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 Line:
 * $\set 3$ is closed in $\R$

and:
 * $\hointl \gets 0$ is closed in $\R$

and:
 * $\hointr 5 \to$ is closed in $\R$

Then from Set is Closed iff Equals Topological Closure:
 * $\set 3^- = \set 3$
 * $\hointl \gets 0^- = \hointl \gets 0$
 * $\hointr 5 \to^- = \hointr 5 \to$

From Closure of Half-Open Real Interval is Closed Real Interval:
 * $\hointr 2 3 = \closedint 2 3$

and:
 * $\hointl 3 4 = \closedint 3 4$

From Closure of Irrational Interval is Closed Real Interval:
 * $\paren {\R \setminus \Q \cap \closedint 4 5}^- = \closedint 4 5$

From Closure of Finite Union equals Union of Closures:

The result follows.