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In the lattice of topologies on a set $X$, the compact topologies are a lower set in the lattice, while the Hausdorff topologies are an upper set. A result of this theorem is that the compact Hausdorff topologies are maximal elements in the set of compact topologies and minimal elements of the set of Hausdorff topologies in this lattice.

I have been failing to construct an example of a maximal compact topology that is not Hausdorff, but I feel like I am just lacking imagination - it seems unlikely that all maximal compact topologies are Hausdorff.

Finite topologies won't work, since the only Hausdorff topology on a finite set is the discrete topology, and, as the maximal element in the lattice, the only maximal compact topology.

My intuition is that there must be such examples, but it seems just possible that if a compact set is not Hausdorff, we might be able to create a new compact topology that is larger in the lattice.

There is the obvious dual question, too: Is there a minimal element of the set of Hausdorff topologies which is not compact?

Just to make the problem self-contained, the result referenced above is:

A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

If $(X,\tau)$ is compact and Hausdorff, and $\tau\subseteq \tau'$ with $\tau'$ also a compact topology, then the identity function $(X,\tau')\to(X,\tau)$ is a continuous bijection from a compact space to a Hausdorff space, so it must be a homeomorphism, which implies $\tau=\tau'$.

Similarly, a compact Hausdorff topology is minimal Hausdorff topologies.

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Both kinds of examples exist: see this abstract. I’ve not yet tracked down any of the relevant papers, though. – Brian M. Scott Jan 8 '13 at 21:32
Might was well make that an answer - it might well be the only answer I get. :) @BrianM.Scott – Thomas Andrews Jan 8 '13 at 21:39
I managed to do a little better: I found a paper by Doug Cameron that includes some of the examples. – Brian M. Scott Jan 8 '13 at 21:46
up vote 8 down vote accepted

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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I particularly like this result, since it clarifies something I was stumbling towards: "the maximal compact spaces [are characterized] as those compact spaces in which the compact subsets are precisely the closed sets." – Thomas Andrews Jan 8 '13 at 22:17
@Thomas: Yes, it’s a nice characterization $-$ not unexpected, when you think about it, but nice. – Brian M. Scott Jan 8 '13 at 22:20

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