TITLE: How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
QUESTION [74 upvotes]: This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
Equivalently, we say in this case that τ is coarser
than σ, that σ is finer than τ or that
σ refines τ. (See wikipedia on comparison of
topologies.)
The least element in this order is the indiscrete topology and the largest topology is the discrete topology. 
One can show that the collection of all topologies on a fixed set is a complete lattice. In the downward direction, for example, the intersection of any collection of
topologies on X remains a topology on X, and this intersection
is the largest topology contained in them all. Similarly,
the union of any number of topologies generates a smallest
topology containing all of them (by closing under finite
intersections and arbitrary unions). Thus, the collection of all topologies on X is a complete
lattice. 
Note that the compact topologies are closed downward in
this lattice, since if a topology τ has fewer open sets than
σ and σ is compact, then τ is compact.
Similarly, the Hausdorff topologies are closed upward,
since if τ is Hausdorff and contained in σ, then
σ is Hausdorff. Thus, the compact topologies inhabit
the bottom of the lattice and the Hausdorff topologies the
top.
These two collections kiss each other in the compact
Hausdorff topologies. Furthermore, these kissing points,
the compact Hausdorff topologies, form an antichain in the
lattice: no two of them are comparable. To see this,
suppose that τ subset σ are both compact
Hausdorff. If U is open with respect to σ, then the
complement C = X - U is closed with respect to σ and
hence compact with respect to σ in the subspace
topology. Thus C is also compact with respect to τ in
the subspace topology. Since τ is Hausdorff, this
implies (an elementary exercise) that C is closed with respect to τ, and so U is
in τ. So τ = σ. Thus, no two distinct compact Hausdorff topologies are comparable, and so these topologies are spread out sideways, forming an antichain of the lattice.
My first question is, do the compact Hausdorff topologies
form a maximal antichain? Equivalently, is every topology
comparable with a compact Hausdorff topology? [Edit:  François points out an easy counterexample in the comments below.] 
A weaker version of the question asks merely whether every
compact topology is refined by a compact Hausdorff
topology, and similarly, whether every Hausdorff topology
refines a compact Hausdorff topology. Under what
circumstances is a compact topology refined by a unique
compact Hausdorff topology? Under what circumstances does a
Hausdorff topology refine a unique compact Hausdorff
topology?
What other topological features besides compactness and
Hausdorffness have illuminating interaction with this
lattice?
Finally, what kind of lattice properties does the lattice
of topologies exhibit? For example, the lattice has atoms,
since we can form the almost-indiscrete topology having
just one nontrivial open set (and any nontrivial subset
will do). It follows that every topology is the least upper
bound of the atoms below it. The lattice of topologies is
complemented.
But the lattice is not distributive (when X has at least
two points), since it embeds N5 by the
topologies involving {x}, {y} and the topology generated by
{{x},{x,y}}.

REPLY [6 votes]: "Under what circumstances does a Hausdorff topology [properly] refine a unique compact Hausdorff topology?"
If $X$ is locally compact (non-compact, Hausdorff), then $X$ refines infinitely many distinct compact Hausdorff topologies. However, there are also examples of (non-compact, Hausdorff) spaces $X$ that refine a unique compact Hausdorff topology.
Theorem: If $X$ is a non-compact, locally compact Hausdorff topology that refines at least one compact Hausdorff topology, then it refines at least $|X|$ compact Hausdorff topologies.
Proof: See the proof of Proposition 4.3 in this paper. The basic idea is to take a one-point compactification of $X$, then take the point at infinity and glue it back down onto any point of $X$. Now you have a topology that's compact, Hausdorff, and refined by $X$. Choosing different targets for the gluing will result in different topologies. QED. [Note: These topologies, while different, may be homeomorphic, for example if $X$ is the real line.]
In contrast, we have the following example. Let $I = [0,1]$ (the set, not the topological space). Let $\sigma$ denote the usual topology on $I$, and let $\langle \sigma,A \rangle$ denote the topology on $I$ with subbasis $\sigma \cup \{A\}$. Let $A = I \setminus \{\frac{1}{n}:n \geq 1\}$. Then I claim that $\sigma$ is the only compact Hausdorff topology refined by $\langle \sigma,A \rangle$.
Let $\tau$ be any compact Hausdorff topology that is refined by $\langle \sigma,A \rangle$.
Notice that if $a > 0$ then $\sigma$ and $\langle \sigma,A \rangle$ agree on $[a,1]$. Furthermore, since $[a,1]$ is compact Hausdorff, passing to $\tau$ will not change its topology, since any strictly coarser topology on $[a,1]$ fails to be Hausdorff. Thus $\sigma$ and $\tau$ agree on every $[a,1]$, and hence on $(0,1]$.
Thus $\tau$ is a (Hausdorff) one-point compactification of $(0,1]$ (with the usual topology). But there's only one of those! So $\langle \sigma,A \rangle$ is refined by only one compact Hausdorff topology, namely $\sigma$.