Some Properties of Ultrafilters of Widely Understood Measurable Spaces

Abstract

Filters and ultrafilters (maximal filters) of a space whose measurable structure is specified by a family of sets that is closed under finite intersections and contains the empty set and an ambient set (unity of the space) are studied. Necessary and sufficient conditions for filters of this space to be maximal are formulated in terms of sets—elements of a dual family, which are called quasi-neighborhoods. These conditions agree with ones known in the theory of Stone spaces, but cover a number of other cases, for example, the situation when the initial set is equipped with a topology (the case of open ultrafilters) or with a family of closed sets of a topological space (i.e., a closed topology in the sense of P.S. Aleksandrov). A key role in these constructions is played by the topology on a space of ultrafilters defined by analogy with the case of a Stone space. Additionally, the case when the above-mentioned measurable space is equipped with a topology admitting a conceptual analogy with the topology used to construct the Wallman extension is considered. As a result, we obtain a bitopological space with comparable topologies, one of which is Hausdorff and the other generates a compact \({{T}_{1}}\)-space. The conditions are indicated under which the topologies coincide, thus yielding a (zero-dimensional) compact set, and the conditions are given under which the topologies differ, thus defining a nondegenerate bitopological space. In the case where the family of sets defining a measurable structure is separable, it is shown that the initial ambient set can be embedded in the above bitopological space in the form of an everywhere dense subset.