Non-Hausdorff Topology and Domain Theory

I am a bit stubborn. In my first post on ideal domains, I thought I would be able to extend Keye Martin’s result from metric to quasi-metric spaces. I have said I had failed, but now I think I have succeeded.  This leads to a notion that I will call a quasi-ideal domain.

Our purpose today is to show that, if X is a continuous Yoneda-complete quasi-metric space, then it embeds into an algebraic dcpo, and in fact, in a very specific way: as the subspace of limit elements of a quasi-ideal domain.  Read the full post.

In my first post on ideal domains, I thought I would be able to extend Keye Martin’s result from metric to quasi-metric spaces. That was more complicated than what I had thought.

Along my journey, I (re)discovered a few results, some old, some new, on ideal completion remainders—namely, the spaces you get by taking the ideal completion of a poset P, and substracting P off—and on the related notion of sobrification remainders.  That may seem like silly notions to you, and I certainly thought so until recently.  But they seem to crop up from time to time.

I will show you that every T₀ space occurs as a sobrification remainder (a result due to R.-E. Hoffmann), and I will give you the rough idea of a proof that the ideal completion remainders of countable posets are exactly the quasi-Polish spaces (a result due to M. de Brecht). I will also describe an intriguing result on wqos and bqos due to Y. Péquignot and R. Carroy. Read the full post.

Last time, we have seen that every completely metrizable space X has an ideal model, that is, that X can be embedded into an ideal domain Y in such a way that we can equate X with the subspace of maximal elements of Y.

We have also seen the converse to that: if X is a metrizable space with an ideal model, then X is completely metrizable.

But we had skipped an essential ingredient: that the set X of maximal elements of an ideal model Y is a G_δ subset of Y.  This is true, but complicated.  As I have already said last time, we shall do something slightly simpler.  See the full post.

I had not posted a crossword puzzle for a long time, so here is one at last: in pdf format, or in AcrossLite format, as usual.  Happy New Year!

A few months ago, Keye Martin drew my attention to his results on so-called ideal models of spaces [1].  Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal.  Despite this, Keye Martin was able to show that: (1) every space that has an ω-continuous model has an ideal model, that is, a model that is an ideal domain; (2) the metrizable spaces that have an ideal model are exactly the completely metrizable spaces.

I will try to expose a few of his ideas here.  I will probably betray him a lot.  For example, I will not talk about measurements (one of Keye’s inventions), and I will not stress the role of Choquet-completeness to go beyond “Lawson at the top” domains, or the role of G_δ subsets so much.

Last minute update: I had also tried to extend whatever I could to the case of quasi-metric, not just metric, spaces, but I did not manage to do so. Read the (corrected) full post.

In my last post, I said I would have trouble finding time to write anything up in August, and sadly, this came out true. Late August, I went to the Domains XII conference, and it may be a good idea if I gave a report on a few of the things I learned there.  Read the full post.

You have probably sweated a lot at trying to understand the constructions of Part IV.  They rest on a lot of topology and domain theory.  Perhaps surprisingly (if you do not know it already), they are completely generic, and work in any category with enough structure.  This is what we learn from Peter Freyd’s adjoint functor theorems, of which there are two: the general one, and the special one.  Read the full post.

Last time, we concluded with a mysterious observation.  There is a theory, that of unital inflationary topological semi-lattices, which plays a fundamental role in the study of the Hoare powerspace.  On the one hand, H(X) is the free sober such thing.  On the other hand, the algebras of the H monad are exactly those things that are sober.  We shall investigate that by looking at theories themselves, and show how those constructions arise from a logical perspective.  In the end, this turned out to be more complicated than what I had thought initially…  Read more.

The last post was late.  Let me compensate by being early this time.

I had promised you that we would see why the theory of the Hoare powerspace monad was given by a small family of axioms, those of unital inflationary semilattices.  I will substantiate this claim in two ways.  Following Andrea Schalk, we shall see that H(X) is the free such (sober) thing over X, and we shall see that those (sober) things are exactly the algebras of the H monad—a nice way to introduce the notion of algebra of a monad.  Read the full post.

Let us deepen our understanding of the Hoare powerspace construction. We shall see that it defines a so-called monad.  There would be many, many things to say about monads!  I will only give a very superficial introduction here, trying to convince you that the Hoare powerspace construction indeed produces a monad.  In part III, I’ll tell you about the inequational theories of monads, and monad algebras.  Read the full post.