Ultrafilters and maximal linked systems: basic properties and topological constructions

Ultrafilters (maximal filters) and maximal linked systems on $\pi$-systems with “zero” and “unit” are considered. Different variants of topological equipment and the resulting bitopological spaces are discussed. It is noted that the bitopological space of ultrafilters can be considered as a subspace of the bitopological space of the maximal linked systems. Necessary and sufficient conditions for maximality of the filters and the properties characterizing maximal linked systems which are not ultrafilters are established. Some conditions sufficient for existence of such systems are clarified. Conditions under which bitopological spaces of ultrafilters and maximal linked systems are degenerate (topologies defining the corresponding bitopological space coincide) and the conditions that guarantee nondegeneracy are found. A new variant of the density property of the initial set in the ultrafilter space with topology of Wallman type is given. This variant can be used in constructing extensions for abstract attainability problems with asymptotic constraints.