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 $f(X)\subseteq\Sigma(L)$.