Here is my proof of that the cofinite topology on an infinite set $X$ is connected.
$X$ is connected $\iff$ There are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$.
Let $U, V \in X$ be non-empty disjoint open subsets of $X$ such that $U \cup V = X$.
As $U \cup V = X$ and $U \cap V = \emptyset$,
$\implies U^c = V$ and $V^c = U$.
$\implies U$ and $V$ are finite sets.
$\implies U \cup V$ is finite.
But $U \cup V = X$ which is infinite so we have a contradiction. Hence there are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$. So $X$ is connected.
Is my understanding correct?
Best Answer
(To remove this from the unanswered list)
Your proof is correct.
There are not even disjoint non-empty open sets in this space.