# Kuratowski's Closure-Complement Problem/Exterior

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## 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 $\R$: $A^{\prime \, \circ}$

From Kuratowski's Closure-Complement Problem: Closure:

\(\displaystyle A^-\) | \(=\) | \(\displaystyle \closedint 0 2\) | 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 |

$\blacksquare$