Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

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.