Non-Hausdorff Topology and Domain Theory

Last time, I explained some of the strange things that happen with projective limits of topological spaces: they can be empty, even if all the spaces in the given projective system are non-empty and all bonding maps are surjective, and they can fail to be compact, even if all the spaces in the projective system are compact.

Steenrod’s Theorem (as Fujiwara and Kato call it) shows that all those pathologies disappear if we work with compact sober spaces.  This rests on a lemma, according to which projective limits of non-empty compact sober spaces are non-empty, which is the subject of this month’s full post.  We will see how Steenrod’s Theorem follows… next time.

This month, let me investigate projective limits of topological spaces.  That is an area of mathematics that is fraught with pitfalls, and I will describe a number of odd situations that can occur in that domain.  You will have to wait until next month (sorry!) to learn about a very nice result on projective limits of compact sober spaces, due to O. Gabber.  Read the full post.

I thought I would devote my blog this month to the Domains workshop, but a sudden health problem prevented me to go there.  Instead, I will talk about a curious alternative to Stone duality, which, instead of an adjunction between Top and the opposite category of the category Frm of frames, is an adjunction between Top and the opposite category of that of something that Frédéric Mynard and I called topological coframes.  Read the full post.

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 Hui Kou, 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 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.

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.