Non-Hausdorff Topology and Domain Theory

Revisiting Stone duality for bitopological spaces

Posted on June 20, 2026 by jgl

Exactly one year ago, I presented a form of Stone duality for bitopological spaces due to Jung and Moshier, and refined by Jakl. This relied on a notion called d-frames, which are a pair of frames linked by a totality and a consistency relation. There are natural relations we would like to be able to express that apparently cannot be stated in the setting of d-frames, and I will show that replacing totality and consistency by a unique relation ⊢ cures this problem. This will give rise to a notion that I have decided to call 2-frames (d-frames and biframes already exist), and we will obtain an expected adjunction between the category of bitopological spaces and the opposite of the category of 2-frames, which is idempotent, and leads to a notion of 2-sobrification of bitopological spaces, and of 2-sober bitopological spaces. I will end this by giving a description of the free 2-frame on a pair of sets. Read the full post.

Posted in Uncategorized | Tagged 2-frame, bitopological space, d-frame, Stone duality | Comments Off

≪-separating domains and FS-domains

Posted on May 20, 2026 by jgl

Achim Jung’s category of FS-domains is one of the two maximal cartesian-closed categories of continuous dcpos. Very recently, Wei Luan proposed a variation on the notion, and which he calls ≪-separating domains. One of his main theorems is that given an FS-domain X and a ≪-separating domain Y, the space of Scott-continuous maps [X → Y] is a ≪-separating domain. Another one is that the same holds if instead X is assumed to be Noetherian. Read the full post.

Posted in Uncategorized | Tagged cartesian closeness, continuous dcpo, FS-domain, RB-domain | Comments Off

Choquet-Wilker spaces

Posted on April 20, 2026 by jgl

A Choquet-Wilker space is a space in which the collection of compact saturated subsets is rich: one where, given a compact saturated set Q included in a union U₁ ∪ U₂ of two open sets, Q is included in a union Q₁ ∪ Q₂ of two compact saturated sets, with Q₁ ⊆ U₁ and Q₂ ⊆ U₂. This condition is sometimes called Wilker’s condition, but dates back to (at least) Choquet. We will see that many topological spaces are Choquet-Wilker, but not all are: we will see that the Alexandroff compactification of the space Q of rational numbers is not Choquet-Wilker. I will conclude with a neat observation, due to M. de Brecht and K. Keimel, on the continuity of the binary union operation on the lattice of open sets. Read the full post.

Posted in Uncategorized | Tagged Alexandroff compactification, Choquet-Wilker space, consonance, KC-space, KT4 space, LCS-complete, locally compact | Comments Off

Appert space

Posted on March 20, 2026 by jgl

Some time ago, I was wondering about the various definitions of a Radon measure, and in particular about the relation between local finiteness and the requirement that the measure of every compact subset be finite. (I will explain everything.) The former implies the latter, but what about the latter? It turns out that the two are not equivalent, and this will be a convenient excuse to introduce a simple counterexample, due to Antoine Appert in 1934, for different reasons. Read the full post.

Posted in Uncategorized | Tagged counterexample, first-countability, zero-dimensional spaces | Comments Off

Characterizing non-ω-well-filtered spaces by forbidden subspaces

Posted on February 20, 2026 by jgl

In a 2023 paper, Hualin Miao, Xiaodong Jia, Chong Shen and Qingguo Li characterized ω-well-filtered spaces as those that do not contain any one of two very simple spaces, S₁ and S_D, as Skula-closed subspaces. This is part of a family of theorems characterizing classes of spaces through forbidden subspaces, and is, I think, a good first step in explaining Matthew de Brecht’s characterization of quasi-Polish spaces through forbidden subspaces, something I aim to be able to do someday. I will also take the opportunity to give a characterization of those first-countable spaces that are sober, due to Xiaoquan Xu, Chong Shen, Xiaoyong Xi, and Dongsheng Zhao in 2020. Read the full post.

Posted in Uncategorized | Tagged forbidden substructures, sobriety, well-filtered space | Comments Off

Stone duality for preordered topological spaces II. Ad-frames

Posted on January 20, 2026 by jgl

A preordered topological space is a topological space X with a preordering ≤. I wish to explain a form of Stone duality for such preordered topological spaces. The idea is pretty simple: last time, we have seen a form of Stone duality for preordered sets, based on the premises that a preordered set is a topological space with an Alexandroff-discrete topology; hence a preordered topological space is a bitopological space, where one of the topologies is Alexandroff-discrete… and we have a Stone-like duality for bitopological spaces, Jung-Moshier duality, based on d-frames, a concept that we have seen in the June 2025 post. Read the full post.