Are there non-Hausdorff examples of maximal compact topologies in the lattice of topologies on a set?

general-topology compactness

Solution 1:

Both kinds of examples exist: see this abstract. A bit more digging turned up this PDF by Douglas Cameron, which has the Smythe and Wilkens example as Example 5.1b, the Hing Tong example as Example 11.1(a). It also points out that the one-point compactification of the rationals is maximal compact but not Hausdorff. See also this paper by Cameron in the Transactions of the AMS.

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