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:
 * $\openint \gets 0^- = \hointl \gets 0$

and:
 * $\openint 5 \to^- = \hointr 5 \to$

From Closure of Union of Adjacent Open Intervals:
 * $\paren {\openint 2 3 \cup \openint 3 4}^- = \closedint 2 4$

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