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Russia Yekaterinburg
Section Mathematics
Title Some ultrafilter properties connected with extension constructions
Author(-s) Chentsov A.G.a
Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa
Abstract General properties of ultrafilters of $\pi$-systems with zero and unit used under extension constructing for abstract attainability problems with the aim of estimation for attraction sets in topological space are considered. Possibilities of employment of the above-mentioned ultrafilters as general elements are considered. Among them, elements admissible with respect to constraints of asymptotic character of the initial problem are selected. Under very general conditions, the goal operator of the given problem extends to the continuous mapping that takes each ultrafilter of $\pi$-system to the limit of corresponding image. The basic attraction set (an asymptotic analog of the attainability domain) is estimated from below by the continuous image of an analogous auxiliary set in the space of ultrafilters. In the particular case of realization of the Stone space (when the used $\pi$-system is an algebra of sets) the above-mentioned estimate is an equality connecting a desired attraction set and an auxiliary one; for the latter a sufficiently simple representation is given. The variant of application (in estimating goals) of the Wallman extension is discussed.
Keywords attraction set, topology, ultrafilter
UDC 519.6
MSC 28A33
DOI 10.20537/vm140108
Received 15 January 2014
Language Russian
Citation Chentsov A.G. Some ultrafilter properties connected with extension constructions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 87-101.
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