Equivalence of Definitions of Exterior

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

The following definitions of the exterior of $H$ are logically equivalent:

Proof
Let $H^e$ be defined as:
 * $H^e$ is the complement of the closure of $H$ in $T$.

Then:

Thus:
 * $H^e$ is the interior of the complement of $H$ in $T$.