Bornological quasi-metrizability in generalized topology

TL;DR

This paper introduces and explores the concept of bornological quasi-metrizability in generalized topological spaces, providing theorems, examples, and categorical frameworks within ZF set theory.

Contribution

It develops the theory of bornological quasi-metrizability, including uniform cases, and introduces new classes of spaces with categorical structures in generalized topology.

Findings

Quasi-metrization theorems for generalized bornological universes

Construction of a weakly locally small space not locally small under ZF+CC

Definition of categories related to generalized bornological universes

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