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The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais)
There is a maximal compact topology on a countable space which is not Hausdorff. See Steen & Seebach 99. (Comment by Gerald Edgar)
There is a minimal Hausdorff topology on a countable space which is not compact. See Steen & Seebach 100. (Comment by François G. Dorais)
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