A theory of generalized filters

Abstract

This manuscript is a collection of many of the known results in the theory of generalized filters (ß-filters) as well as an extension of some of the work in this area. The relation of ß-filters to compactifications and realcompactifications is given special attention. Special emphasis is also given to the concept of tracing. After the necessary preliminaries are disposed of, Section I motivates the study of ß-filters by discussing the collection of zero-sets and then the Wallman compactification. The concept of realcompactification is also mentioned. Section II introduces the concept of ß-filters and exhibits many of the elementary facts about these generalized filters. The beginning of Section III deals with the convergence of ß-filters. The Frink or Wallman-type compactification of Tychonoff space is presented in detail. The generalization of this method in constructing realcompactifications in Tychonoff spaces is also discussed. The section closes with the presentation of some tracing results. Finally , Section IV investigates the concept of ß-continuity.