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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.

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

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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

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