Quasi-Metrizability of Bornological Biuniverses in ZF

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.