International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 1, Pages 1-12
-topological and -regular: dual notions in convergence theory
Department of Mathematics, Washington State University, Pullman 99164-3113, WA, USA
Received 3 July 1997; Revised 20 August 1997
Copyright © 1999 Scott A. Wilde 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 natural duality between topological and regular, both considered as convergence space properties, extends
naturally to -regular convergence spaces, resulting in the new concept of a -topological convergence space. Taking advantage of this duality, the behavior of -topological and -regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.