Cauchy sequences in quasi-pseudo-metric spaces

Abstract

Abstract. This paper considers the problem of defining Cauchy sequence and completeness in quasi-pseudo-metric spaces. The definitions proposed allow versions of such classical theorems as the Baire Category Theorem, the Contraction Principle and Cantor's characterization of completeness to be formulated in the quasi-pseudo-metric setting.

Zabrejko's lemma and the fundamental principles of functional analysis in the asymmetric case

“…In the case of a quasi-metric space (X, ρ) there are several notions of Cauchy sequence and more notions of completeness, see [11]. We present only the following two notions of Cauchy sequence.…”

Fuzzy quasi-metrics for the Sorgenfrey line

“…Many authors have contributed to the study of completeness and completion of quasi-metrics based on the notion of Cauchy sequence, for instance [36,20,29,33,4,1,31,22,23], and in some cases this study has been extended to the fuzzy context [12,19]. Obviously, these studies are particularly interesting in the case of quasi-metrizable non metrizable topological spaces.…”

“…Then, we study the (sequential) completeness of these fuzzy quasi-metrics in the sense of Doitchinov [4,12] and the different notions of Cauchy sequence in [29]. We prove that all these fuzzy quasi-metrics are balanced and complete in the sense of Doichinov.…”

Quasi-pseudometric properties of the Nikodym-Saks space

“…Following the terminology introduced in [10], if (X, q) is a quasi-pseudometric space, a sequence (x n ) ∞ n=1 in X is said to be right-k-Cauchy whenever, given ε > 0, there is k ∈ N such that, for n ≥ m ≥ k, we have q(x n , x m ) < ε. We say that (X, q) is right-k-sequentially complete provided every right-k-Cauchy sequence converges.…”