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:
 * $A_1 := \left({0 \,.\,.\, 1}\right)$
 * $A_2 := \left({1 \,.\,.\, 2}\right)$
 * $A_3 := \left\{ {3} \right\}$
 * $A_4 := \left({\Q \cap \left({4 \,.\,.\, 5}\right)}\right)$

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

By De Morgan's Laws:


 * $\displaystyle A' := \R \setminus A = \bigcap_{i \mathop = 1}^4 \left({\R \setminus A_i}\right)$

from which the result follows by inspection.