Abstract
Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.
Highlights
The theory of metric spaces initiated in 1906 by Frechet [16] has developed into a huge branch of mathematics
As a simple consequence, we get a further generalization of Cantor’s theorem in which we allow sets to be non-closed. This enables us to derive from that result a quasimetric version of the Banach Contraction Principle established by Bakhtin [4]
A novelty here is that we provide quite a lot of error estimates for sequences of approximate fixed points and iterates of a mapping
Summary
The theory of metric spaces initiated in 1906 by Frechet [16] has developed into a huge branch of mathematics. The three axioms of a metric can be considered as three pillars of this theory Some of these conditions might be substituted by others or even omitted completely. The first section is devoted to introducing the notions used in the article, as well as recalling a few selected results in the field of semimetric spaces. The third section starts with establishing Cantor’s intersection theorem for semimetric spaces satisfying one of Wilson’s [27] axioms. As a simple consequence, we get a further generalization of Cantor’s theorem in which we allow sets to be non-closed This enables us to derive from that result a quasimetric version of the Banach Contraction Principle established by Bakhtin [4]. Some of them seem to be new even in a metric setting
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