Regular convergence spaces

Probabilistic convergence spaces and generalized metric spaces

“…For x E X, denote by the fixed ultrafilter generated by The next proposition summarizes previously mentioned results pertaining to these axioms. The first assertion is proved in [14], the second in [1] and [5]. For all/2 6-U(X), R z for all U 6-/, there exists y 6-U such that -L z.…”

p‐regular Cauchy completions

“…A form of "duality" between the properties "regular" and "topological" in convergence space theory was first observed by Cook and Fischer [2] in 1967. Both properties have "diagonal" characterizations and if the concluding implication in the characterizations of either property is reserved, the resulting axiom characterizes the other property.…”

p‐topological and p‐regular: dual notions in convergence theory

“…We define a filter Ᏼ ∈ F(J) as follows: for each F ∈ Ᏺ, let H F = {(Ᏻ,y) ∈ J: F ∈ Ᏻ}, and let Ᏼ be the filter on J generated by Proof. The first assertion is proved in [6], the second by combining results from [2,1].…”