Kuratowski's Closure-Complement Problem/Interior of Complement of Interior of Closure

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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 complement of the interior of the closure of $A$ in $\R$ is given by:

\(\ds A^{- \, \circ \, \prime \, \circ}\) \(=\) \(\ds \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\ds \) \(\) \(\, \ds \cup \, \) \(\ds \left({2 \,.\,.\, 4}\right)\) Definition of Open Real Interval
\(\ds \) \(\) \(\, \ds \cup \, \) \(\ds \left({5 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntCompIntClos.png


Proof

From Kuratowski's Closure-Complement Problem: Interior of Closure:

\(\ds A^{- \, \circ}\) \(=\) \(\ds \left({0 \,.\,.\, 2}\right)\) Definition of Open Real Interval
\(\ds \) \(\) \(\, \ds \cup \, \) \(\ds \left({4 \,.\,.\, 5}\right)\) Definition of Open Real Interval


By inspection:

\(\ds A^{- \, \circ \, '}\) \(=\) \(\ds \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\ds \) \(\) \(\, \ds \cup \, \) \(\ds \left[{2 \,.\,.\, 4}\right]\) Definition of Closed Real Interval
\(\ds \) \(\) \(\, \ds \cup \, \) \(\ds \left[{5 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval


From Interior of Closed Real Interval is Open Real Interval:

$\left({\gets \,.\,.\, 0}\right]^\circ = \left({\gets \,.\,.\, 0}\right)$

and:

$\left[{2 \,.\,.\, 4}\right]^\circ = \left({2 \,.\,.\, 4}\right)$

and:

$\left[{5 \,.\,.\, \to}\right)^\circ = \left({5 \,.\,.\, \to}\right)$




$\blacksquare$