## Localic completion of generalized metric spaces II: Powerlocales (2009)

Citations: | 12 - 3 self |

### BibTeX

@MISC{Vickers09localiccompletion,

author = {Steven Vickers},

title = { Localic completion of generalized metric spaces II: Powerlocales},

year = {2009}

}

### OpenURL

### 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 generalized 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.

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Citation Context ...in other words that ⊥ is less than all generalized points. Hence ⊥ is a bottom point of X in a strong sense. (This is a special case of the concept of local topos, which has been studied in [11]; see =-=[10]-=-.) To prove that a locale is local, we shall normally give a geometric definition of the bottom point and show that it is less than every point. The geometricity allows us to deduce that the bottom po... |

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Citation Context ... is classically trivial but constructively important. 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 then its compactness (the Heine–Borel Theorem) follows immediately from the way points of VX (for any locale X) correspond to certain compact sublocales... |

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Citation Context ...ic dL on sets, and define an upper quasimetric dU(A, B) = sup b∈B infa∈A d(a, b) (which would equal m(B, A) if d were symmetric), then we can take dH(A, B) = max(dL(A, B), dU(A, B)). Bonsangue et al. =-=[2]-=- have already studied hyperspaces for Lawvere’s generalized metric spaces, in which the only assumptions are zero self-distance and the triangle inequality: the metric may be asymmetric, may take infi... |

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Citation Context ...topology on PX. This is given by a subbase of opens, comprising sets of the form □U = {A | A ⊆ U} ♦U = {A | A ∩ U ̸= ∅} where U is open in X. This Vietoris topology motivated Johnstone’s construction =-=[9]-=- of what we now call the Vietoris powerlocale. (It is also described in preliminary form – with an unnecessary restriction to the compact regular case – in [7], together with detailed historical notes... |

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