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

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

 $\displaystyle A^{\prime \, \circ \, -}$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right]$ Definition of Unbounded Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{2 \,.\,.\, 4}\right]$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{5 \,.\,.\, \to}\right)$ Definition of Unbounded Closed Real Interval ## Proof

 $\displaystyle A^e$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right)$ Definition of Unbounded Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)$ Definition of Union of Adjacent Open Intervals $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left({5 \,.\,.\, \to}\right)$ Definition of Unbounded Open Real Interval
$\left({\gets \,.\,.\, 0}\right)^- = \left({\gets \,.\,.\, 0}\right]$

and:

$\left({5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$
$\left({\left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)}\right)^- = \left[{2 \,.\,.\, 4}\right]$

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

$\blacksquare$