Abstract
We give a point-free characterisation of Bishop compact metric spaces in terms of formal topology. We show that the notion of overt compact enumerably completely regular formal topology is a point-free counterpart of that of Bishop compact metric space. Specifically, a formal topology is isomorphic to an overt compact enumerably completely regular formal topology if and only if it is isomorphic to the image of a compact metric space under the localic completion of metric spaces into formal topologies. The result is obtained in Bishop constructive mathematics with the axiom of Dependent Choice.
Highlights
Bishop [2] developed a large body of analysis constructively, but he did not develop general topology beyond the theory of metric spaces
The study of general topology in a constructive setting was initiated by Sambin [15], when he introduced the notion of formal topology, a point-free approach to general topology based on the impredicative theory of locale (Johnstone [11])
Formal topologies and formal topology maps form a category, which we denote by FTop
Summary
Bishop [2] developed a large body of analysis constructively, but he did not develop general topology beyond the theory of metric spaces He found it difficult to find a useful topological notion of compactness which is compatible with the corresponding metric notion defined by completeness and totally boundedness. In this paper, building on these previous works, we characterise the image of compact metric spaces under the Palmgren’s embedding (ie a full and faithful functor) in terms of formal topology. We obtain a preliminary characterisation of formal topologies that are isomorphic to the image of some compact metric space under the localic completion (Corollary 4.17).
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