Kuratowski's Closure-Complement Problem/Closure of Interior of Closure
Jump to navigation
Jump to search
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 closure of $A$ in $\R$ is given by:
| \(\ds A^{- \, \circ \, -}\) | \(=\) | \(\ds \closedint 0 2\) | Definition of Closed Real Interval | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \closedint 4 5\) | Definition of Closed Real Interval |
Proof
From Kuratowski's Closure-Complement Problem: Interior of Closure:
| \(\ds A^{- \, \circ}\) | \(=\) | \(\ds \openint 0 2\) | Definition of Open Real Interval | |||||||||||
| \(\ds \) | \(\) | \(\, \ds \cup \, \) | \(\ds \openint 4 5\) | Definition of Open Real Interval |
From Closure of Open Real Interval is Closed Real Interval:
- $\openint 0 2^- = \closedint 0 2$
and:
- $\openint 4 5^- = \closedint 4 5$
The result follows from Closure of Finite Union equals Union of Closures.
$\blacksquare$