Discussion
Let A and B be connected sets. Then :
Answer: B
The true statement is A ∪ B is connected if A ∩ B ≠ φ.
If A and B are connected and share a point p, suppose \(A \cup B\) is split into two disjoint open pieces. Each of A and B, being connected, lies entirely in one piece, and both contain p, so both lie in the same piece and the other piece is empty. Hence the union is connected.
The others fail. In the plane, an annulus and a horizontal strip are connected, but their intersection is two separate pieces. A \ B can be disconnected: take A = [0, 2] and B = {1}, with B inside A, which leaves [0, 1) and (1, 2]. Taking B not inside A does not help either: in the plane, a disc minus a line segment that crosses it and runs out beyond it falls into two pieces.
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