Since "being connected" is defined in terms of open sets, I don't think there is a way around talking of open sets and the like in a proof of the above proposition.

To show that the union of connected sets is connected, if , it suffices to show that given any two open sets with , and , it follows that either or .

And why is that? Well, suppose that , then we have either or (but not both, because U and V are disjoint).

If , then, because are both connected, it follows thatand, and therefore .

Similarly, if it follows that .

Finally, to the question as to whether is connected if are connected and . The answer is no, for consider the intersection of a line and a circle in that consists of two isolated points. In such a case, the intersection is clearly not connected.