Kuratowski's Closure-Complement Problem/Closure of Interior of 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 closure of the interior of the complement of $A$ in $\R$ is given by:
\(\ds A^{\prime \, \circ \, -}\) | \(=\) | \(\ds \hointl \gets 0\) | Definition of Unbounded Closed Real Interval | |||||||||||
\(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \closedint 2 4\) | Definition of Closed Real Interval | ||||||||||
\(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \hointr 5 \to\) | Definition of Unbounded Closed Real Interval |
Proof
From Kuratowski's Closure-Complement Problem: Exterior:
\(\ds A^e\) | \(=\) | \(\ds \openint \gets 0\) | Definition of Unbounded Open Real Interval | |||||||||||
\(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \openint 2 3 \cup \openint 3 4\) | Definition of Union of Adjacent Open Intervals | ||||||||||
\(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \openint 5 \to\) | Definition of Unbounded Open Real Interval |
From Closure of Open Real Interval is Closed Real Interval:
- $\openint \gets 0^- = \hointl \gets 0$
and:
- $\openint 5 \to^- = \hointr 5 \to$
From Closure of Union of Adjacent Open Intervals:
- $\paren {\openint 2 3 \cup \openint 3 4}^- = \closedint 2 4$
The result follows from Closure of Finite Union equals Union of Closures.
$\blacksquare$