Locales as spaces in outer models

TL;DR

This paper extends Zapletal's framework to interpret locales, a generalization of topological spaces, within set theory models, ensuring that localic properties correspond precisely to their interpretations in the ambient universe.

Contribution

It develops a new interpretation method for locales that preserves key properties and aligns localic notions with their set-theoretic interpretations, improving upon previous models.

Findings

Localic products interpret as spatial products.

Locales' properties like $T_U$, $T_1$, $T_2$, regularity, and compactness match their interpretations.

The extended framework behaves well on a broad class of locales.

Abstract

Let M be a transitive model of set theory and X be a space in the sense of M. Is there a reasonable way to interpret X as a space in V? A general theory due to Zapletal provides a natural candidate which behaves well on sufficiently complete spaces (for instance \v{C}ech complete spaces) but behaves poorly on more general spaces - for instance, the Zapletal interpretation does not commute with products. We extend Zapletal's framework to instead interpret locales, a generalization of topological spaces which focuses on the structure of open sets. Our extension has a number of desirable properties; for instance, localic products always interpret as spatial products. We show that a number of localic notions coincide exactly with properties of their interpretations; for instance, we show a locale is $T_U$ if and only if all its interpretations are $T_1$, a locale is $I$-Hausdorff if and only if all its interpretations are $T_2$, a locale is regular if and only if all its interpretations are $T_3$, and a locale is compact if and only if all its interpretations are compact.