Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University Mathematical Sciences Matematik Bornological quasi-metrizability in generalized topology https://orcid.org/0000-0002-7515-2418 Piękosz Artur Cracow University of Technology https://orcid.org/0000-0003-1864-2303 Wajch Eliza Siedlce University of Natural Sciences and Humanities 12 08 2019 48 6 1653 1666 05 27 2016 06 07 2018 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform quasi-metrizability with respect to a bornology is studied. The class of locally small spaces is considered and a possibly larger class of weakly locally small spaces is defined. The proofs and numerous examples are given in ZF. An example of a weakly locally small space which is not locally small is constructed under ZF+CC. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.

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