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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?

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    $\begingroup$ The one-point space. Probably you should ask about infinite spaces. $\endgroup$ – Joel David Hamkins Mar 30 '15 at 13:53
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There is a maximal connected Hausdorff topology for the reals, according to this paper of Guthrie, Stone and Wage.

I haven't read it, some googling led me to the paper "Problems on (ir)resolvability" of Oleg Pavlov in "Open problems in Topology II" which lists some open questions and cites the above paper (among others).

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