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

\(\displaystyle A^{\circ \, \prime \, \circ}\) \(=\) \(\displaystyle \openint \gets 0\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \openint 2 \to\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntCompInt.png


Proof

From Complement of Interior equals Closure of Complement:

$A^{\circ \, \prime} = A^{\prime \, -}$

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

\(\displaystyle A^{\prime \, -}\) \(=\) \(\displaystyle \hointl \gets 0\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \set 1\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \hointr 2 \to\) Definition of Unbounded Closed Real Interval


From Interior of Closed Real Interval is Open Real Interval:

$\hointl \gets 0^\circ = \openint \gets 0$

and:

$\hointr 2 \to^\circ = \openint 2 \to$

From Interior of Singleton in Real Number Line is Empty:

$\set 1^\circ = \O$



$\blacksquare$