Given a connected space, it is easy to tell if there is a connected expansion because maximal connected spaces (those admitting no finer connected topology) have the property that every dense subset is open. If you can find a dense subset $D$ which is not open (typically this is very easy to find), then the topology generated by $\tau$ with $D$ is a connected expansion.
My question is a little more specific.
Let $X$ be infinite connected Hausdorff space with collection of open sets $\tau$. Given a nowhere dense infinite set $S\subseteq X$, I would like to know when it is possible to find a subset of $S$ which can be added to $\tau$ to produce a topology that is still connected. Are there conditions depending $\tau$ and $S$ which will guarantee this is possible?
