Exterior of Exterior of Union of Adjacent Open Intervals
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Theorem
Let $A$ be the union of the two adjacent open intervals:
- $A := \openint a b \cup \openint b c$
Then:
- $A^{ee} = \openint a c$
where $A^e$ is the exterior of $A$.
Proof
By definition of exterior, $A^e$ is the complement relative to $\R$ of the closure of $A$ in $\R$.
Thus:
\(\ds A^{ee}\) | \(=\) | \(\ds \paren {\relcomp \R {A^-} }^e\) | Definition of Exterior | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\relcomp \R {\closedint a c} }^e\) | Closure of Interior of Closure of Union of Adjacent Open Intervals | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\openint \gets a \cup \openint c \to}^e\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp \R {\paren {\openint \gets a \cup \openint c \to}^-}\) | Definition of Exterior | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp \R {\openint \gets a^- \cup \openint c \to^-}\) | Closure of Finite Union equals Union of Closures | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp \R {\hointl \gets a \cup \hointr c \to}\) | Closure of Open Ball in Metric Space | |||||||||||
\(\ds \) | \(=\) | \(\ds \openint a c\) | Definition of Relative Complement |
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $5 \ \text{(b)}$