Scott locales

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 $L$ equipped with the Scott topology $\Omega(L)$, in particular studying a canonical frame homomorphism $\phi:\Omega(L)\to L$ which is motivated by an application to cognitive science. Such a topological locale $L$ is called a Scott locale if the inclusion of primes $p:\Sigma(L)\to L$ is continuous. We prove that the spectrum $\Sigma(L)$ of a Scott locale $L$ is necessarily $T_1$, 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 $f:X\to L$ such that $f^{-1}=\phi$ and compare it with the points-to-primes map $p:X\to L$, 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 $T_1$ and $f(X)\subseteq\Sigma(L)$.