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

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## Theorem

Let $\R$ be the real number line with the usual (Euclidean) topology.

Let $A \subseteq \R$ be defined as:

\(\displaystyle A\) | \(:=\) | \(\displaystyle \openint 0 1 \cup \openint 1 2\) | Definition of Union of Adjacent Open Intervals | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \cup \, \) | \(\displaystyle \set 3\) | Definition of Singleton | |||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \cup \, \) | \(\displaystyle \paren {\Q \cap \openint 4 5}\) | Rational Numbers from $4$ to $5$ (not inclusive) |

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

\(\displaystyle A^{\circ \, - \, \circ \, \prime}\) | \(=\) | \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) | Definition of Unbounded Closed Real Interval | ||||||||||

\(\displaystyle \) | \(\) | \(\, \displaystyle \cup \, \) | \(\displaystyle \left[{2 \,.\,.\, \to}\right)\) | Definition of Unbounded Closed Real Interval |

## Proof

From Kuratowski's Closure-Complement Problem: Interior of Closure of Interior:

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

The result follows by inspection.

$\blacksquare$