Kuratowski's Closure-Complement Problem/Closure of Interior

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 $A$ in $\R$ is given by:

\(\ds A^{\circ \, -}\)

\(=\)

\(\ds \closedint 0 2\)

Definition of Closed Real Interval

Proof

From Kuratowski's Closure-Complement Problem: Interior:

$A^\circ = \openint 0 1 \cup \openint 1 2$

From Closure of Union of Adjacent Open Intervals:

$A^{\circ \, -} = \closedint 0 2$

$\blacksquare$