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

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

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

\(\displaystyle A^{\prime \, -}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {1} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{2 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval


From Interior of Closed Real Interval is Open Real Interval:

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

and:

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

From Interior of Singleton in Real Number Space is Empty:

$\left\{ {1} \right\}^\circ = \varnothing$



$\blacksquare$