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

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

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


Kuratowski-Closure-Complement-Theorem-ClosIntClos.png


Proof

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

\(\displaystyle A^{- \, \circ}\) \(=\) \(\displaystyle \left({0 \,.\,.\, 2}\right)\) Definition of Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({4 \,.\,.\, 5}\right)\) Definition of Open Real Interval


From Closure of Open Real Interval is Closed Real Interval:

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

and:

$\left({4 \,.\,.\, 5}\right)^- = \left[{4 \,.\,.\, 5}\right]$


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

$\blacksquare$