On bornology of extended quasi-metric spaces

Beer studied the structure of sets equipped with the extended metrics with a focus on bornologies. In the paper [A. Piekosz and E. Wajch, Quazi-metrizability of bornological biuniverses inZF, J. Convex Anal. 2015], Piekosz and Wajch extended the well-known Hu's Theorem on boundedness in a topological space (see [S.-T. Hu, Boundedness in a topological space, J. Math. Pures Appl. 1949 ]) to the framework of quasi-metric spaces. In this note, we continue the work of Piekosz and Wajch. We show that many results on bornology of extended metric spaces due to Beer do not use the symmetry axiom of the extended metric, with appropriate modifications they still hold in the context of extended T-0-quasi-metric spaces.