Localic completion of uniform spaces

TL;DR

This paper generalizes the localic completion process from metric spaces to uniform spaces, providing a constructive, point-free framework that extends classical results and simplifies existing characterizations.

Contribution

It introduces a functorial localic completion for uniform spaces, extending previous metric space results to a broader setting within constructive set theory.

Findings

The localic completion lifts uniform structures to point-free structures.

It provides a simplified cover characterization for locally compact uniform spaces.

The completion is equivalent to the point-free completion of the associated formal topology.

Abstract

We extend the notion of localic completion of generalised metric spaces by Steven Vickers to the setting of generalised uniform spaces. A generalised uniform space (gus) is a set X equipped with a family of generalised metrics on X, where a generalised metric on X is a map from the product of X to the upper reals satisfying zero self-distance law and triangle inequality. For a symmetric generalised uniform space, the localic completion lifts its generalised uniform structure to a point-free generalised uniform structure. This point-free structure induces a complete generalised uniform structure on the set of formal points of the localic completion that gives the standard completion of the original gus with Cauchy filters. We extend the localic completion to a full and faithful functor from the category of locally compact uniform spaces into that of overt locally compact completely regular formal topologies. Moreover, we give an elementary characterisation of the cover of the localic completion of a locally compact uniform space that simplifies the existing characterisation for metric spaces. These results generalise the corresponding results for metric spaces by Erik Palmgren. Furthermore, we show that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus. We work in Aczel's constructive set theory CZF with the Regular Extension Axiom. Some of our results also require Countable Choice.