Some questions connected with implementation of attraction sets accurate to a predetermined neighborhood

Questions connected with implementation of attraction sets (AS) in attainability problem with constraints of asymptotic nature (CAN) are considered. It is investigated the possibility of AS implementation accurate to arbitrary neighborhood in class of closures of attainability sets corresponding to concrete sets from the family generating CAN. Moreover, some relations for AS generated by different CAN are considered (disjunction conditions of AS are investigated). General constructions of neighborhood implementation of AS were applied in the case when these AS were considered in the space of ultrafilters of broadly understood measurable space (MS). In particular, the case when CAN are defined by a filter was investigated in detail; for this case, under non-restrictive conditions on the original MS, the set of ultrafilters majorizing the original filter is implemented as AS. In this case (of ultrafilter space) variants of equipment of ultrafilter set with topologies of Stone and Wallman types are investigated separately.