TITLE: Dense and co-dense subsets in connected $T_2$-spaces
QUESTION [7 upvotes]: Is there a connected $T_2$-space $(X,\tau)$ with more than 1 point and with the following property?

Whenever $D\subseteq X$ is dense, $X\setminus D$ is not dense.

REPLY [4 votes]: There is such a thing as a submaximal topology, in which every dense subset is open.  These obviously satisfy your condition.
Take any connected Hausdorff space $(X,\tau)$.  Let $\mathscr F$ be an ultrafilter of $\tau$-dense sets.  Let $\tau'$ be the topology generated by $\tau\cup \mathscr F$. Then $(X,\tau')$ is submaximal Hausdorff connected.