We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.


If $(X,\tau)$ is connected, is there a topology $\tau' \supseteq \tau$ such that $(X,\tau')$ is maximal connected?



Maximal connected topologies
No. You can find a counterexample here: Baggs, Ivan. A connected Hausdorff space which is not contained in a maximal connected space. Pacific J. Math. 51 (1974), no. 1, 11--18 


