Kuratowski's Closure-Complement Problem/Closure of Complement

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


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

Proof
From Kuratowski's Closure-Complement Problem: Complement:

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

and:
 * $\left({\gets \,.\,.\, 0}\right]$ is closed in $\R$

and:
 * $\left[{5 \,.\,.\, \to}\right)$ is closed in $\R$

Then from Set is Closed iff Equals Topological Closure:
 * $\left\{ {3} \right\}^- = \left\{ {3} \right\}$
 * $\left({\gets \,.\,.\, 0}\right]^- = \left({\gets \,.\,.\, 0}\right]$
 * $\left[{5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$

From Closure of Half-Open Real Interval is Closed Real Interval:
 * $\left[{2 \,.\,.\, 3}\right) = \left[{2 \,.\,.\, 3}\right]$

and:
 * $\left({3 \,.\,.\, 4}\right] = \left[{3 \,.\,.\, 4}\right]$

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

From Closure of Finite Union equals Union of Closures:

The result follows.