NDLI: Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Content Provider	Springer Nature Link
Author	Schauerte, Anneliese Frith, John
Copyright Year	2012
Abstract	Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.
Ending Page	701
Page Count	21
Starting Page	681
File Format	PDF
ISSN	09272852
e-ISSN	15729095
Journal	Applied Categorical Structures
Issue Number	6
Volume Number	21
Language	English
Publisher	Springer Netherlands
Publisher Date	2012-05-17
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) Extensions of spaces (compactifications, supercompactifications, completions, etc.) Frame Uniform structures and generalizations Quasi-nearness biframe Quasi-uniform biframe Order topology biframe Frames, locales Bitopologies Regular reduction Theory of Computation Strong Geometry Enough regular Cauchy bifilters Totally bounded Quasi-completion Quasi-complete Convex and Discrete Geometry Mathematical Logic and Foundations
Content Type	Text
Resource Type	Article
Subject	Theoretical Computer Science Algebra and Number Theory Computer Science

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