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:
 * $\left({\gets \,.\,.\, 0}\right]^\circ = \left({\gets \,.\,.\, 0}\right)$

and:
 * $\left[{2 \,.\,.\, \to}\right)^\circ = \left({2 \,.\,.\, \to}\right)$

From Interior of Singleton in Real Number Space is Empty:


 * $\left\{ {1} \right\}^\circ = \varnothing$