Many kinds of structures are given by families of subsets of a given set: Moore collections (df-mre), topologies (df-top), pi-systems, rings of sets, delta-rings, lambda-systems/Dynkin systems, algebras/fields of sets, sigma-algebras/sigma-fields/tribes (df-siga), sigma rings, monotone classes, matroids/independent sets, bornologies, filters.
There is a natural notion of structure induced on a subset. It is often given by an elementwise intersection, namely, the family of intersections of sets in the original family with the given subset. In this subsection, we define this notion and prove its main properties. Classical conditions on families of subsets include being nonempty, containing the whole set, containing the empty set, being stable under unions, intersections, subsets, supersets, (relative) complements. Therefore, we prove related properties for the elementwise intersection.
We will call the elementwise intersection on the family by the class .
REMARK: many theorems are already in set.mm: "MM> SEARCH *rest* / JOIN".