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…

# Coherence of Dcpos

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.

# Open Problems

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.

# Pointwise directed families of maps

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.

# Some new, easy facts on consonant spaces

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 T_{0} spaces. The results are going to be easy consequences of results from the book. Read the full post.

# The Dolecki-Greco-Lechicki Theorem

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.

# Summer Topology Conference 2016

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!