  We study topological aspects of the category of abstract Cuntz semigroups,
termed Cu. We provide a suitable setting in which we are able to uniformly
control how to approach an element of a Cu-semigroup by a rapidly increasing
sequence. This approximation induces a semimetric on the set of Cu-morphisms,
generalizing Cu-metrics that had been constructed in the past for some
particular cases. Further, we develop an approximate intertwining theory for
the category Cu. Finally, we give several applications such as the
classification of unitary elements of any unital AF-algebra by means of the
functor Cu.
