Kuratowski's Closure-Complement Problem/Closure of Interior of Complement

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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 closure of the interior of the complement of $A$ in $\R$ is given by:

\(\displaystyle A^{\prime \, \circ \, -}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{2 \,.\,.\, 4}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{5 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval


Kuratowski-Closure-Complement-Theorem-ClosIntComp.png


Proof

From Kuratowski's Closure-Complement Problem: Exterior:

\(\displaystyle A^e\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) 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

From Closure of Open Real Interval is Closed Real Interval:

$\left({\gets \,.\,.\, 0}\right)^- = \left({\gets \,.\,.\, 0}\right]$

and:

$\left({5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$


From Closure of Union of Adjacent Open Intervals:

$\left({\left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)}\right)^- = \left[{2 \,.\,.\, 4}\right]$


The result follows from Closure of Finite Union equals Union of Closures.

$\blacksquare$