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