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Category Archives: Uncategorized
Ldomains, stable open sets, and stable Stone duality
Stone duality relates topological spaces and locales (or frames). But there are really many sorts of Stone dualities. In 1997, Yixiang Chen studied Stone dualities that relate socalled Ldomains to socalled distributive Dsemilattices. This was refined later in a common … Continue reading
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Tagged Ldomain, Stone duality
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Irredundant families, the Smyth powerdomain, the LyuJia theorem, and the baby Groemer theorem
A ∩semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its nonempty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give … Continue reading
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Tagged compactness, corecompactness, hyperspace, powerdomain
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Sheaves and streams II: sheafification, and stratified étale maps
In part I, I explained how one can build the étale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an étale map, leading to a nice adjunction and … Continue reading
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Tagged étale map, sheaf, stream
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Sheaves and streams I: sheaves of locally monotone maps
Sheaves are a fundamental notion. In this post and later posts, I would like to explain some of the basic theory of the most mundane notion of sheaves: sheaves of sets over a topological space. My real goal is really … Continue reading
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Tagged étale map, sheaf, stream
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Statures of Noetherian spaces I: maximal order types of wpos
I have had a very gifted masters 2 student from midMay to late July, Bastien Laboureix. He mostly solved the questions I had left open here. I wanted to report on his work, but that is a lot too technical … Continue reading
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Tagged noetherian, ordinal, wqo
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Topological functors I: definition, duality, limits and colimits
I have briefly mentioned topological functors in a recent post. It is time for me to explain what they are. This is a truly wonderful concept, which abstracts topological spaces away and concentrates on the key properties of the forgetful … Continue reading
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Tagged category theory, topological functor
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From prestreams to streams
Last time, I had described two satisfactory models of topological spaces with a local direction of time: Marco Grandis’ dspaces, and Sanjeevi Krishnan’s prestreams. The two kinds form categories that are related by an adjunction S ⊣ D, discovered by … Continue reading
Prestreams and dspaces
How do you model a topological space with a direction of time? That should seem easy; for example, a topological space with a preordering should be enough. But how do you model the directed circle, where times goes counterclockwise? That … Continue reading
Firstcountable spaces and their Smyth powerdomain
This month, we will look at certain conditions recently found by He, Li, Xi and Zhao in 2019, and then by Xu and Yang in 2021, in order to ensure that the Smyth powerdomain Q(X) (with the Scott topology) of … Continue reading
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Tagged firstcountability, powerdomain
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The Sorgenfrey line is not consonant
In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, Rℓ, is consonant. But the proofs I had in mind were much too simpleminded to stand any … Continue reading
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Tagged consonance, counterexample, powerdomain, valuation
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