The definition of connected sets is:
A topological space $X$ is connected iff there do not exist sets $U, V \subset X$ such that: $U, V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$, with both $U$ and $V$ both open and closed.
I am having trouble applying this definition to certain cases--for example, the union of two intervals in the real number line with the usual topology.
Intuitively, $C=(0,1) \cup (2,3)$ should be disconnected (and I found a special definition of connectedness for open sets that allows me to prove that), but I don't see how to apply the actual definition of connectedness to prove that (or to prove, for example, the same problem with closed sets).
$C$ being disconnected should imply the existence of $U$ and $V$ satisfying the above property, but I can't find any.