Let S be a set. Then the improper subset of S is S itself, viewed as a subset of itself.
As the subsets of S correspond to the predicates on S and to the unary relations on S, so the improper subset corresponds to the predicate that is always true and the relation which always holds.
As the subsets of S correspond to the injections to S (up to isomorphism in the slice category Set/S), so the improper subset corresponds to the identity function id S:S→S (or to any bijection to S, since these are all isomorphic in Set/S).
The improper subset is the top element of the power set 𝒫S, viewed as a lattice.
The improper subset is so called because it is not a proper subset; by excluded middle, it is the only subset that is not proper.