Kuratowski's Closure-Complement Problem/Exterior

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Theorem

Let $\R$ be the real number space under the usual (Euclidean) topology.

Let $A \subseteq \R$ be defined as:

\(\displaystyle A\) \(:=\) \(\displaystyle \left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)\) Definition of Union of Adjacent Open Intervals
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {3} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({\Q \cap \left({4 \,.\,.\, 5}\right)}\right)\) Rational Numbers from $4$ to $5$ (not inclusive)


The exterior of $A$ in $\R$ is given by:

\(\displaystyle A^e\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)\) Definition of Union of Adjacent Open Intervals
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({5 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-Ext.png


Proof

By definition, the exterior of $A$ in $\R$ can be defined either as:

the complement of the closure of $A$ in $\R$: $A^{- \, \prime}$

or as:

the interior of the complement of $A$ in $\R$: $A^{\prime \, \circ}$


From Kuratowski's Closure-Complement Problem: Closure:

\(\displaystyle A^-\) \(=\) \(\displaystyle \left[{0 \,.\,.\, 2}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {3} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{4 \,.\,.\, 5}\right]\) Definition of Closed Real Interval


It follows by inspection that:

\(\displaystyle A^e = A^{- \, \prime}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)\) Definition of Union of Adjacent Open Intervals
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({5 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval

$\blacksquare$