Uniformity, generalized from TYCH to TOP
by
Darrell W. Hajek
University of Puerto Rico

In 1974, Herrlich defined the category NEAR, of nearness spaces and uniformly continuous functions as a generalization of UNIF, the category of uniform spaces and uniformly continuous functions. (UNIF was, itself, a generalization of metric spaces and uniformly continuous functions.) Meanwhile, Castillo, Cuevas, DiCristina, Hajek, Perlis and Wilson (in various combinations at various times) were working at generalizing the idea of metric spaces. This work resulted in the category DST, of distance spaces and continuous functions. DST contains a full subcategory ZDST, of zeroed distance spaces and continuous functions, and the (metric) definition of uniform continuity can be applied without change to functions in ZDST, thus producing yet another category, ZUNIF, of zeroed distance spaces and uniformly continuous functions. The definitions of uniform continuity in NEAR and in ZUNIF are very different, but Castillo and Hajek have been able to show that they are at least consistent, in that NEAR can be embedded as a full subcategory of ZUNIF. The form that the definition of uniform continuity takes in ZUNIF suggests a very natural generalization which can be applied for all distance spaces. This produces another category, DUNIF, of distance spaces and uniformly continuous functions, and the category TOP, of all topological spaces, can be shown to have exactly the same relationship with DUNCIF as TYCH, the category of Tychonoff spaces has with UNIF.

Date received: February 9, 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 # caak-16.