Entourages, Density, Cauchy Maps, and Completion
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We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one).
KeywordsFrame Locale Sublocale Uniform frame Quasi-uniform frame Entourage Uniform map Uniform dense embedding Cauchy map Complete Completion
Mathematics Subject Classification06D22 18B30 54E15 54D35
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We are most grateful to the referee for all valuable comments and suggestions that improved the paper and, in particular, for alerting us for a mistake in the previous version of 6.2.1. Further, we acknowledge support from grants P202/12/G061 (Grant Agency of the Czech Republic) and MTM2015-63608-P (Ministry of Economy and Competitiveness of Spain) and from the Centre for Mathematics of the University of Coimbra (UID/MAT/00324/2013 funded by FCT/MCTES and FEDER through the Partnership Agreement PT2020).
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