Kuratowski's Closure-Complement Problem/Closure of Complement

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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 the complement of $A$ in $\R$ is given by:

\(\displaystyle A^{\prime \, -}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {1} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{2 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval


Kuratowski-Closure-Complement-Theorem-ClosComp.png


Proof

From Kuratowski's Closure-Complement Problem: Complement:

\(\displaystyle A'\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {1} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{2 \,.\,.\, 3}\right)\) Definition of Half-Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({3 \,.\,.\, 4}\right]\) ... adjacent to Half-Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({\R \setminus \Q \cap \left[{4 \,.\,.\, 5}\right]}\right)\) Irrational Numbers from $4$ to $5$
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{5 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval

From Real Number is Closed in Real Number Space:

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

and:

$\left({\gets \,.\,.\, 0}\right]$ is closed in $\R$

and:

$\left[{5 \,.\,.\, \to}\right)$ is closed in $\R$

Then from Set is Closed iff Equals Topological Closure:

$\left\{ {3} \right\}^- = \left\{ {3} \right\}$
$\left({\gets \,.\,.\, 0}\right]^- = \left({\gets \,.\,.\, 0}\right]$
$\left[{5 \,.\,.\, \to}\right)^- = \left[{5 \,.\,.\, \to}\right)$


From Closure of Half-Open Real Interval is Closed Real Interval:

$\left[{2 \,.\,.\, 3}\right) = \left[{2 \,.\,.\, 3}\right]$

and:

$\left({3 \,.\,.\, 4}\right] = \left[{3 \,.\,.\, 4}\right]$


From Closure of Irrational Interval is Closed Real Interval:

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


From Closure of Finite Union equals Union of Closures:

\(\displaystyle A^{\prime \, -}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) Definition of Unbounded Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left\{ {1} \right\}\) Definition of Singleton
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{2 \,.\,.\, 3}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{3 \,.\,.\, 4}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{4 \,.\,.\, 5}\right]\) Definition of Closed Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left[{5 \,.\,.\, \to}\right)\) Definition of Unbounded Closed Real Interval

The result follows.

$\blacksquare$