The union of non-disjoint connected is connected

Suppose $\displaystyle S_{1},S_{2}$ are connected sets in $\displaystyle \mathbb{R}^{n}$ and $\displaystyle S_{1}\cap S_{2} \ne \emptyset$. Show that $\displaystyle S_{1}\cup S_{2}$ is connected. Does this hold true for $\displaystyle S_{1} \cap S_{2}$?

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 $\displaystyle U,V$ are open in $\displaystyle A$ connected, and $\displaystyle U \cup V = A$, then $\displaystyle U \cap V \ne \emptyset$. 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.