Category Archives: Uncategorized

Compactly Choquet-complete spaces III: products and continuous almost open images

Posted on October 20, 2025 by jgl

We continue our study of the class of compactly Choquet-complete spaces. We will show two things. First, that this class is closed under finite and countable products, but not under arbitrary products. While the technique for finite products consists in … Continue reading →

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Compactly Choquet-complete spaces II: completely Baire spaces

Posted on September 20, 2025 by jgl

A completely Baire space is a space that is not only Baire, but whose closed subspaces are all Baire. We will continue our exploration of compactly Choquet-complete spaces, and we will see that they are all completely Baire. In fact, … Continue reading →

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Compactly Choquet-complete spaces I: LCS-complete and Gδ subspaces

Posted on August 20, 2025 by jgl

A Choquet-complete space is one in which player α has a winning strategy in the strong Choquet game. By winning, it is meant that the intersection of all the open sets played by α is non-empty. If you change the … Continue reading →

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Proper maps, quasi-adjoints and the Smyth hyperspace

Posted on July 20, 2025 by jgl

This month, I would like to talk about proper maps, a nifty equivalent characterization through something I call quasi-adjoints and which involves the Smyth hyperspace monad, with a few non-trivial applications. Read the full post.

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Bitopological spaces, d-frames, and Jung-Moshier duality

Posted on June 20, 2025 by jgl

Stone duality is an adjunction between the category Top of topological spaces and the category Loc of locales, namely the opposite of the category Frm of frames. Is there a similar-looking adjunction between the category biTop of bitopological spaces—namely, sets … Continue reading →

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All countable continuous dcpos are algebraic

Posted on May 20, 2025 by jgl

A little pearl this month: as the title says, all countable continuous dcpos are algebraic. Read the full post.

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Jimmie Lawson’s compact interval theorem

Posted on April 20, 2025 by jgl

We will see a relatively recent theorem of Jimmie Lawson’s: consider a locally compact space X with a semiclosed ordering ≤, and a closed connected chain C inside X; then every closed interval inside C is compact in X. The … Continue reading →

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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 … Continue reading →

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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 … Continue reading →

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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 … Continue reading →

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