# 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

 $\ds A^-$ $=$ $\ds \closedint 0 2$ 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
$\closedint 0 2^\circ = \openint 0 2$

and:

$\closedint 4 5^\circ = \openint 4 5$
$\set 3^\circ = \O$

$\blacksquare$