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:
\(\ds H^e\) | \(=\) | \(\ds T \setminus H^-\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {T \setminus H}^\circ\) | Complement of Interior equals Closure of Complement |
Thus:
- $H^e$ is the interior of the complement of $H$ in $T$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors