Non-Hausdorff Topology and Domain Theory

I have already said I would be at the Galway Colloquium in Leicester, UK, on Monday, August 1st, 2016. Right after that, I will participate to the 2016 Summer Topology Conference—same place, from Tuesday to Friday.

I will give a talk on Noetherian spaces. They form Section 9.7 of the book, and I will try to make an accessible introduction to the view expounded there, that they form a natural topological generalization of the notion of well-quasi-ordering. The pièce de résistance will be the topological Higman lemma, of which I will attempt to give a complete proof during the talk.  I will also talk about applications in computer science, finite representations of open and closed subsets, and the curious case of the powerset of a Noetherian space.  You can find the slides here.  See you there!

I will be at the Galway Colloquium at Leicester, UK, on Monday, August 1st, 2016.  I will give an introduction to T₀ topology and domain theory there, aimed mostly at students in mathematics.

I will try to illustrate some of the key concepts in the field by progressing through a proof of Dana S. Scott’s theorem (1972) that the injective T₀ spaces are the continuous lattices. This will be an opportunity to touch various subjects: dcpos, the specialization ordering, Sierpiński space, retracts, Stone duality, sober spaces and sobrification, b-spaces and algebraic posets, c-spaces and continuous posets, notably.

Have a look at the slides! Note that there are several grayed areas, which hide some of the proofs.  This is on purpose: I am expecting the attendees to find the missing arguments by themselves.  If you plan to come to Leicester, you may try to think about them in advance.  Otherwise, have fun and try to find the arguments by yourselves!  Note that most of the material is scattered through the book, too…

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.