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

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


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

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

From Closure of Open Real Interval is Closed Real Interval:
 * $\left({0 \,.\,.\, 2}\right)^- = \left[{0 \,.\,.\, 2}\right]$

and:
 * $\left({4 \,.\,.\, 5}\right)^- = \left[{4 \,.\,.\, 5}\right]$

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