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.
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With respect and admiration to Professor Karol Baron on the occasion of his jubilee.
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Chrząszcz, K., Jachymski, J. & Turoboś, F. Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces. Aequat. Math. 93, 277–297 (2019). https://doi.org/10.1007/s00010-018-0597-9
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DOI: https://doi.org/10.1007/s00010-018-0597-9