# Category Archives: Uncategorized

## We’ve moved!

Dear all, the site has moved… there (follow the link). Sorry for the inconvenience. If you had subscribed before Monday, November 27th, 2023, 15h UTC+1, then you are still subscribed on the new site, and you have nothing else to … Continue reading

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## Compact semilattices without small semilattices II: Gierz’s counterexample

We pursue last month’s post, written with Zhenchao Lyu, and we describe Gierz’s example of a compact semilattice — namely, a compact Hausdorff topological semilattice — which does not have small semilattices. It is a bit simpler than Jimmie Lawson’s … Continue reading

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## Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace

I have already talked about compact semilattices before, but there is a lot more to say, especially on the subject of having small semilattices or not. Zhenchao Lyu is joining me this month, and we will pursue this next month. … Continue reading

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## Scott’s formula

There is a well-known formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f’ defined on the whole of X and … Continue reading

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## The fundamental theorem of compact semilattices

Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains. They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties. One characterization of bc-domains which I have not included in the book … Continue reading

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## Exponentiable locales II: the exponentiable locales are the continuous frames

Two months ago, we have seen that every exponentiable locale had to be continuous, as a frame. We will see that the converse holds: the continuous frames are exactly the exponentiable locales. The result is due to Martin Hyland in … Continue reading

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## The Banaschewski-Lawson-Ershov observation on separate vs. joint continuity

Joint continuity is a stronger property than separate continuity. In what cases are those properties equivalent? The question was solved, partially, by Yuri Ershov in 1997, and completely by Bernhard Banaschewski in 1977 (apparently with a gap in the proof) … Continue reading

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## Exponentiable locales I: every exponentiable locale is continuous

The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed … Continue reading

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## The Seminar on Continuity in Semilattices

Recently, Achim Jung sent me a message from Jimmie Lawson, and suggested that I might be interested in posting the information on this blog. The red book [1] is a precious source of information on domain theory, and if you … Continue reading

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## Topological Functors II: the Cartesian-closed category of C-maps

Some time ago, I gave an introduction to topological functors. They form a pretty brilliant categorical generalization of topological spaces. The point of today’s post is to give one particular example of the fact that you can somehow generalize some … Continue reading

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