Author Archives: jgl

A core-compact, non-locally compact space

Last time, I had announced that we would do Exercise V-5.25 of the red book, constructing a core-compact, yet not locally compact, space.  And this is exactly what we shall do: read the full post.

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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 … Continue reading

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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 … Continue reading

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The locale of random elements of a space

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 … Continue reading

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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 … Continue reading

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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 … Continue reading

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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 … Continue reading

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Another form of Stone duality

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 … Continue reading

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Dcpos and convergence spaces II: preserving products

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 … Continue reading

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Dcpos and convergence spaces I: Scott and Heckmann convergences

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 … Continue reading

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