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

## Theorem

Let $\R$ be the real number line with the usual (Euclidean) topology.

Let $A \subseteq \R$ be defined as:

 $\displaystyle A$ $:=$ $\displaystyle \openint 0 1 \cup \openint 1 2$ Definition of Union of Adjacent Open Intervals $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \set 3$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \paren {\Q \cap \openint 4 5}$ Rational Numbers from $4$ to $5$ (not inclusive)

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

 $\displaystyle A^{\circ \, \prime \, \circ}$ $=$ $\displaystyle \openint \gets 0$ Definition of Unbounded Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \openint 2 \to$ Definition of Unbounded Open Real Interval ## Proof

$A^{\circ \, \prime} = A^{\prime \, -}$
 $\displaystyle A^{\prime \, -}$ $=$ $\displaystyle \hointl \gets 0$ Definition of Unbounded Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \set 1$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \hointr 2 \to$ Definition of Unbounded Closed Real Interval
$\hointl \gets 0^\circ = \openint \gets 0$

and:

$\hointr 2 \to^\circ = \openint 2 \to$
$\set 1^\circ = \O$

$\blacksquare$