Interior is Subset of Exterior of Exterior

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Let $H^e$ denote the exterior of $H$, and let $H^\circ$ denote the interior of $H$.

Then:

$H^\circ \subseteq \paren {H^e}^e$

Proof

\(\ds \paren {H^e}^e\)

\(=\)

\(\ds \paren {T \setminus H^e}^\circ\)

Definition of Exterior

\(\ds \)

\(=\)

\(\ds \paren {T \setminus \paren {T \setminus H^-} }^\circ\)

Equivalence of Definitions of Exterior

\(\ds \)

\(=\)

\(\ds \paren {H^-}^\circ\)

Relative Complement of Relative Complement

\(\ds \)

\(\supseteq\)

\(\ds H^\circ\)

Interior is Subset of Interior of Closure

$\blacksquare$

Also see

• Interior may not equal Exterior of Exterior

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