2025/11/18 by Resende, Pedro, Santos, João Paulo
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2511.14892
We prove some facts about locales L equipped with the Scott topology Ω(L), in particular studying a canonical frame homomorphism ϕ:Ω(L)→ L which is motivated by an application to cognitive science. Such a topological locale L is called a Scott locale if the inclusion of primes p:Σ(L)→ L is continuous. We prove that the spectrum Σ(L) of a Scott locale L is necessarily T1, and that preregular locales (a generalization of regular locales) are Scott locales. If L is the topology of a topological space X we find a (necessarily unique) continuous map f:X→ L such that f-1=ϕ and compare it with the points-to-primes map p:X→ L, showing that f=p if and only if X is preregular, and that a sober space X is Hausdorff if and only if X is T1 and f(X)⊆Σ(L).