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)

Sources

• 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: $9$