To be able to consider convenient hulls of $(q)\cat{AUnif}$,

% bracket split

the category of \text{(quasi-)} approach uniform spaces 

and uniform contractions,

one needs to have (in particular finally dense) topological 

universe extensions of these categories available.

To this end, the categories $(q)\cat{SAUConv}$,

$(q)\cat{SAULim}$ and $(q)\cat{PsAULim}$ are introduced

and are shown to be topological universes

and appropriate generalizations (quantifications)

of the categories $(q)\cat{SUConv}$, $(q)\cat{SULim}$ and 

$(q)\cat{PsULim}$ introduced earlier by Preu\ss\ and Behling 

(\cite{Be 92}) (as extensions of $(q)\cat{Unif}$, 

the category of (quasi-)uniform spaces

and uniformly continuous maps), in the same way that

$(q)\cat{AUnif}$ generalizes $(q)\cat{Unif}$.

By also describing the final hulls of $(q)\cat{AUnif}$ 

in the previously mentioned topological universes,

some finally dense topological universe extensions are 

eventually obtained.