Abstract

We give a characterisation of Bishop locally compact metric spaces in terms of formal topology. We identify inhabited enumerably locally compact regular formal topologies as point-free counterpart of Bishop locally compact metric spaces. Specifically, we show that a formal topology is isomorphic to an inhabited enumerably locally compact regular formal topology if and only if it is isomorphic to the localic completion of a Bishop locally compact metric space. The result is obtained in Bishop constructive mathematics with the Dependent Choice.

Highlights

  • In locale theory (Johnstone [13]), the standard adjunction between the category of topological spaces and that of locales restricts to an equivalence between the category of sober spaces and that of spatial locales

  • We introduce the notion of inhabited enumerably locally compact regular formal topology and show that the class of inhabited enumerably locally compact regular formal topologies characterises the image of Bishop locally compact metric spaces under the embedding M up to isomorphism

  • We show that the category of Bishop locally compact metric spaces is equivalent to the full subcategory of formal topologies consisting of those objects which are isomorphic to some inhabited enumerably locally compact regular formal topology (Theorem 7.6)

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Summary

Introduction

In locale theory (Johnstone [13]), the standard adjunction between the category of topological spaces and that of locales restricts to an equivalence between the category of sober spaces and that of spatial locales. Since the Fan theorem is not acceptable in Bishop constructive mathematics [3], the current situation prevents us from applying the results in formal topology to Bishop’s theory of metric spaces [3, Chapter 4]. In our previous work [14, Chapter 4], we characterised the image of the category of compact metric spaces under the embedding M using the notion of compact overt enumerably completely regular formal topology. To this end, we introduce the notion of the open complement of a located subtopology.

Formal topologies
Inductively generated formal topologies
Products
Open subtopologies and closed subtopologies
Localic completion of metric spaces
Open complements of located subtopologies
Enumerably completely regular formal topologies
Point-free one-point compactification
Point-free characterisation
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