  We use the ultrafilter-convergence axiomatics for topological spaces to
motivate in detail a gentle categorical introduction, first to Barr's Set-based
relational T-algebras, and then to Burroni's T-preorders internal to a category
C, here called T-spaces in C, for a monad T on C that substitutes the
ultrafilter monad on Set. Within these settings one finds not only the notions
of compactness and Hausdorff separation, originally due to Manes, but also that
of complete regularity. Based on a somewhat hidden result by Burroni, the main
theorem of this paper establishes an external fibrational characterization of
the category of completely regular T-spaces with its reflexive subcategory of
compact Hausdorff T-spaces, under modest assumptions on C and T.
