Category Archives: Uncategorized

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 →

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 … Continue reading →

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 … Continue reading →

Although I am trying to post about every month, I have not posted anything for two months.  My first idea was to talk about synthetic topology, after I read some papers by Weng Kin Ho and by Martín Hötzel Escardó. … Continue reading →

Both filter spaces and equilogical spaces form Cartesian-closed categories that contain Top as a full subcategory.  Is there any connection between them?  Very much so, as found by Reinhold Heckmann in 1998, following Martin Hyland (1977).  Read the full post.