Uniforme Konvergenz in topologischen Kategorien
by
Dieter Leseberg
FU Berlin + TU Braunschweig

It has been a trend in Categorical Topology to embed topological categories into topological universes, i.e. topological categories which are set-like enough to allow constructions of special objects such as function spaces or universal one-point extensions. The topological universe of " semiuniform convergence spaces" introduced by Preuss form a common generalization of symmetric limit spaces (and thus also symmetric topological spaces) and uniform limit spaces (and thus also uniform spaces). Moreover, this universe has the property that product of quotients are again quotients. Adamék and Reiterman had described the (cartesian closed topological) extension of the category UNIF of uniform spaces. This is the category of so called "bornological uniform spaces". These are uniform spaces endowed with a bornology, naturally related to the uniformity; the morphisms are uniformly continuous maps which preserve bounded sets. The hom-object HOM(A,B) of two uniform spaces A,B is their function space with the uniform convergence and the bornology of all equicontinuous sets.The " supertopologies" introduced by Doítchinov or, more generally, the "neighborhood structures" defined by Tozzi and Wyler, are a common generalization of topological and non-symmetrical proximities. But they fail to be cartesian closed. Now, we define and study the so called "filterunitopic spaces" which are a common generalization of all the above mentioned categories and, moreover, the antitonic ones are set-like enough to allow to construct function space structures (in a very natural way). We will show that the generalized convergence spaces in the sense of Kent are nothing else than special filterunitopic spaces.

http://at.yorku.ca/h/a/a/a/12.htm

Date received: January 27, 2000

Copyright © 2000 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 # caej-02.