@article{article_656663, title={Bornological quasi-metrizability in generalized topology}, journal={Hacettepe Journal of Mathematics and Statistics}, volume={48}, pages={1653–1666}, year={2019}, author={Piękosz, Artur and Wajch, Eliza}, keywords={Delfs-Knebusch generalized topological space,quasi-metric,bornology,topological category,ZF}, abstract={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.}, number={6}, publisher={Hacettepe University}