Is there a connected $T\_2$ space $(X,\tau)$ with more than one point, such that the singletons and $X$ are the only connected subspaces of $X$?



Connected $T_2$ space with essentially no connected subspaces
There is no such space. For if $x$ is any point and $X\setminus \{x\}=U|V$ then $\{x\}\cup U$ and $\{x\}\cup V$ are connected sets, each with more than one point and different from $X$.


The closest thing you can get is a connected set whose connected subsets are cofinite. The axiom CH implies there is a countable connected Hausdorff space with this property [here](http://www.ams.org/journals/proc/1994-122-03/S0002-9939-1994-1287102-2/S0002-9939-1994-1287102-2.pdf) that is not too complicated (relative to the completely regular example). I don't think a Hausdorff example has ever been constructed in ZFC.


