Idempotency of Extensions via the Bicompletion

Let Top0 be the category of topological T0-spaces, QU0 the category of quasi-uniform T0-spaces, T : QU0 → Top0 the usual forgetful functor and K : QU0 → QU0 the bicompletion reflector with unit k : 1 → K. Any T -section F : Top0 → QU0 is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF . The literature considers important T -sections F that enjoy all three, or just one, or none of these properties. It is known that T (K, k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T (K, k)F is the reflection to Fix(TkF ) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T (K, k)F is not idempotent, and a K-true F for which the finest associated T -preserving bireflector in QU0 is not stable under K. AMS (2000) Subject Classifications: 54B30, 54D35, 54E15, 18A40