Kuratowski's Closure-Complement Problem/Closure

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


 * Kuratowski-Closure-Complement-Theorem-Clos.png

Proof
From Closure of Union of Adjacent Open Intervals:
 * $\left({\left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)}\right)^- = \left[{0 \,.\,.\, 2}\right]$

From Real Number is Closed in Real Number Space:
 * $\left\{ {3} \right\}$ is closed in $\R$

From Set is Closed iff Equals Topological Closure:
 * $\left\{ {3} \right\}^- = \left\{ {3} \right\}$

From Closure of Rational Interval is Closed Real Interval:
 * $\left({\Q \cap \left({4 \,.\,.\, 5}\right)}\right)^- = \left[{4 \,.\,.\, 5}\right]$

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