Kuratowski's Closure-Complement Problem/Complement

Theorem

Let $\R$ be the real number line with the usual (Euclidean) topology.

Let $A \subseteq \R$ be defined as:

$\ds A$

$:=$

$\ds \openint 0 1 \cup \openint 1 2$

Definition of Union of Adjacent Open Intervals

$\ds $

$\, \ds \cup \, $

$\ds \set 3$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \paren {\Q \cap \openint 4 5}$

Rational Numbers from $4$ to $5$ (not inclusive)

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

$\ds A'$

$=$

$\ds \hointl \gets 0$

Definition of Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 1$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \hointr 2 3$

Definition of Half-Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \hointl 3 4$

... adjacent to Half-Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \paren {\paren {\R \setminus \Q} \cap \closedint 4 5}$

Irrational Numbers from $4$ to $5$

$\ds $

$\, \ds \cup \, $

$\ds \hointr 5 \to$

Definition of Unbounded Closed Real Interval

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:

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

$\ds \R \setminus A_1$

$=$

$\ds \hointl \gets 0 \cup \hointr 1 \to$

$\ds \R \setminus A_2$

$=$

$\ds \hointl \gets 1 \cup \hointr 2 \to$

$\ds \R \setminus A_3$

$=$

$\ds \openint \gets 3 \cup \openint 3 \to$

$\ds \R \setminus A_4$

$=$

$\ds \hointl \gets 4 \cup \paren {\R \setminus \Q} \cup \hointr 5 \to$

De Morgan's Laws

from which the result follows by inspection.

$\blacksquare$