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Journal of Convex Analysis 22 (2015), No. 4, 1041--1060
Copyright Heldermann Verlag 2015

Quasi-Metrizability of Bornological Biuniverses in ZF

Artur Piekosz
Institute of Mathematics, Cracow University of Technology, Warszawska 24, 31-155 Cracow, Poland

Eliza Wajch
Institute of Mathematics and Physics, Siedlce University of Natural Sciences and Humanities, 3 Maja 54, 08-110 Siedlce, Poland

Hu's metrization theorem for bornological universes is shown to hold in ZF and it is adapted to a quasi-metrization theorem for bornologies in bitopological spaces. The problem of uniform quasi-metrization of quasi-metric bornological universes is investigated and several relevant results independent of ZF are obtained. In particular, some theorems of Garrido and Meroņo that contain necessary and sufficient conditions for a bornology in a metric space to be uniformly metrizable are generalized to quasi-metric bornological universes in ZF, while some other theorems of Garrido and Meroņo are shown to fail in a model for ZF.

Keywords: Bornology, bitopological space, quasi-metric, ZF.

MSC: 54E55, 54A35; 54E35, 54E99

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