On countability: the compact completed sequence

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.

Isbell’s non sober complete lattice

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.

Countably presented locales

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.

Bernstein subsets of R

This month, we will start to do Exercise V-5.25 of the red book (Continuous Lattices and Domains), which gives an example of a core-compact, not locally compact space.  That is pretty hard to obtain, really.  This month, we will do the first step of that exercise by constructing Bernstein sets, which are very, very pathological subsets of R.  Read the full post.

On countability

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.

Projective limits of topological spaces III: finishing the proof of Steenrod’s theorem

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.


Projective limits of topological space II: Steenrod’s theorem

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.

Projective limits of topological spaces I: oddities

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.