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

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Theorem

Let $\R$ be the real number line with the usual (Euclidean) topology.

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

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


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

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


Kuratowski-Closure-Complement-Theorem-CompIntClosInt.png


Proof

From Kuratowski's Closure-Complement Problem: Interior of Closure of Interior:

\(\displaystyle A^{\circ \, - \, \circ}\) \(=\) \(\displaystyle \left({0 \,.\,.\, 2}\right)\) Definition of Open Real Interval


The result follows by inspection.

$\blacksquare$