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

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

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

$\blacksquare$