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 that and , 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.