Equivalence of Definitions of Exterior

Theorem

The following definitions of the concept of Exterior in the context of Topology are equivalent:

Let $T$ be a topological space.

Let $H \subseteq T$.

Definition 1

The exterior of $H$ is the complement of the closure of $H$ in $T$.

Definition 2

The exterior of $H$ is the interior of the complement of $H$ in $T$.

Proof

Let $H^e$ be defined as:

$H^e$ is the complement of the closure of $H$ in $T$.

Then:

 $\displaystyle H^e$ $=$ $\displaystyle T \setminus H^-$ $\displaystyle$ $=$ $\displaystyle \paren {T \setminus H}^\circ$ Complement of Interior equals Closure of Complement

Thus:

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

$\blacksquare$