Open AccessSome properties of ultrafilters of widely understood measurable spaces
Author(s) -
А. Г. Ченцов
Publication year - 2019
Publication title -
doklady akademii nauk. rossijskaâ akademiâ nauk
Language(s) - English
Resource type - Journals
ISSN - 0869-5652
DOI - 10.31857/s0869-5652486124-29
Subject(s) - mathematics , network topology , degeneracy (biology) , space (punctuation) , set (abstract data type) , pure mathematics , topology (electrical circuits) , nowhere dense set , character (mathematics) , unit (ring theory) , ultrafilter , zero (linguistics) , discrete mathematics , combinatorics , computer science , geometry , bioinformatics , mathematics education , biology , programming language , operating system , linguistics , philosophy
Filters and ultrafilters (maximal filters) on the π-system with “zero” and “unit” are considered (here, π-system is a nonempty family of sets closed with respect to finite intersections); so, our π-system contains including and empty sets. Characteristic properties of ultrafilters obtained from special representations for bases of two typical topologies connected with construction of Wallman extension and Stone compactums are investigated. The topology of Wallman type on the ultrafilters set of arbitrary π-system with “zero” and “unit” is defined. In addition, initial set is transformed in a compact T1-space with points in the form of ultrafilters of above-mentioned π-system. Under equipment of the resulting ultrafilter set with two topologies (by sense, Stone and Wallman topologies), bitopological space with comparable topologies is obtained; for this space, the degeneracy (in the sense of coincidence for above-mentioned topologies) and nondegeneracy conditions are indicated. The initial set is immersed in above-mentioned bitopological space as everywhere dense subset. Resulting construction is oriented on application in extensions of abstract attainability problems with constraints of asymptotic character (we keep in mind the possible application of ultrafilters as generalized elements).