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

Abstract

Abstract. The natural duality between "topological" and "regular," both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior of p-topological and p-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.

p‐regular Cauchy completions

“…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. This dual behaviour is shown in [11] to extend to the properties "p-regular" and "p-topological" in the convergence space setting. We begin this section by showing that properties are likewise dual in the setting of Cauchy spaces.…”

“…Additional information about p-regular and p-topological Cauchy and convergence spaces may be found in [4,10,11]. If (X, Ꮿ) is a Cauchy space, the associated convergence structure is denoted by q Ꮿ .…”

“…The discovery of a dual relationship between the properties "pregular" and "p-topological" in convergence space theory led to an investigation of p-topological convergence spaces [11] and p-topological Cauchy completions [10]. These, in turn, raised some questions about p-regular completion theory that had not been previously considered, such as the possible duality between p-regular and p-topological Cauchy spaces, the role of diagonal axioms in the study of p-regular completions, the existence of p-topological Reed completions, and the relevance of p-regular completions to the study of regular completions.…”

p-Regularity and p-Regular Modification in ⊤-Convergence Spaces

“…One approach is stated through a diagonal condition of filters [2,3], the other approach is represented through a closure condition of filters [4]. In [5,6], for a pair of convergence structures p, q on the same underlying set, Wilde-Kent-Richardson considered a relative regularity (called p-regularity) both from two equivalent approaches. When p = q, p-regularity is nothing but regularity.…”

“…When p = q, p-regularity is nothing but regularity. Wilde-Kent [6] further presented a theory of lower and upper p-regular modifications in convergence spaces. Said precisely, for convergence structures p, q on a set X, the lower (resp., upper) p-regular modification of q is defined as the finest (resp., coarsest) p-regular convergence structure coarser (resp., finer) than q.…”