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Open sublocales of localic completions

Palmgren

2010
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Journal of Logic and Analysis
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We give a constructive characterization of morphisms between open sublocales of localic completions of locally compact metric (LCM) spaces, in terms of continuous functions. The category of open subspaces of LCM spaces is thereby shown to embed fully faithfully into the category of locales (or formal topologies). 2 Erik Palmgren particular, that there is a bijection between the continuous maps X / / Y (i.e. locally uniformly continuous functions) and the continuous morphisms of formal
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... of formal topologies M(X) / / M(Y) (approximable mappings). To study point-free versions of topological manifolds it is of interest to characterize the maps between open sublocales of formal Euclidean spaces We consider a more general version of this problem where the Euclidean spaces have been replaced by localic versions of LCM spaces. In this paper we study the correspondence between maps and morphisms when the localic completions are restricted to open sublocales This correspondence is not trivial from a constructive point of view. As shown in [5] the maps M(X) / / M(R) |(0,∞) correspond to continuous functions X / / R that on each open ball has a positive uniform lower bound, rather than positive functions. Constructively, there is a distinction: Specker [7] gives a recursive example of a continuous positive function [0, 1] / / R that has no uniform positive lower bound. These considerations make it clear that the set of maps U * / / V * between open subspaces of LCM spaces has to meet some extra conditions to be in 1-1 correspondence to maps in (1). In Section 2 we introduce and study the appropriate categories of metric spaces, called OLCM and FLCM. In Section 3 the open sublocales of M(X) are studied. Section 4 establishes full and faithful functors OLCM / / FLCM / / FTop. The whole development is constructive in the sense of Bishop [1] and may be formalized within constructive set theory CZF with dependent choice, or in Martin-Löf type theory. Open subspaces of LCM spaces Bishop and Bridges [1] define a metric space X to be locally compact if it is inhabited and every bounded subspace is contained in a compact subspace. It follows that such a space X is complete (and separable). Below we define a category OLCM of open subspaces of locally compact metric (LCM) spaces. It is partly suggested by Definition 2.2.4 of [1], but its enunciation appears to be new. The category of open subspaces of LCM spaces is given as follows. The objects are pairs (X, U) where X = (X, d) is a LCM space and U is an open subset of X . A

doi:10.4115/jla.2010.2.1
fatcat:ey4uwfhpgzfylb7etlrepztdfm