  Bornological universes were introduced some time ago by Hu and obtained
renewed interest in recent articles on convergence in hyperspaces and function
spaces and optimization theory. One of Hu's results gives us a necessary and
sufficient condition for which a bornological universe is metrizable. In this
article we will extend this result and give a characterization of uniformizable
bornological universes. Furthermore, a construction on bornological universes
that the author used to find the bornological dual of function spaces endowed
with the bounded-open topology will be used to define realcompactness for
bornological universes. We will also give various characterizations of this new
concept.

General Topology