Let A, B be two connected subsets of a topological space X such that A intersects the closure of B . Prove that A ∪ B is connected. I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to treat this problem? Also, for the specific case, the union of the open ball B((−1, 0), 1) and closed ball B((1, 0), 1) should be connected, right? I can see it, but I'm not sure how to word the proof. Any help is greatly appreciated!