Sequential compactness and minors

This is potentially another rambling post. It relates what I’ve been doing on a cute but rather strange corner of point-set topology. For now, when I say ‘space’ I mean ‘Hausdorff space’. Most of what I’m doing here isn’t very interesting in the non-Hausdorff case. Some of it is, but in those cases the Hausdorff assumption makes no real difference either way.
I’ve been playing with the structure of compact spaces which aren’t sequentially compact for a while. Primarily with the aim of showing that in the absence of the full axiom of choice these don’t neccesarily exist (specifically I wanted to show it was consistent with ZF + DC that they didn’t exist). I’ve got to the point where if I really put a lot of effort into it I could probably get a solution, but I’ve rather lost steam on that. It’s not all that interesting a result even if it is true.

In playing with this in general I’ve noticed something a bit more interesting, and related to my original reasons for forming this conjecture.

We’ve basically got two ‘classic’ examples of compact non-sequentially compact spaces. [tex]beta mathbb{N}[/tex] and [tex]{ 0, 1 }^{2^{mathbb{N}} }[/tex]. The first is not sequentially compact for any one of a number of reasons, the second is not sequentially compact because the sequence [tex]x_n : a to a_n[/tex] has no convergent subsequence.

One thing I noticed recently: This is actually only one example. We can identify [tex]beta mathbb{N}[/tex] with the set of ultrafilters on [tex]mathbb{N}[/tex]. i.e. a subset of [tex]P(P(mathbb{N}))[/tex], which is in turn identified in a natural way with [tex]{ 0, 1 }^{2^{mathbb{N}} }[/tex]. Under this identification, the subset topology and the normal topology on [tex]beta mathbb{N}[/tex] agree. Further, the [tex]x_n[/tex] defined above are the principal ultrafilters at [tex]n[/tex]. So their closure is precisely a copy of [tex]beta mathbb{N}[/tex].

So, our two classic examples are really only one example!

I’ve searched the literature for examples of compact spaces which are not sequentially compact, and turned up surprisingly short. Other than these two, there seems to only be one other class of examples (I’ll come to that in a minute). At this point I started getting suspicious – does every compact space which isn’t sequentially compact contain a copy of [tex]beta mathbb{N}[/tex]?

Well, no. At least, not quite. So far the only counterexampe I have depends on the value of the reaping number, [tex]tau[/tex]. It’s fairly standard (I’ll write a post on it at some point) that [tex]{0, 1}^{tau}[/tex] is not sequentially compact. If [tex]tau < 2^{aleph_0}[/tex] then this is an example of a compact non-sequentially compact space of weight [tex]< 2^{aleph_0}[/tex], so can't contain a copy of [tex]beta mathbb{N}[/tex], which has weight [tex]2^{aleph_0}[/tex]. So, if the answer isn't simply no then the problem depends on the underlying combinatorial structure of the universe (or, if you prefer, is independent of ZFC). I suspect that if [tex]2^{aleph_0} = aleph_1[/tex], or possibly if diamond holds, then the answer will be yes. You can at least pare down any example into a compactification of [tex]mathbb{N}[/tex] for which the sequence [tex](n)[/tex] does not have any convergent subsequences. Suppose [tex]x_n[/tex] is some non-convergent sequence. We can find a subsequence which has the discrete topology on it. The closure of this subsequence together with the map [tex]k to x_{n_k}[/tex] is our desired compactification of [tex]mathbb{N}[/tex]. Allow me to phrase the problem in what may be a slightly more suggestive way. Let [tex]X, Y[/tex] be topological spaces. Say [tex]Y[/tex] is a minor of [tex]X[/tex] if it is the continuous image of a closed subset of [tex]X[/tex], and write [tex]Y prec X[/tex]. Also say [tex]X[/tex] is a major of [tex]Y[/tex]. Note that [tex]prec[/tex] forms a preorder on topological spaces. This is not a partial order on isomorphism classes. For example, [tex][0, 1][/tex] and [tex]S^1[/tex] are both minors of eachother and are not isomorphic. Also, [tex]beta mathbb{N}[/tex] and [tex]{0, 1}^{2^{mathbb{N}}}[/tex] are minors of eachother which are not isomorphic.

A standard result is that every compact hausdorff space of weight [tex]leq kappa[/tex] is a minor of [tex]{0, 1}^{kappa}[/tex]. So given a minor closed class in order to show that it contains all spaces of weight [tex]leq kappa[/tex] it suffices to show that it includes [tex]{0, 1}^{kappa}[/tex]. e.g. in order to show all compact spaces of weight [tex]< tau[/tex] are sequentially compact, it suffices to show that [tex]{0, 1}^{kappa}[/tex] is for every [tex]kappa < tau[/tex]. Note: The 'minor' notation is totally nonstandard as far as I'm aware. I've stolen it from graph theory in order to suggest certain analogies (but I know very little graph theory, so these analogies may be totally wrong). We'll be interested in looking at classes of spaces which are closed under taking minors. Examples of these include:

  • The class of compact spaces.
  • The class of sequentially compact spaces.
  • The compact spaces of weight [tex]leq kappa[/tex] (note: The compactness assumption is neccesary here. In general weight is not neccesarily non-decreasing on continuous images).
  • For any space [tex]X[/tex], the class of minors of [tex]X[/tex].

A class is closed under taking minors iff its complement is closed under taking majors. This is important!

[tex]beta mathbb{N}[/tex] has the following interesting property. If [tex]X[/tex] is compact (Hausdorff) and [tex]f : X to beta mathbb{N}[/tex] is a continuous surjection then there is an embedding [tex]g : beta mathbb{N} to X[/tex] such that [tex]fg = id[/tex]. The proof is fairly straightforward, but the important part to take home is that if [tex]beta mathbb{N}[/tex] is a minor of [tex]X[/tex] then it is a subspace of [tex]X[/tex]. So the class of spaces which contain [tex]beta mathbb{N}[/tex] is major closed.

Final point: Every compact space which is not sequentially compact contains a separable subspace which is not (indeed a compactification of [tex]mathbb{N}[/tex]. There are only [tex]2^{2^{aleph_0}}[/tex] separable Hausdorff spaces, so there is a set of representatives for the isomorphism classes of such subspaces. Thus we can find some set [tex]R[/tex] such that [tex]X[/tex] is not sequentially compact iff it has a minor in [tex]R[/tex]. So far we’ve shown we can choose [tex]|R| leq 2^{2^{aleph_0}}[/tex], and we want to show we can choose [tex]|R| = 1[/tex]. Oh well, a little way to go…
Maybe this isn’t that suggestive, but I think it’s a neat way of looking at it and may prove useful. Certainly I’d like to study results about minor closed spaces some more – obviously I’ve been motivated by the excluded minor theorem, that a minor closed set of graphs is precisely the set of graphs which exclude some finite set of forbidden minors. I really doubt this is true in topological version, even restricting ourself to compact Hausdorff spaces, (I suspect the succesor ordinals form a counterexample), but some analogue of it might be.

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