A THEORY OF GENERALIZED FILTERS.

This manuscript is a collection of many of the known results in the theory of generalized filters Sigma-filters as well as an extension of some of the work in this area. The relation of Sigma-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 Sigma-filters by discussing the collection of zero-sets and then the Wallman compactification. The concept of realcompactification is also mentioned. Section 2 introduces the concept of Sigma-filters and exhibits many of the elementary facts about these generalized filters. The beginning of Section 3 deals with the convergence of Sigma-filters. The Frink or Wallman-type compactification of a 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 4 investigates the concept of Sigma-continuity. Author