More on upper bicompletion-true functorial quasi-uniformities

Abstract

Let T : QU 0 → Top 0 denote the usual forgetful functor from the category of quasi-uniform T 0 -spaces to that of the topological T 0 -spaces. We regard the bicompletion reflector as a (pointed) endofunctor K : QU 0 → QU 0 . For any section F : Top 0 → QU 0 of T we consider the (pointed) endofunctor R = T K F : Top 0 → Top 0 . The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in Top 0 . An important known characterisation is that F is upper K-true iff the canonical embedding X → R X is an epimorphism in Top 0 for every X in Top 0 . We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T : QU 0 → Top 0 , we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX.