# 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

From Kuratowski's Closure-Complement Problem: Complement:

\(\displaystyle A'\) | \(=\) | \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) | 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 |

From Real Number is Closed in Real Number Space:

- $\left\{ {3} \right\}$ is closed in $\R$

and:

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

and:

- $\left[{5 \,.\,.\, \to}\right)$ is closed in $\R$

Then from Set is Closed iff Equals Topological Closure:

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

From Closure of Half-Open Real Interval is Closed Real Interval:

- $\left[{2 \,.\,.\, 3}\right) = \left[{2 \,.\,.\, 3}\right]$

and:

- $\left({3 \,.\,.\, 4}\right] = \left[{3 \,.\,.\, 4}\right]$

From Closure of Irrational Interval is Closed Real Interval:

- $\left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)^- = \left[{4 \,.\,.\, 5}\right]$

From Closure of Finite Union equals Union of Closures:

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

$\blacksquare$