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.
In Exercise 8.4.23 of the book, I said: “Exercise 8.4.21 may give you the false impression that the O functor preserves binary products. This is wrong, although an explicit counterexample seems too complicated to study here: see Johnstone (1982, 2.14).” O, here and as usual on these pages, is the open subset functor from Top to Loc. My purpose here is to show that that is not that complicated after all.
My initial plan was to follow John Isbell’s Product spaces in locales 1981 paper (Theorem 2). The proof is only 5 lines, so that should be doable… or so I thought. But Isbell used to be very terse, and my explanation will be much longer. Read the full post.
I have just returned from the International Symposium on Domain Theory, which took place in Shijiazhuang, Hebei, China. That was a fine conference indeed. There, I met Xiaoyong Xi and Jimmie Lawson, who just happened to publish a remarkable result, related to a very recent post on coherence of dcpos: every complete lattice, and more generally every bounded-complete dcpo is well-filtered in its Scott topology. Read the full post.
The nice thing about colleagues is that, sometimes, they give me a primer on their latest results. I would like to talk about a strange result by Dongsheng Zhao and Xiaoyong Xi, which, while accepted for publication, does not seem to be out yet. (Thanks to D. Zhao for letting me know about this!) I have already talked about models of topological spaces. Following earlier results by Zhao, Xi, and Erné, one can show that every T1 space has a bounded complete, and even algebraic, poset model, and that every T1 space has a (not bounded complete) dcpo model, but can we have both at the same time? In other words, does every T1 space have a bounded complete dcpo model? Answer (and explanations) in the full post…
It had been a long time since I wanted to explain a nifty result by Jia, Jung, and Li (2016), which gives a simple test for whether a given well-filtered dcpo is coherent. The proof, in particular, is extremely nifty. Read the full post.
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.
— Jean Goubault-Larrecq