# Kuratowski's Closure-Complement Problem/Exterior

## 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 exterior of $A$ in $\R$ is given by:

 $\displaystyle A^e$ $=$ $\displaystyle \openint \gets 0$ Definition of Unbounded Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \openint 2 3 \cup \openint 3 4$ Definition of Union of Adjacent Open Intervals $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \openint 5 \to$ Definition of Unbounded Open Real Interval

## Proof

By definition, the exterior of $A$ in $\R$ can be defined either as:

the complement of the closure of $A$ in $\R$: $A^{- \, \prime}$

or as:

the interior of the complement of $A$ in $\R$: $A^{\prime \, \circ}$
 $\displaystyle A^-$ $=$ $\displaystyle \closedint 0 2$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \set 3$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \closedint 4 5$ Definition of Closed Real Interval

It follows by inspection that:

 $\displaystyle A^e = A^{- \, \prime}$ $=$ $\displaystyle \openint \gets 0$ Definition of Unbounded Open Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \openint 2 3 \cup \openint 3 4$ Definition of Union of Adjacent Open Intervals $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \openint 5 \to$ Definition of Unbounded Open Real Interval

$\blacksquare$