Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in [1] a completion (called a quasi-completion) was constructed for such structures and in [2] completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this talk, 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 quasiuniform. The talk will conclude with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.