Kuratowski's Closure-Complement Problem/Complement

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


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

Proof
For ease of analysis, let:
 * $A_1 := \openint 0 1$
 * $A_2 := \openint 1 2$
 * $A_3 := \set 3$
 * $A_4 := \Q \cap \openint 4 5$

Thus:
 * $\ds A = \bigcup_{i \mathop = 1}^4 A_i$

By De Morgan's Laws:


 * $\ds A' := \R \setminus A = \bigcap_{i \mathop = 1}^4 \paren {\R \setminus A_i}$

from which the result follows by inspection.