There are a few questions that I would like to solve, and I’ve decided to share them with you. See the new open problems page. I am starting with three questions now, but I will update that from time to time. Some of these problems are pretty tough, and I’m asking for help here. Some others are more doable, and indeed would be nice subjects for brilliant young M2 or PhD students.
Happy New Year 2017! Sorry I have not posted for some time… Today, let me talk about a curious construction of Lawson and Xi on families of continuous maps that are not directed, rather pointwise directed. They use that to show that the dcpo of all Scott-continuous maps from a core-compact, core-coherent space to an RB-domain is a continuous dcpo; but the idea of pointwise directed families of maps, and the way they use it to find a basis of dcpos of continuous maps, is intriguing. Read the full post.
Last time, we had stated and proved the Dolecki-Greco-Lechicki theorem: every regular Čech-complete space is consonant. I would like to show that there are some other classes of consonant spaces, among T0 spaces. The results are going to be easy consequences of results from the book. Read the full post.
I have met Szymon Dolecki at the Summer Topology Conference 2016, and he is a charming person. In 1995, with Gabriele Greco and Alojzy Lechicki, he proved a very nice theorem — every Čech-complete space is consonant — that deserves to be well-known. As a coincidence, this very theorem was mentioned by Jimmie Lawson at the same conference. It is also the subject of Exercise 8.3.4 in the book, but I am afraid that, as stated, it is way too hard. My purpose here is to give a complete solution to the exercise. The proof is elementary, that is, it does not require any deep knowledge of topology. Read the full post.
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…
My goal today is to describe two elegant proofs of the fact that nuclei form a frame. There are many proofs of that.
The main difficulty is that, while meets (infima) of nuclei are taken pointwise, joins (suprema) are much harder to describe. That is certainly an obstacle if one ever tries to prove that binary meets distribute over joins. A similar difficulty occurs with sublocales.
I will present a short proof due to Picado and Pultr, using sublocales. I will also present an entirely different proof due to M. H. Escardó, which works on nuclei. The latter will make use of a fixed point theorem we have already seen on this site! Read the full post.
Last time, I promised you we would explore another way of defining sublocales. We shall again use the naive approach that consists in imagining how we would encode subspaces of a T0 topological space X by looking at open subsets only, and certainly not at points. We shall encode a localic version of the notion of subspaces through what I call sieves, which are certain sets of formal crescents. I’ll then show you that this gives you a complete lattice that is isomorphic to the lattice of sublocales that we have seen last time. See the full post.
Stone duality leads naturally to the idea of locale theory. Quickly said, the idea is that, instead of reasoning with topological spaces, we reason with frames. The two concepts are not completely interchangeable, but the O ⊣ pt adjunction shows that they are close. My purpose today is to give an introduction to what the analogue of a topological subspace should be on the localic side. Read the full post.
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.