Kuratowski's Closure-Complement Problem/Closure

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Theorem

Let $\R$ be the real number space under the usual (Euclidean) topology.

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

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


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

\(\displaystyle A^-\) \(=\) \(\displaystyle \left[{0 \,.\,.\, 2}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {3} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{4 \,.\,.\, 5}\right]\) Definition of Closed Real Interval


Kuratowski-Closure-Complement-Theorem-Clos.png


Proof

From Closure of Union of Adjacent Open Intervals:

$\left({\left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)}\right)^- = \left[{0 \,.\,.\, 2}\right]$


From Real Number is Closed in Real Number Space:

$\left\{ {3} \right\}$ is closed in $\R$

From Set is Closed iff Equals Topological Closure:

$\left\{ {3} \right\}^- = \left\{ {3} \right\}$


From Closure of Rational Interval is Closed Real Interval:

$\left({\Q \cap \left({4 \,.\,.\, 5}\right)}\right)^- = \left[{4 \,.\,.\, 5}\right]$


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

$\blacksquare$