**This is a community wiki of the answers in the comments.**

* 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 <a href="http://books.google.com/books?id=DkEuGkOtSrUC&lpg=PA118&ots=3hHBTJE-k7&dq=maximal%20compact%20topology&pg=PA118#v=onepage&q=&f=false">Steen & Seebach 99</a>. (Comment by Gerald Edgar)

* There is a minimal Hausdorff topology on a countable space which is not compact. See <a href="http://books.google.com/books?id=DkEuGkOtSrUC&lpg=PA118&ots=3hHBTJE-k7&dq=maximal%20compact%20topology&pg=PA119#v=onepage&q=&f=false">Steen & Seebach 100</a>. (Comment by François G. Dorais)

* Those examples can be lifted to any cardinality space, simply by using the disjoint sum with any given compact Hausdorff space. (Comment by Gerald Edgar)

* The Axiom of Choice implies that every set admits a compact Hausdorff topology, using the order topology of a successor well-ordering of it. (Comments by François G. Dorais and Gerald Edgar)

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