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Bornological quasi-metrizability in generalized topology. (English) Zbl 1488.54011

Summary: 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.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
54B30 Categorical methods in general topology
54E35 Metric spaces, metrizability
54E55 Bitopologies
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