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

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


 * 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:

From Interior of Closed Real Interval is Open Real Interval:
 * $\hointl \gets 0^\circ = \openint \gets 0$

and:
 * $\hointr 2 \to^\circ = \openint 2 \to$

From Interior of Singleton in Real Number Line is Empty:


 * $\set 1^\circ = \O$