About topological properties of attraction set in ultrafilter space

A. G. Chentsov

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 143, pp. 335-356

Résumé

The representation of attraction set (AS) in the class of nets in the ultrafilter space on the broadly understood measurable space (MS) with topologies of Stone and Wallman types is considered. Representation of the interior of AS and some of its implications are obtained. Possibilities of the choice of usual solutions are defined by specifying constraints of asymptotic nature (CAN). The mentioned CAN can be connected with weakening of standard constraints (in control problems, boundary and intermediate conditions, phase restrictions; in problems of mathematical programming, constraints of inequality type), but they may appear initially in the form of nonempty directed (usually) families of sets. In article, some set families connected with construction of ultrafilters (maximal filters) of MS majorizing a given a priory filter are treated as CAN. Shown, that in this case, under condition of the void intersection of all sets of the given filter, the resulting CAN variant is closed, but not canonically closed set for each of topologies Wallman and Stone types. This is connected with the fact established in the article that, for initial filter with property of the empty intersection of all its sets, the interior of generated by this filter AS is empty (at the same time, there are examples of control problems with opposite property: under empty intersection of sets for the family defining CAN, the interior of arising AS is not empty).

Détail

Keywords: interior, attraction set, ultrafilter