Extending constructions from the T₀-spaces to all topological spaces
by
G.C.L. Brümmer
University of Cape Town

It is commonly taken for granted that if one restricts one's attention from the topological spaces to the T₀-spaces, one loses no structural information. Conversely it is assumed that if one can do a construction for the T₀-spaces, then one can extend it to all topological spaces. However, uniqueness of the extension is not always automatically clear, even though one knows that Top is the MacNeille completion of Top₀.

Here we shall consider the construction of compatible quasi-uniformities on all topological spaces in such a way that continuous maps become quasi-uniform maps (i.e. uniformly continuous) with respect to the imposed quasi-uniformities. This means constructing a functor F from Top to the category QU of quasi-uniform spaces and quasi-uniform maps in such a way that TF is the identity functor on Top, where T : QU --> Top is the usual forgetful functor. We shall call such F a section (i.e. right inverse) of T. It is known that T has very many sections. Now let QU₀ denote the category of T₀ quasi-uniform spaces and quasi-uniform maps and consider the restriction of T, say T⁰ : QU₀ --> Top₀. Further let q_X : X --> QX denote the T₀-quotient of any quasi-uniform space X.

Lemma. Let F be a section of T : QU --> Top. If X is coarser than FTX, then QX is coarser than FTQX.

Theorem. Every section of T⁰ : QU₀ --> Top₀ extends uniquely to a section of T : QU --> Top. Thus we have a bijection (given by restriction) between the sections of T and the sections of T⁰.

The corresponding result is true for the forgetful functor Unif --> CregTop and its restriction Unif₀ --> Tych. We can even state and prove it for topological categories which are universal in the sense of Marny [2], meaning that Marny's T₀-quotient q_X is an initial map. However, the category UnifLim of uniform limit spaces is not universal [2].

The author acknowledges conversations about the above Theorem with Sergio Salbany more than twenty years ago. The reason for revisiting it now is that the author and H.-P.A. Künzi have applied the theorem to facilitate the proof of [1, Proposition 3.5].

[1] G.C.L. Brümmer and H.-P.A. Künzi, Bicompletion and Samuel bicompactification, Appl. Categ. Struct., to appear. http://www.sun.ac.za/maths/CatTop/Output

[2] Th. Marny, On epireflective subcategories of topological categories, General Topology Appl. 10 (1979), 175-181.

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

Date received: July 10, 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 # cagx-33.