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Category Archives: Uncategorized
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’ d-spaces, and Sanjeevi Krishnan’s prestreams. The two kinds form categories that are related by an adjunction S ⊣ D, discovered by … Continue reading
Prestreams and d-spaces
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
First-countable 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 first-countability, 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 simple-minded to stand any … Continue reading
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Tagged consonance, counterexample, powerdomain, valuation
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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 order-preserving maps) … Continue reading
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Tagged bitopological space, duality, stably compact space
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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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Tagged counterexample, powerdomain
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Quasi-continuous domains and the Smyth powerdomain
“Quasi-continuous 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 quasi-continuous domains in this post. Rather, I will mention three pearls that this paper … Continue reading
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Tagged powerdomain, quasi-continuous domain
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Quasi-uniform spaces IV: Formal balls—a proposal
Formal balls are an extraordinarily useful notion in the study of quasi-metric, and even hemi-metric spaces. Is there any way of extending the notion to the case of quasi-uniform spaces? This is what I would like to start investigating. This … Continue reading
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Tagged formal ball, quasi-uniform space
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Quasi-uniform spaces III: Smyth-completeness, symcompactness
We embark on the study of notions of completeness for quasi-uniform spaces, and we concentrate on Smyth-completeness. We will see that at least two familiar theorems from the realm of quasi-metric spaces generalize to quasi-uniform spaces: all Smyth-complete quasi-uniform spaces … Continue reading
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Tagged completeness, quasi-uniform space
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