite sure where to start, I think I have to do something with continuity, but I'm not sure...

Answer 1

If one of the sets A or B is emty, the conclusion is trivial. So, I'll suppose neither is empty.

Since A and B are closed, so is A inter B.

If A is not connected, then there are 2 disjoint non empty subsets A1 and A2 of A, closed with respect to A, such that A1 U A2 = A. Since A is closed with respect to X, so are A1 and A2. Then,

X = A1 U A2 U B

If B does not intersect A1, then A1 and (A2 U B) are nonempty closed and disjoint subsets of X whose union is X. Hence, contrarily to the hypothesizes, X is disconnected, so that B must intersect A1. By a similar reasoning, B intersects A2.

We have that

A inter B = (A1 inter B) U (A2 inter B).

As we have seen, (A1 inter B) and (A2 inter B) are non empty and disjoint, for A1 and A2 are disjoint. Since A1, A2 and B are closed, so are (A1 inter B) and (A2 inter B). This shows the closed A inter B is given by the union of 2 disjoint non empty closed subsets, so that A inter B, contrarily to one of the hypothesizes, is disconnectd. From this contradiction, if follows A is connected.

By a similar reasoning, we conclude B is connected, too.