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

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

\(\displaystyle A^{- \, \circ}\) | \(=\) | \(\displaystyle \left({0 \,.\,.\, 2}\right)\) | Definition of Open Real Interval | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \cup \, \) | \(\displaystyle \left({4 \,.\,.\, 5}\right)\) | Definition of Open Real Interval |

## Proof

From Kuratowski's Closure-Complement Problem: Closure:

\(\displaystyle A^-\) | \(=\) | \(\displaystyle \left[{0 \,.\,.\, 2}\right]\) | 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 |

From Interior of Closed Real Interval is Open Real Interval:

- $\left[{0 \,.\,.\, 2}\right]^\circ = \left({0 \,.\,.\, 2}\right)$

and:

- $\left[{4 \,.\,.\, 5}\right]^\circ = \left({4 \,.\,.\, 5}\right)$

From Interior of Singleton in Real Number Space is Empty:

- $\left\{ {3} \right\}^\circ = \varnothing$

$\blacksquare$