Filters and linked families of sets

Abstract

Properties of ultrafilters (u/f) and maximal linked systems (MLS) on the widely understood measurable space (MS) and representations of linked (not necessarily maximal) families and filters on this MS are investigated. Conditions realizing maximality of linked families (systems) and natural representations for bitopological spaces (BTS) of u/f and MLS are established. Equipments of sets of linked families and filters corresponding to Wallman and Stone schemes are studied; the connection of these equipments with analogous equipments (with topologies) for u/f and MLS leading to above-mentioned BTS is studied too. Properties of linked family products for two (widely understood) MS are investigated. It is shown that MLS on the π-system product (that is, on the family of 'measurable' rectangles) are limited to products of corresponding MLS on initial spaces. © 2020 Udmurt State University. All rights reserved.