Scott locales
Pedro Resende, Jo\~ao Paulo Santos

TL;DR
This paper investigates Scott locales, a special class of topological locales with properties related to the Scott topology, primes, and preregular locales, with implications for topology and cognitive science.
Contribution
It introduces Scott locales, explores their properties, and establishes connections between their spectrum, preregularity, and classical topological notions.
Findings
Spectrum of a Scott locale is T1.
Preregulary locales are Scott locales.
Characterization of sober spaces as Hausdorff via Scott locales.
Abstract
We prove some facts about locales equipped with the Scott topology , in particular studying a canonical frame homomorphism which is motivated by an application to cognitive science. Such a topological locale is called a Scott locale if the inclusion of primes is continuous. We prove that the spectrum of a Scott locale is necessarily , and that preregular locales (a generalization of regular locales) are Scott locales. If is the topology of a topological space we find a (necessarily unique) continuous map such that and compare it with the points-to-primes map , showing that if and only if is preregular, and that a sober space is Hausdorff if and only if is and .
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras
