How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more complicated question. All colimits indeed exist in Dcpo, and this has been shown by various authors over time, but this is complex. I will explain why. Then I will explain what the definition of quotients should be in Dcpo, and how one can build them. Funnily, this is related to other questions, such as the existence of d-completions, for example. Read the full post.
Last time, I motivated the construction of the well-filterification Wf(X) of a space X of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every core-compact well-filtered T0 space is sober, and hence also locally compact. This solves a question asked by X. Jia, and first solved positively by J. Lawson and X. Xi. I then realized that their proof contains a (somewhat concealed) gem; polishing it reveals an interesting new variant of the Hofmann-Mislove theorem, which applies to well-filtered spaces rather than to sober spaces, and rests on a funny countability assumption. We will see that the latter is satisfied in all core-compact spaces, and this will allow us to show that all core-compact well-filtered T0 spaces are locally compact, hence sober. Read the full post.
Xiaodong Jia once asked the following question: is every core-compact, well-filtered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, X. Xu, Ch. Shen, X. Xi and D. Zhao found a simpler solution, and I have changed my plans. My new plan for this time, and next time, is to explain what they have done. This time, we will concentrate on well-filterifications of topological space, which are just like sobrifications except ‘sober’ is replaced by ‘well-filtered T0‘. Building one is not completely obvious. Xu, Shen, Xi and Zhao show that the will-filterification of X can be defined as its set of closed WD sets, a new notion that is intermediate between directed sets and irreducible sets; the proof also relies on a refinement of R. Heckmann and K. Keimel’s topological version of Rudin’s Lemma, which is interesting in its own right. Read the full post.
It often happens that one wishes to show that a certain subspace A of a given sober space X is sober. The following is a pearl due to Keimel and Lawson, which was mentioned to me by Zhenchao Lyu in July: the sober subspaces of a sober space are exactly its subsets that are closed in the Skula topology. Read the full post.
Bounded-complete domains (bc-domains) are an incredibly useful form of continuous dcpos. Given a continuous dcpo X, is there a completion of X as a bc-domain, for example a free bc-domain on X? That does not exist in general, but Yuri Ershov showed that one can build a so-called bc-hull of any continuous dcpo in 1997. I will describe what that is in the full post. My point is really to show that, despite the fact that it is a complicated construction in general, this really becomes a very simple, and familiar, one when X is coherent.
Marcus Tressl just found a horrible mistake in the book: I had claimed that the T0 quotient of the topological quotient of any sober space by any equivalence relation is sober, but that is completely wrong. In fact, Moshe Shimrat had shown in 1956 that you can get absolutely any topological space as a topological quotient of a Hausdorff space. In the full post, I will explain why Shimrat’s theorem directly contradicts my claim, and I will explain Shimrat’s proof. I will also comment of my own personal (hence biased) selection of the papers I think were the best among those presented at the 8th International Symposium on Domain Theory, in Yangzhou, Jiangsu province, China, from which I have just come back.
Recently, Matthew de Brecht sent me a proof of a neat and rather surprising result: the product and the Scott topologies coincide for products of first-countable, not necessarily continuous, posets. This rests on a clever argument, inspired by techniques invented by Matthias Schröder, and a simple observation: if you take all the elements of a convergent sequence, plus one (any) of its limits, what you get is a compact set. The latter fails if you take a net instead of a sequence. Read the full post.
Johnstone space J (1981) is the most famous example of a non-sober dcpo. In 1982, Isbell came up with a non-sober complete lattice. His construction is so complex that most authors use it as a black box. I would like to explain how Isbell’s non-sober complete lattice is constructed. As you can expect, this is a pretty clever construction, but I claim this is understandable. I will conclude with remarkable results of Xu, Xi, and Zhao (2019), who proved that there even exists a non-sober frame. Read the full post.
Reinhold Heckmann showed the following in a very nice paper of 2014: every countably presented locale is spatial. What makes it even nicer is that he shows how tightly this is connected with the Baire property. This also gives a localic description of Matthew de Brecht’s quasi-Polish spaces, and of Ruiyuan Chen’s more recent countably correlated spaces. Read the full post.
Last time, I had announced that we would do Exercise V-5.25 of the red book, constructing a core-compact, yet not locally compact, space. And this is exactly what we shall do: read the full post.