The union of non-disjoint connected is connected

Suppose are connected sets in and . Show that is connected. Does this hold true for ?

Okay, so I have seen proofs that rely on topological properties that we have not really discussed or proven in my class (like for instance, if are open in connected, and , then . Assuming that, I understand the proof for this theorem pretty well, but how do I prove that assumption? Or is there another way to solve this question without invoking open sets/topology in general? Thanks.