Kuratowski's Closure-Complement Problem/Interior of Closure

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

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

$=$

$\ds \openint 0 2$

Definition of Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \openint 4 5$

Definition of Open Real Interval

Proof

From Kuratowski's Closure-Complement Problem: Closure:

$\ds A^-$

$=$

$\ds \closedint 0 2$

Definition of Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 3$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \closedint 4 5$

Definition of Closed Real Interval

From Interior of Closed Real Interval is Open Real Interval:

$\closedint 0 2^\circ = \openint 0 2$

and:

$\closedint 4 5^\circ = \openint 4 5$

From Interior of Singleton in Real Number Line is Empty:

$\set 3^\circ = \O$

This needs considerable tedious hard slog to complete it.
In particular: Get past this: Interior of Union is not necessarily Union of Interiors
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$\blacksquare$