Equivalence of Definitions of Exterior

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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:

\(\displaystyle H^e\) \(=\) \(\displaystyle T \setminus H^-\)
\(\displaystyle \) \(=\) \(\displaystyle \left({T \setminus H}\right)^\circ\) Complement of Interior equals Closure of Complement

Thus:

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

$\blacksquare$


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