Author Archives: jgl

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →

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 … Continue reading →