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Category Archives: Uncategorized
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
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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
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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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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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Bitopological spaces and stable compactness
A while back (in March 2019, to be precise), Tomáš Jakl told me that he had a nice, short proof of the fact that the categories of stably compact spaces (and perfect maps) and compact pospaces (and continuous orderpreserving maps) … Continue reading
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Plotkin’s powerdomain and the hedgehog
There are three classical powerdomains in domain theory, named after Hoare, Smyth, and Plotkin. The first two are natural and well studied, and the third one is intricate and intriguing. To start with, there are several possible definitions for a … Continue reading
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Quasicontinuous domains and the Smyth powerdomain
“Quasicontinuous domains and the Smyth powerdomain” is the title of a very nice 2013 paper by Reinhold Heckmann and Klaus Keimel. I will not talk about quasicontinuous domains in this post. Rather, I will mention three pearls that this paper … Continue reading
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Quasiuniform spaces IV: Formal balls—a proposal
Formal balls are an extraordinarily useful notion in the study of quasimetric, and even hemimetric spaces. Is there any way of extending the notion to the case of quasiuniform spaces? This is what I would like to start investigating. This … Continue reading
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Quasiuniform spaces III: Smythcompleteness, symcompactness
We embark on the study of notions of completeness for quasiuniform spaces, and we concentrate on Smythcompleteness. We will see that at least two familiar theorems from the realm of quasimetric spaces generalize to quasiuniform spaces: all Smythcomplete quasiuniform spaces … Continue reading
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Quasiuniform spaces II: Stably compact spaces
There is a standard result in the theory of uniform spaces that shows (again) how magical compact Hausdorff space can be: for every compact Hausdorff space X, there is a unique uniformity that induces the topology of X, and its … Continue reading
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