Tag Archives: Scott topology

When is ℒ(X) locally compact?

Posted on November 21, 2024 by jgl

Let ℒ(X) be the dcpo of all lower semicontinuous maps from a space X to R+ ∪ {∞}. We give it the Scott topology. Under which conditions on X is it locally compact? We will see that this happens exactly … Continue reading →

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The complete lattice Lfan (part II)

Posted on June 20, 2024 by jgl

Last time, we had started to study the complete lattice Lfan, namely just N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an … Continue reading →

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On products of dcpos, the Miao-Xi-Li-Zhao lemma, and the complete lattice Lfan (part I)

Posted on May 20, 2024 by jgl

The product of a poset P with itself can be given two topologies: the Scott topology of the product, or the product of the Scott topologies. Those two topologies differ in general, but they coincide when P is a continuous … Continue reading →

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When do the upper (a.k.a., lower Vietoris) and Scott topologies coincide on the Hoare hyperspace of a space?

Posted on March 20, 2022 by jgl

I would like to talk about a nifty, recent result due to Yu Chen, Hui Kou, and Zhenchao Lyu. There are two natural topologies on the Hoare hyperspace of a space X, the Scott and the lower Vietoris topology, and … Continue reading →

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