Let us continue last month’s story. We had define various structures of convergence spaces on a dcpo, which were all admissible in the sense that their topological modification is the Scott topology. We shall see that equipping dcpos with their Heckmann, or with their Scott convergence structures, defines a product-preserving functor from Dcpo to Conv. The result is due to Reinhold Heckmann, and contrasts with the fact that the similar functor from Dcpo to Top does not preserve products—a very nasty source of mistakes. Read the full post.
Every dcpo can be seen as a topological space, once we equip it with the Scott topology. And every topological space can be seen as a convergence space, so every dcpo can be seen as a convergence space. In 2003, Reinhold Heckmann observed that we could see dcpos as convergence spaces in another way, with some serendipitous properties. We shall see what serendipitous properties next time. This month, we shall prepare the grounds for that piece of work, by investigating various convergences that can be put on dcpos, in particular one introduced by Dana S. Scott way earlier. Read the full post.
Only a short post this month: I would like to explain Lawson’s construction of an FS-domain that is not known to be an RB-domain. Roughly speaking, this is the domain of closed discs of the under with reverse inclusion, and one can generalize it to the domain of formal balls of certains (quasi-)metric spaces. Read the full post.
Characterizing properties of graphs, posets, and even dcpos by forbidden substructures is an intriguing approach. Xiaodong Jia managed to show that every CCC of quasi-continuous domains must consist of continuous domains exclusively, and I would like to explain how this rests on the very ingenious idea that one should study meet-continuous dcpos, and specifically, that one can characterize non-meet-continuous dcpos through certain forbidden substructures. Read the full post.
Meet-continuous dcpos were defined and studied by new proof of that theorem through Stone duality. Today, I would like to talk about yet another proof, which I had the pleasure to read in Xiaodong Jia‘s remarkable PhD thesis. Read the full post.Ying-Ming Liu, and Mao-Kang Luo about 14 years ago, and their importance only starts to be appreciated now. One of the leading results in the theory of meet-continuous dcpos is that a dcpo is continuous if and only if it is quasi-continuous and meet-continuous. Weng Kin Ho, Achim Jung and Dongsheng Zhao’s gave a
I have already mentioned Markowsky’s Theorem (1976): every chain-complete poset is a dcpo. This is a non-trivial theorem, and I’ve given you a proof of it based on Iwamura’s Lemma and ordinals in a previous post. Maurice Pouzet recently pointed me to P. M. Cohn’s book Universal algebra (1965), where you can find the same theorem already! Cohn’s proof is very different and does not rely on Iwamura’s Lemma. Let me describe it in the full post.
At the start of the book, I had stated: “Topological convexity, topological measure theory, hyperspaces, and powerdomains will be treated in further volumes.” The book got out in 2013, but I wrote that in 2011, almost seven years ago now. What happened?
Well, nothing went according to plan, but I in fact wrote plenty of things during the period. Let me tell you what happened… with a surprise in the middle of the full post.
Klaus Keimel passed away on Saturday, November 18th, 2017, and this is sad news. I would like to pay homage to his memory, through a partial recollection of my own path with Klaus.
In the open problem section, I defined a FAC space as a topological space in which every closed subspace is a finite union of irreducible closed subspaces. FAC is for “finite antichain property”, since it generalizes the following theorem, due to Erdős and Tarski (1943): a poset has the finite antichain property (namely, all its antichains are finite) if and only if its downwards-closed subsets are finite unions of ideals. I asked about a similar characterization of FAC spaces. Let me give a positive answer to that in the full post!
When I wrote my latest blog post, there were many things I thought would be useful to know about sublocales. Those eventually turned out to be useless in that context. However, I think they should be known, in a more general context. In particular, I would like to stress Isbell’s amazing density theorem, an easy but rather counterintuitive result in locale theory, and its consequence on intersections of sublocales. Read the full post.