Kuratowski's Closure-Complement Problem/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 interior of the closure of the interior of $A$ in $\R$ is given by:

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


Kuratowski-Closure-Complement-Theorem-IntClosInt.png


Proof

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

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


From Interior of Closed Real Interval is Open Real Interval:

$\left[{0 \,.\,.\, 2}\right]^\circ = \left({0 \,.\,.\, 2}\right)$

Hence the result.

$\blacksquare$