Asymmetric filter convergence and completeness

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. Quasi-uniformities and quasi-nearnesses on biframes provide appropriate structures with which to investigate uniform and nearness ideas in the asymmetric context. In [9] a notion of completeness (called “quasi-completeness”) was presented for quasi-nearness biframes in terms of suitable strict surjections being isomorphisms, and a quasi-completion was constructed for any quasi-nearness biframe. In this paper we show that quasi-completeness can indeed be viewed in terms of the convergence of certain filters, namely, the regular Cauchy bifilters. We use the notion of a T -valued bifilter, which generalizes the characteristic function of a filter. An important tool is an appropriate composition for such bifilters. We show that the right adjoint of the quasi-completion is the universal regular Cauchy bifilter and use it to prove this characterization of quasi-completeness. We also construct the so-called Cauchy filter quotient for a biframe using a quotient of the downset biframe that involves only the Cauchy, and not the regularity, condition. Like the quasi-completion, this provides a universal Cauchy bifilter; unlike the quasi-completion, this construction is functorial.

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

Quaestiones Mathematicae 36(2013), 291–308