Kuratowski's Closure-Complement Problem/Complement of Interior of Closure of Interior
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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 the interior of the closure of the interior of $A$ in $\R$ is given by:
| \(\ds A^{\circ \, - \, \circ \, \prime}\) | \(=\) | \(\ds \left({\gets \,.\,.\, 0}\right]\) | Definition of Unbounded Closed Real Interval | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \left[{2 \,.\,.\, \to}\right)\) | Definition of Unbounded Closed Real Interval |
Proof
From Kuratowski's Closure-Complement Problem: Interior of Closure of Interior:
| \(\ds A^{\circ \, - \, \circ}\) | \(=\) | \(\ds \left({0 \,.\,.\, 2}\right)\) | Definition of Open Real Interval |
The result follows by inspection.
$\blacksquare$