nLab
improper subset

Let SS be a set. Then the improper subset of SS is SS itself, viewed as a subset of itself.

As the subsets of SS correspond to the predicates on SS and to the unary relations on SS, so the improper subset corresponds to the predicate that is always true and the relation which always holds.

As the subsets of SS correspond to the injections to SS (up to isomorphism in the slice category Set/SSet/S), so the improper subset corresponds to the identity function id S:SS\id_S\colon S \to S (or to any bijection to SS, since these are all isomorphic in Set/SSet/S).

The improper subset is the top element of the power set 𝒫S\mathcal{P}S, 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 (64.89.48.241)