A finite connected set with more than one element Does such a set exist? I've tried some examples and it seems to me that there always exists disjoint open sets $U$ and $V$ such that the original set is in their union. So would this mean any finite set is connected iff it has only 1 element?

As other answers have said, there are spaces for which this is possible, but it cannot happen in $\mathbb R$.
Here's the proof:
Suppose $X \subset \mathbb R$ is a finite set with more than one element. Let $a, b \in X$. Since $X$ is finite, there exists $c \notin X$ such that $a < c < b$. Then $(X \cap (-\infty, c)) \cup (X \cap (c, \infty))$ is a separation of $X$.
