# Is there a maximal connected Hausdorff space?

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.

Is there a maximal connected Hausdorff space with more than 1 point?

• The one-point space. Probably you should ask about infinite spaces. – Joel David Hamkins Mar 30 '15 at 13:53