How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

Question

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}}.

Answer 1

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) Addendum: Without sufficient Choice, the infinite set $X$ may be amorphous. Amorphous sets are precisely the infinite sets for which this approach doesn't work. Very little Choice is needed to ensure that no such beast exists. (Edit by Cameron Buie)

• 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)

• 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)

• Every set admits a compact Hausdorff topology, by topologizing it as the one-point compactification of the discrete space structure on the complement of any point. (Answer below by Cameron Buie)

(Feel free to edit and expand)

Answer 2

I see that I'm rather late to the party. Here's an answer to the following question that you asked in the comments above:

"[I]s it conceivable that it is a weak AC principle that every set has a compact Hausdorff topology?"

In fact, there is no need for any choice principle at all, if by finite, we mean in bijection with a natural number (or something equivalent), not Dedekind-finite. Obviously, the empty set's only topology is compact Hausdorff.

Suppose $X$ is a non-empty set, fix $x\in X$, and let $Y:=X\setminus\{x\}.$ Now let $\mathcal T$ be the set of all subsets $U$ of $X$ such that either (1) $U\subseteq Y$ or (2) $x\in U$ and $X\setminus U$ is finite. Then $\mathcal T$ is a compact Hausdorff topology on $X.$ In particular, if $X$ is infinite, then $\langle X,\mathcal T\rangle$ is homeomorphic to the Alexandrov one-point compactification of $Y$ in the discrete topology; if $X$ is finite, then $\mathcal T$ is discrete.

Answer 3

Most of this is classical, starting with the memoir by Alexandrov and Urysohn, in which they introduced their notion of the compact Hausdorff space (as bicompact), and also of the absolutely closed space (closed in any Hausdorff superspace) including an extensive discussion of them. This and the minimal Hausdorff spaces, and the related stuff, is very nicely presented as exercises in the Bourbaki General Topology; also Engelking covers these topics in his classical monography (which had several editions). Needless to say, a number of research papers was devoted to minimal Hausdorff spaces and similar.

It's easy to see why the standard Euclidean topology in the space of rational numbers cannot be weakened to a compact topology. The key is: Baire property.

A pretty general result of this type appeared in my paper, Minimal Hausdorff Spaces and $T_1$-Bicompacta, Soviet DAN 1968, v.178, pp 24-26. Let's talk about $T_1$-spaces only, so that complete regularity implies Hausdorff. Theorem 1' states:

Let a completely regular space be a countable union of its nowhere dense (i.e. having empty interior) compact subsets. Then its topology does not dominate any minimal Hausdorff topology.

Still more general (but easier to prove) is Theorem 1 there:

Let a Hausdorff space   $X$ be a countable union of its nowhere dense (i.e. closed and having empty interior) compact subsets. Assume also that it is a dense subspace of a Hausdorff space which has the Baire property. Then topology of   $X$   does not dominate any minimal Hausdorff topology.

The formulation of Theorem 1 suggests how to prove Theorem 1'.

NOTE:   in the published paper the formulation of Theorem 1' missed word countable (it appears in Theorem 1).

Answer 4

In fact there are spaces which are "minimal Hausdorff" -- they have no coarser Hausdorff topology -- but are not compact. It turns out that these spaces are "H-closed" (every open cover has a finite subfamily whose closures cover) and semi-regular (the collection of regular open sets form a base). A minimal Hausdorff space is compact exactly when it is Urysohn. Spaces which have coarser minimal Hausdorff topologies are called Katĕtov. A "nice" example of a space which is not Katĕtov is the space of rational numbers $\mathbb{Q}$.

I'm not sure about compact spaces, but I suspect that a Hausdorff space has a unique coarser minimal Hausdorff topology exactly when it is H-closed. One direction I'm sure of -- the semi-regularization of an H-closed space is minimal Hausdorff.

By the way, (one of) THE BOOK(s) on this topic is Extensions and absolutes of Hausdorff spaces by Porter and Woods, however it discusses Hausdorff spaces almost exclusively.

Answer 5

"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 $\sigma$.