Corpus ID: 235266073

Uniform locales and their constructive aspects

@inproceedings{Manuell2021UniformLA,
  title={Uniform locales and their constructive aspects},
  author={Graham Manuell},
  year={2021}
}
Much work has been done on generalising results about uniform spaces to the pointfree context. However, this has almost exclusively been done using classical logic, whereas much of the utility of the pointfree approach lies in its constructive theory, which can be interpreted in many different toposes. Johnstone has advocated for the development of a constructive theory of uniform locales and wrote a short paper on the basic constructive theory via covering uniformities, but he never followed… Expand

References

SHOWING 1-10 OF 23 REFERENCES
Localic completion of uniform spaces
  • Tatsuji Kawai
  • Mathematics, Computer Science
  • Log. Methods Comput. Sci.
  • 2017
TLDR
It is shown that the localic completion of a symmetric gus is equivalent to the point-free completion of the uniform formal topology associated with the gus, which simplifies the existing characterisation for metric spaces. Expand
A structural investigation on formal topology: coreflection of formal covers and exponentiability
TLDR
The coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact openLocales are exponentiable in the category ofopen locales. Expand
Quantalic spectra of semirings
Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a localic semiring R (the pointfree analogue of a topological semiring), weExpand
Uniform Spaces, I
Publisher Summary Uniform spaces can be defined in various equivalent ways. Every uniform space carries a natural topology; it is defined using neighborhood bases. A uniformly continuous map is alsoExpand
Frames and Locales - Topology without points
TLDR
This chapter discusses the structure of localic morphisms in Spaces and lattices of open sets, which describes the construction of frames and locales in the real world. Expand
The congruence biframe as a quasi-uniform bicompletion
Abstract Kunzi and Ferrario have shown that a T 0 space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictlyExpand
Localic Metric spaces and the localic Gelfand duality
In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localicExpand
The Stone–Čech compactification of locales, III
The Stone-Cech compactification of a locale L is shown to be obtained constructively by taking the Lindenbaum locale of the theory of almost prime completely regular filters on L. Modifying theExpand
On the completeness of localic groups
The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number ofExpand
On the collection of points of a formal space
  • G. Curi
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 2006
Abstract The collection of points of a locally compact regular formal space is shown to be isomorphic to a set in the context of Martin-Lof type theory. By introducing the notion of uniform formalExpand
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