Abstract
Abstract A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.
Highlights
The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density
Quasi-metrics became a subject of intensive research in the context of functional analysis, topology and theoretical computer science
[3] was the first to give a definition of a Cauchy sequence in a quasi-metric space and the corresponding completeness as given in the items (1) below
Summary
The convergence of a sequence (xn) in a quasi-metric space X with respect to τρ (τρ ) is called left (right) ρ-convergence which means xn l-ρ. [3] was the first to give a definition of a Cauchy sequence in a quasi-metric space and the corresponding completeness as given in the items (1) below. In the case of a quasi-metric space, there are several completeness notions by asking that each corresponding Cauchy sequence converges with respect to the topologies τρ or τρ. We give some basic definitions and obtain some fundamental results related to statistical convergence on quasi-metric spaces. Some interesting results are obtained related to completeness (in some sense), compactness and precompactness in this setting by using these statistically Cauchy sequences. In the last part of the paper, some summability type theorems are presented
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