Equivalence of Definitions of Exterior

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

The two definitions of the exterior of $H$:
 * $(1): \quad H^e$ is the complement of the closure of $H$ in $T$
 * $(2): \quad H^e$ is the interior of the complement of $H$ in $T$

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 $T$.