Kuratowski's Closure-Complement Problem/Complement
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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.
$\blacksquare$