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

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


 * Kuratowski-Closure-Complement-Theorem-ClosIntComp.png

Proof
From Kuratowski's Closure-Complement Problem: Exterior:

From Closure of Open Real Interval is Closed Real Interval:
 * $\left({\gets \,.\,.\, 0}\right)^- = \left({\gets \,.\,.\, 0}\right]$

and:
 * $\left({5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$

From Closure of Union of Adjacent Open Intervals:
 * $\left({\left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)}\right)^- = \left[{2 \,.\,.\, 4}\right]$

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