I think I kind of get it....

But if I just had 2 sets, like A and B, I would say suppose AUB is disconnected.

So there exist sets X and Y such that

XnAUB != YnAUB != empty

XnAUB U YnAUB = AUB

XnAUB n YnAUB = empty

so some part of A, B is in X, and also some other part of A, B is in Y since XnAUB, YnAUB is not empty and their intersection is empty (so the same part of A, B can't be in both Y and X)

so XnA, YnA is not empty

so XnA U YnA = A??

So A is disconnected because XnA and YnA are disjoint?

And same goes for B

Is this correct?