Kuratowski's Closure-Complement Problem/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 interior of $A$ in $\R$ is given by:

$\ds A^\circ$

$=$

$\ds \openint 0 1 \cup \openint 1 2$

Union of Adjacent Open Intervals

$A^\circ = A^{\prime \, - \, \prime}$

$\ds A^{\prime \, -}$

$=$

$\ds \hointl \gets 0$

Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 1$

Singleton

$\ds $

$\, \ds \cup \, $

$\ds \hointr 2 \to$

Unbounded Closed Real Interval

It can be determined by inspection that:

$A^{\prime \, - \, \prime} = \openint 0 1 \cup \openint 1 2$

Hence the result.

$\blacksquare$