# Contents

## Idea

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 $\id_S\colon S \to S$ (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 $\mathcal{P}S$, 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.

## Generalizations

The notion of improper subset could be generalized from Set to any category $C$ as the notion of improper subobject.

If a subobject of an object $A \in \mathrm{Ob}(C)$ is an isomorphism class of monomorphisms into $A$, then an improper subobject is of an object $A \in \mathrm{Ob}(C)$ is an isomorphism class of isomorphisms into $A$.