The structure of extended real-valued quasi-metric spaces

An extended quasi-metric q on a nonempty set X without any assumed structure is a distance
functional that satis es the usual properties of a quasi-metric except that it can assume
values of in nity, in addition to non-negative real values. Given a quasi-metrizable space
X we exhibit a universal space for all extended quasi-metric spaces compatible with the
asymmetric topologies of X. De ning a set in an extended quasi-metric space (X; q) to be
bounded if it is contained in an intersection of the left-q and right-q open (or closed)-balls, we
characterize these kinds of bornologies on X and, obtain necessary and su cient conditions
in order for the same bornologies to be realized as those for quasi-metrically bounded sets.
We also consider in this setting a second possible de nition of bounded sets involving quasicomponents.
Keywords: Quasi-metric, Extended real-valued quasi-metric, uniform equivalent quasimetrics,
Bounded set, Partial function, Bornology, Quasi-metric bornology, Quasi isometry,
Free union bitopology, Generalized Hus Theorem.