Let me first wish you a Merry Christmas, and since I will not post again next week, a Happy New Year 2019 as well. I have no specific present this year, sorry… This month’s post is about a few thoughts I have had about the role of countability is a few situations other than the expected ones. This will culminate in a recent result of de Brecht and Kawai—the Scott and the upper Vietoris topologies coincide on the Smyth powerdomain of a well-filtered second-countable space— and its clever proof: see the full post.
Alex Simpson has a lot of slides with very interesting ideas. One of them is what he calls the locale of random sequences. This is a terribly clever idea that aims at solving the question “what are random sequences?”, using locale theory. He obtains a very big locale, but without points in general… because every single random sequence is essentially not random. Read the full post.
Last time, we embarked on proving that the projective limit of a projective system of compact sober (resp., and non-empty) spaces is compact and sober (resp., and non-empty), a theorem that Fujiwara and Kato call Steenrod’s Theorem. However, instead, we merely proved that a projective limit of a projective system of non-empty compact sober spaces is non-empty. Do not despair: this is the essential argument in the proof of Steenrod’s Theorem, which we complete this month. Read the full post.
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.