Original Articles

Asymmetric filter convergence and completeness

Published in: Quaestiones Mathematicae
Volume 36 , issue 2 , 2013 , pages: 291–308

DOI: 10.2989/16073606.2013.779972

Author(s): John Frith Department of Mathematics and Applied Mathematics, South Africa , Anneliese Schauerte Department of Mathematics and Applied Mathematics, South Africa

Keywords: Regular Cauchy bifilter , universal bifilter , convergent bifilter , quasi-complete , quasi-nearness biframe , downset biframe , congruence , nucleus , frame , Regular Cauchy bifilter , universal bifilter , convergent bifilter , quasi-complete , quasi-nearness biframe , downset biframe , congruence , nucleus , frame

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Abstract

Completeness for metric spaces is traditionally presented in terms of convergence of Cauchy sequences, and for uniform spaces in terms of Cauchy filters. Somewhat more abstractly, a uniform space is complete if and only if it is closed in every uniform space in which it is embedded, and so isomorphic to any space in which it is densely embedded. This is the approach to completeness used in the point-free setting, that is, for uniform and nearness frames: a nearness frame is said to be complete if every strict surjection onto it is an isomorphism.

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