Kuratowski's Closure-Complement Problem/Example

Example for Kuratowski's Closure-Complement Problem

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)

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $9$