Let be a set. Then the improper subset of is itself, viewed as a subset of itself.
As the subsets of correspond to the predicates on and to the unary relations on , so the improper subset corresponds to the predicate that is always true and the relation which always holds.
As the subsets of correspond to the injections to (up to isomorphism in the slice category ), so the improper subset corresponds to the identity function (or to any bijection to , since these are all isomorphic in ).
The improper subset is the top element of the power set , 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.
Revised on August 9, 2010 22:40:52
by Toby Bartels