Interior is Subset of Exterior of Exterior
Jump to navigation
Jump to search
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
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