Koch’s arc theorem

Posted on March 20, 2025 by jgl

Koch’s arc theorem is a famous theorem saying that, under a few conditions, given any point x of an open subset U of a pospace (X, ≤), one can draw an arc chain from some point on the boundary of U to x. The conditions are that: (i) the closure of U should be compact and (ii) U should not contain any local minimum. That may seem clear: starting from x, go down inside U, and continue forever; this cannot stop while in U because U has no local minimum, hence must go all the way to its boundary. The actual proof requires quite some ingenuity, and builds the desired arc chain as a maximal chain for an ordering that is itself minimal in some well-crafted family, using Zorn’s Lemma twice. Read the full post.

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Arcs and arc chains II: every arc is an arc chain

Posted on February 20, 2025 by jgl

Last time, we have seen that every arc chain is an arc. This month, we will see that every arc is an arc chain. The proof proceeds by inventing a suitable partial ordering, the cut-point ordering. We will also see that all second-countable arcs (or arc-chains) are isomorphic, in particular they are all isomorphic to [0, 1], and that there are non-second-countable arcs (or arc-chains) of the same cardinality that are not isomorphic. Read the full post.

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Arcs and arc chains

Posted on January 20, 2025 by jgl

Jimmie Lawson had presented some results about chains in partially ordered spaces at the ISDT conference in 2022, which I wanted to talk about. But this requires Koch’s arc theorem, so I decided to talk about the latter first. That theorem is about arcs, and I decided to explain a few basic things about arcs, and their close cousins, the arc chains. This topic expanded so much that, in the end, this is the only thing I will talk about this month. In short: every arc chain is an arc, and arc chains can be characterized purely order theoretically. Read the full post.

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When is ℒ(X) locally compact? II. The non-consonant case and the non-second-countable case

Posted on December 20, 2024 by jgl

Last month, we have seen that ℒ(X) is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact) if and only if OX is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact), and that those properties are equivalent to X being core-compact, provided that X is a poset with its Scott topology or a ⊙-consonant second-countable space. Matthew de Brecht observed that the latter condition can be relaxed. We will see that we only need X to be second-countable or consonant for the equivalence to hold. Read the full post.

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When is ℒ(X) locally compact?

Posted on November 21, 2024 by jgl

Let ℒ(X) be the dcpo of all lower semicontinuous maps from a space X to R+ ∪ {∞}. We give it the Scott topology. Under which conditions on X is it locally compact? We will see that this happens exactly when X is core-compact, in two situations: when X is a poset with its Scott topology, and when X is an ⊙-consonant second countable space. I don’t know anything outside of these two cases. The case of posets has a remarkably short and clever proof, due to Xiaodong Jia. The case of ⊙-consonant second countable spaces has a much more tortuous proof, and relies on deep properties of hyperspaces discovered by Matthew de Brecht and Tatsuji Kawai. Read the full post.

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Compact closed subsets in the patch topology

Posted on October 20, 2024 by jgl

Given a coherent, well-filtered space X, one can give a pretty explicit description of the compact closed sets in Xpatch, the space obtained by equipping X with its patch topology—and we do not need local compactness or compactness. They are exactly the non-empty intersections of finite unions of lenses. This wonderful result lies somewhere in a paper on measure extension theorems for T0 space of Klaus Keimel and Jimmie Lawson, and has a very clever proof, which J. Lawson sent me recently. Read the full post.

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Posets determined by countably many core-compact subspaces

Posted on September 21, 2024 by jgl

Let me return to a topic that I have addressed a few times in recent posts: under which conditions does the Scott topology of the product of two posets coincide with the product topology of the Scott topologies? We have seen quite a few cases already, and this month we will see that this holds for ccω-posets, namely posets whose Scott topology is determined by the Scott topology of a countably ascending sequence of sub-d-posets, each of them being core-compact in its Scott topology. The construction is due to Lawson and Xu. The ccω-posets are exactly certain forms of colimits of posets in the category Posσ of posets and Scott-continuous maps, and the whole point is that not only are those colimits also colimits in Top, but those form colimits in Top over which products distribute (see the post of previous month). Read the full post.

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When do products distribute over colimits in Top?

Posted on August 20, 2024 by jgl

In a recent paper, Lawson and Xu give a new class of posets on which the Scott topology of the (poset) product of two posets coincides with the product topology (of each poset with its Scott topology). I will explain that more in detail next month. The following question is at the heart of their approach, and this is what I will focus on this month. Given two diagrams F and G in Top, we can form their colimits colim F and colim G. We can also consider the product diagram F × G. Is it true that colim (F × G) = colim F × colim G? We will see that this is rarely the case—I will give counterexamples—and that one of the most general situations where that is actually true is given by what I will call ω-rigid diagrams of core-compact spaces. Read the full post.

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Happy summer holidays 2024!

Posted on July 16, 2024 by jgl

I will not post anything this month, sorry: I am on holidays, starting in a few days, and for about one month. Said otherwise: I have tried finding some time to write up on something before leaving off, but I will not manage to do so this time. Hence there will be no serious July 20th, 2024 post, sorry. I will definitely try to post something on August 20th, and I already have something in mind.

I am taking the opportunity to say that, after more than 10 years of posting every month, I may start slowing down a bit. Each post takes me between three days and one week of work, and it is getting harder and harder to find the time to write them.

In the meantime, you might want to attend the 10th International Symposium on Domain Theory, which will take place in Qufu, Shandong, China, 2-5 August 2024. I won’t be there (remember, I will be on holidays), but many others will. Otherwise, let me wish you a happy summer holiday!

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The complete lattice Lfan (part II)

Posted on June 20, 2024 by jgl

Last time, we had started to study the complete lattice Lfan, namely just N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an extra top element ⊤. This is a complete lattice with the Chen-Kou-Lyu property, and we have seen a few consequences of that. This time, we will see that Lfan is a maximal limit space (in particular, it is sober and weakly Hausdorff); we knew that it is not core-compact, but it is consonant; we knew that it is not first-countable, but its lattice of open sets is second-countable in its Scott topology; and the upper Vietoris and Scott topologies coincide on its Smyth hyperspace. Read the full post.

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