# Kuratowski's Closure-Complement Problem/Closure

## Theorem

Let $\R$ be the real number line with the usual (Euclidean) topology.

Let $A \subseteq \R$ be defined as:

 $\displaystyle A$ $:=$ $\displaystyle \openint 0 1 \cup \openint 1 2$ Definition of Union of Adjacent Open Intervals $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \set 3$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \paren {\Q \cap \openint 4 5}$ Rational Numbers from $4$ to $5$ (not inclusive)

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

 $\displaystyle A^-$ $=$ $\displaystyle \closedint 0 2$ Definition of Closed Real Interval $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \set 3$ Definition of Singleton $\displaystyle$  $\, \displaystyle \cup \,$ $\displaystyle \closedint 4 5$ Definition of Closed Real Interval ## Proof

$\paren {\openint 0 1 \cup \openint 1 2}^- = \closedint 0 2$
$\set 3$ is closed in $\R$
$\set 3^- = \set 3$
$\paren {\Q \cap \openint 4 5 }^- = \closedint 4 5$

The result follows from Closure of Finite Union equals Union of Closures.

$\blacksquare$