nLab
improper subset

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 $\mathrm{Set}/S$), so the improper subset corresponds to the identity function ${id}_S:S\to S$ (or to any bijection to $S$, since these are all isomorphic in $\mathrm{Set}/S$).

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.