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}}.
