Kuratowski's Closure-Complement Problem/Exterior
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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 exterior of $A$ in $\R$ is given by:
| \(\ds A^e\) | \(=\) | \(\ds \openint \gets 0\) | 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 |
Proof
By definition, the exterior of $A$ in $\R$ can be defined either as:
- the complement of the closure of $A$ in $\R$: $A^{- \, \prime}$
or as:
- the interior of the complement of $A$ in $\R$: $A^{\prime \, \circ}$
From Kuratowski's Closure-Complement Problem: Closure:
| \(\ds A^-\) | \(=\) | \(\ds \closedint 0 2\) | Definition of Closed Real Interval | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \set 3\) | Definition of Singleton | ||||||||||
| \(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \closedint 4 5\) | Definition of Closed Real Interval |
It follows by inspection that:
| \(\ds A^e = A^{- \, \prime}\) | \(=\) | \(\ds \openint \gets 0\) | 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 |
$\blacksquare$