When is the bicompletion reflective with respect to a functorial quasi-uniformity?
by
G. C. L. Brümmer
University of Cape Town, South Africa

Let QU₀ be the category of quasi-uniform spaces with T₀ topology, Top₀ the category of T₀ topological spaces and T : QU₀ --> Top₀ the usual forgetful functor. For any Y in QU₀ let kY : Y --> KY denote its bicompletion. Let F : Top₀ --> QU₀ be any section of T , i.e. a functorial assignment of compatible quasi-uniformities to the T₀ topological spaces. An open question is:

Under what conditions on F is Tk_FX : X --> TKFX a reflection in Top₀ at every X in Top₀ ?

It is known that a large class of reflections in Top₀ are representable in the above form (more briefly, let us say ïn the form TkF "), and that, e.g., for the Pervin quasi-uniformity functor P, TkP is not a reflection. A recent joint result with Eraldo Giuli and David Holgate is:

TkF is a reflection if and only if F is finer than the wellmonotone quasi-uniformity functor and TkF is simple in the sense of [Cassidy, Hebert, Kelly 1985].

(The cited notion of simplicity is applicable to pointed endofunctors which are not necessarily reflections). The open question is thus reduced to the problem of characterizing those F for which TkF is simple. We discuss this problem and partial answers to it.

Date received: July 1, 1997

Copyright © 1997 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 # caao-65.