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- Hoover’s maximal limit spaces II: products, liftings, retracts, function spaces, and hyperspaces
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Author Archives: jgl
Hoover’s maximal limit spaces II: products, liftings, retracts, function spaces, and hyperspaces
Last time, we had introduced Hoover’s maximal limit spaces: spaces in which every convergent filter has a unique largest limit. That notion is closed under many constructions, as we will see: products, liftings, retracts, notably, and that is elementary. The … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, function space, hyperspace, sobriety, strong sobriety, weak Hausdorffness
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Hoover’s maximal limit spaces I: local strong sobriety, bounded sup-completeness and weak Hausdorffness
In 1995, Douglas Hoover introduced and studied a notion of maximal limit spaces: spaces in which every convergent net has a unique largest limit. This has connections with many other kinds of spaces that we have explored already, and I … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, sobriety, strong sobriety, weak Hausdorffness
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Skula spaces II: the Dow-Watson counterexample
Last time, we have started to explain some results due to A. Dow and S. Watson, and we have seen that every compact T0 scattered space of scattering height at most 3 is Skula, namely can be obtained from a … Continue reading
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Tagged scattered, skula topology
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Skula spaces I: clopen selectors
A Skula space is a space that is obtained from another space X by giving it the Skula topology instead, which is generated by the open sets and the closed sets of the original space X. In 1990, Alan Dow … Continue reading
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Tagged compactness, connectedness, noetherian, scattered, skula topology
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The (positive) unique structure theorem
There is an amazing theorem in the theory of analytic spaces, called the unique structure theorem: any two countable families F1 and F2 of Borel subsets of an analytic space X that separate the points of X generate the same … Continue reading
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Tagged Borel set, omega-continuous dcpo, quasi-analytic space, quasi-polish space, sigma-algebra, sigma-lattice
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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
Posted in Uncategorized
Tagged compactness, continuous dcpo, Hoare hyperspace, powerdomain, semilattice, Urysohn
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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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Tagged compact pospace, semilattice
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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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Tagged continuous lattice, exponentiability, locale
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