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 spaces

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# Markowsky or Cohn?

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.

# Integration

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.

# In memoriam: Klaus Keimel

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.

# A characterization of FAC spaces

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!

# Isbell’s density theorem and intersection of sublocales

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.

# The O functor does not preserve binary products

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.

# Well-filtered dcpos

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.

# Bounded complete and dcpo models of T1 spaces

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 T_{1} space has a bounded complete, and even algebraic, *poset* model, and that every T_{1} space has a (not bounded complete) *dcpo* model, but can we have both at the same time? In other words, does every T_{1} space have a bounded complete dcpo model? Answer (and explanations) in the full post…