More on upper bicompletion-true functorial quasi-uniformities

Let T:QU0→Top0 denote the usual forgetful functor from the category of quasi-uniform T0-spaces to that of the topological T0-spaces. We regard the bicompletion reflector as a (pointed) endofunctor K:QU0→QU0. For any section F:Top0→QU0 of T we consider the (pointed) endofunctor R=TKF:Top0→Top0. 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 Top0. An important known characterisation is that F is upper K-true iff the canonical embedding X→RX is an epimorphism in Top0 for every X in Top0. 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:QU0→Top0, 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.