Natural extensions of the T₀ spaces, their idempotency, and the quasi-uniform bicompletion.
by
Guillaume Brümmer
University of Cape Town
Coauthors: Hans-Peter Künzi (University of Cape Town)

We denote the usual forgetful functor from the category of T₀ quasi-uniform spaces to the category of T₀ topological spaces by T:QU₀ --> Top₀. By a T-section we mean any functor F : Top₀ --> QU₀ such that TF is the identity functor on Top₀. This means that F imposes compatible quasi-uniformities on the T₀ topological spaces in such a way that the continuous maps become quasi-uniformly continuous. There is a vast class of T-sections, including familiar constructions such as the fine, the fine transitive, the well-monotone, the semicontinuous, the point-finite, the locally finite or the Pervin quasi-uniformity.

The T₀ bicompletion of a T₀ quasi-uniform space will be denoted by k_Y : Y --> KY. Considering a given T-section F, we assign to each X in Top₀ the dense topological embedding Tk_FX : X --> TKFX, denoted for brevity by r_X : X --> RX with the endofunctor R = TKF : Top₀ --> Top₀.

The pair (R, r) is called idempotent iff r_RX : RX --> R² X is a homeomorphism for each X in Top₀. It is remarkable that this is the case iff (R, r) is a reflection, and implies that each r_X is b-dense, i.e. epi in Top₀. We shall give new results relating properties of F (such as KF being coarser or finer than FR) to properties of (R, r) (such as idempotency or the existence of a monad (R, r, \mu)). Background may be found in [1], [2].

[1] G.C.L. Brümmer, Categorical aspects of the theory of quasi-uniform spaces, Rend. Istit. Mat. Univ. Trieste, 30 Suppl. (1999), 45-74. Free online http://mathsun1.univ.trieste.it/Rendiconti/

[2] G.C.L. Brümmer and H.-P.A. Künzi, Bicompletion and Samuel bicompactification, Preprint (2001). http://www.sun.ac.za/maths/CatTop/Output/

http://www.sun.ac.za/maths/CatTop/Output/

Date received: May 25, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-84.