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

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.