# 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:

 $\displaystyle A^e$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right)$ 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 ## 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 $\R$: $A^{\prime \, \circ}$
 $\displaystyle A^-$ $=$ $\displaystyle \left[{0 \,.\,.\, 2}\right]$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left\{ {3} \right\}$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \left[{4 \,.\,.\, 5}\right]$ Definition of Closed Real Interval

It follows by inspection that:

 $\displaystyle A^e = A^{- \, \prime}$ $=$ $\displaystyle \left({\gets \,.\,.\, 0}\right)$ 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

$\blacksquare$