Attraction sets in the abstract problem of reachability in topological space

Abstract reachability problems with constraints of asymptotic nature (CAN) are considered; for these problems, the result (analog of reachability set) is defined every time in the form of attraction set (AS) in topological space. The CANs themselves are generated by non-empty families of subsets of the original set of ordinary (available for implementation) solutions. Among these families, filters are distinguished: the family of all possible AS is realized by adding an empty set to a similar family of AS corresponding to each CAN generated by a filter; in addition, a unique attraction element is assigned to ultrafilters each time. This allows us to establish a number of important properties of the family of all AS generated by filters. So, in particular, it is established that given family is closed under finite unions; conditions are indicated under which a finite union of filters generates an AS of the above-mentioned family. A family of singletons, which are AS generated by filters, is indicated. The very appearance of non-empty non-singletonic AS can be interpreted in terms of the non-maximality of the filter generated the CAN: nonempty ASs that are not singletons necessarily correspond to CAN generated by filters that are not ultrafilters.