Kuratowski's Closure-Complement Problem/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 interior of the closure of the interior of $A$ in $\R$ is given by:
\(\ds A^{\circ \, - \, \circ}\) | \(=\) | \(\ds \openint 0 2\) | Definition of Open Real Interval |
Proof
From Kuratowski's Closure-Complement Problem: Closure of Interior:
\(\ds A^{\circ \, -}\) | \(=\) | \(\ds \closedint 0 2\) | Definition of Closed Real Interval |
From Interior of Closed Real Interval is Open Real Interval:
- $\closedint 0 2^\circ = \openint 0 2$
Hence the result.
$\blacksquare$