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

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

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

Proof

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

By inspection:

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

and:

$\left[{2 \,.\,.\, 4}\right]^\circ = \left({2 \,.\,.\, 4}\right)$

and:

$\left[{5 \,.\,.\, \to}\right)^\circ = \left({5 \,.\,.\, \to}\right)$

$\blacksquare$