Bornological quasi-metrizability in generalized topology

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 $\mathbf{\text{ZF}}$. An example of a weakly locally small space which is not locally small is constructed under $\mathbf{\text{ZF+CC}}$. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.