Abstract

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 Meroño 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 Meroño are shown to fail in a model for ZF.

Author information

Contact details are reproduced from the original publication and may be historical.

Artur Piękosz

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

pupiekos@cyfronet.pl

Eliza Wajch

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

eliza.wajch@wp.pl

Keywords

  • Bornology
  • bitopological space
  • quasi-metric
  • ZF

Mathematics Subject Classification

54E55, 54A35; 54E35, 54E99

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Suggested citation

A. Piękosz, E. Wajch. “Quasi-Metrizability of Bornological Biuniverses in ZF.” Journal of Convex Analysis 22 (2015), No. 4, 1041–1060. https://doi.org/10.68381/jca22055

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.