International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 497-509
-topological Cauchy completions
Department of Pure and Applied Mathematics, Washington State University, Pullman 99164-3113, WA, USA
Received 13 March 1998; Revised 15 June 1998
Copyright © 1999 J. Wig and D. C. Kent. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “-regular” and “-topological.” Since earlier papers have investigated regular, -regular, and topological Cauchy completions, we hereby initiate a study of -topological Cauchy completions. A -topological Cauchy space has a -topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing a -topological completion, it is shown that a certain class of Reed completions preserve the -topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsest -topological completions. However, not all -topological completions are Reed completions. Several extension theorems for -topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowing -topological and -topological completions, respectively, can always be extended to a -continuous map between any -topological completion of the first space and any -topological completion of the second.