# Kuratowski's Closure-Complement Problem/Closure

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## Theorem

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

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

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

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

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

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

\(\ds A^-\) | \(=\) | \(\ds \closedint 0 2\) | Definition of Closed Real Interval | |||||||||||

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

\(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \closedint 4 5\) | Definition of Closed Real Interval |

## Proof

From Closure of Union of Adjacent Open Intervals:

- $\paren {\openint 0 1 \cup \openint 1 2}^- = \closedint 0 2$

From Real Number is Closed in Real Number Line:

- $\set 3$ is closed in $\R$

From Set is Closed iff Equals Topological Closure:

- $\set 3^- = \set 3$

From Closure of Rational Interval is Closed Real Interval:

- $\paren {\Q \cap \openint 4 5 }^- = \closedint 4 5$

The result follows from Closure of Finite Union equals Union of Closures.

$\blacksquare$