Non-Hausdorff Topology and Domain Theory

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.

While a topological space is a space of points, a hyperspace is a space of subsets, with a suitable topology.  Examples abound in the literature.  For example, the so-called Smyth powerdomain (Proposition 8.3.25) is one.  To start the series, let me look at the Hoare hyperspace instead.  We shall see that it is a space that has many nice properties.  I won’t say to which purpose it has been put in denotational semantics yet, or that it defines a monad with a very natural inequational theory… all that will have to wait!  Read the full post.

On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets.  This is a theorem by George Markowsky (1976).  It is time I explained seriously how this worked.  The first step is Iwamura’s Lemma (1944), which states that every directed subset decomposes as the union of a small chain of small directed subsets.

The reason I did not put the proof of that result in the book is because it rests on using ordinals, and I did not want to introduce ordinals, specially if they served for only one result.  I’m attacking the problem head-front in the full post.

From time to time, we happen to discover that several distinct notions are in fact the same, and this is exactly what happened in 2014, in two papers that appeared about at the same time, with similar discoveries. One is due to Jimmie Lawson and Xiaoyong Xi, the other one is due to Achim Jung and myself.  The theorem says that QRB-domains, QFS-domains, and stably compact, locally finitary spaces, are exactly the same thing.  The first arose in probabilistic domain theory.  See the full post.

I hope you’ve had a Merry Christmas, and wish you a Happy New Year 2015.  To renew with an old habit, I’ve produced a new crosswords puzzle on the occasion (#14).  Sorry, it does not have anything particular related to Christmas, or with any other season-related celebration.  I may do better next time.