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Abstract
The work investigates the powerlocales (lower, upper, Vietoris) of localic completions of generalized metric spaces. The main result is that all three are localic completions of generalized metric powerspaces, on the Kuratowski finite powerset. This is a constructive, localic version of spatial results of Bonsangue et al. and of Edalat and Heckmann. As applications, a localic completion is always overt, and is compact iff its gener- alized metric space is totally bounded. The representation is used to discuss closed intervals of the reals, with the localic Heine-Borel Theorem as a consequence. The work is constructive in the topos-valid sense. 2000 Mathematics Subject Classification 54B20 (primary); 06D22, 03G30,54E50,03F60 (secondary)
Highlights
In the geometric approach to point-free topology, as outlined in some detail in [32], a prominent place in the reasoning style is occupied by the powerlocales
The present paper studies powerlocales when applied to localic completions [33] of metric spaces, and shows that the powerlocales too are localic completions, got by taking appropriate generalized metrics on the finite powersets of the original spaces
The description there is in terms of “spaces”, on the understanding that classical mathematicians can interpret it in the conventional way, while locale theorists can read in a localic interpretation as described above
Summary
In the geometric approach to point-free topology, as outlined in some detail in [32], a prominent place in the reasoning style is occupied by the powerlocales (or point-free hyperspaces). A locale is compact iff its upper powerlocale is colocal (has, in a certain universal sense, a top point) This makes it easy to characterize compactness of X in terms of a total boundedness property on the generalized metric space X. The second set of applications is to the real line R, as completion Q of the rationals Our techniques make it easy to define the closed interval [0, 1] as a point of the Vietoris powerlocale VR, and its compactness (the Heine–Borel Theorem) follows immediately from the way points of VX (for any locale X ) correspond to certain compact sublocales of X.
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