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

## Theorem

Let $\R$ be the real number space under the usual (Euclidean) topology.

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

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

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

 $\displaystyle A^{\prime \, -}$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right]$ Definition of Unbounded Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left\{ {1} \right\}$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{2 \,.\,.\, \to}\right)$ Definition of Unbounded Closed Real Interval

## Proof

 $\displaystyle A'$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right]$ Definition of Unbounded Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left\{ {1} \right\}$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{2 \,.\,.\, 3}\right)$ Definition of Half-Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left({3 \,.\,.\, 4}\right]$ ... adjacent to Half-Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)$ Irrational Numbers from $4$ to $5$ $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{5 \,.\,.\, \to}\right)$ Definition of Unbounded Closed Real Interval
$\left\{ {3} \right\}$ is closed in $\R$

and:

$\left({\gets \,.\,.\, 0}\right]$ is closed in $\R$

and:

$\left[{5 \,.\,.\, \to}\right)$ is closed in $\R$
$\left\{ {3} \right\}^- = \left\{ {3} \right\}$
$\left({\gets \,.\,.\, 0}\right]^- = \left({\gets \,.\,.\, 0}\right]$
$\left[{5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$
$\left[{2 \,.\,.\, 3}\right) = \left[{2 \,.\,.\, 3}\right]$

and:

$\left({3 \,.\,.\, 4}\right] = \left[{3 \,.\,.\, 4}\right]$
$\left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)^- = \left[{4 \,.\,.\, 5}\right]$
 $\displaystyle A^{\prime \, -}$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right]$ Definition of Unbounded Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left\{ {1} \right\}$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{2 \,.\,.\, 3}\right]$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{3 \,.\,.\, 4}\right]$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{4 \,.\,.\, 5}\right]$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{5 \,.\,.\, \to}\right)$ Definition of Unbounded Closed Real Interval

The result follows.

$\blacksquare$