Kuratowski's Closure-Complement Problem

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Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A \subseteq S$ be a subset of $T$.

By successive applications of the operations of complement relative to $S$ and the closure, there can be as many as $14$ distinct subsets of $S$ (including $A$ itself).


Example

Kuratowski-Closure-Complement-Theorem.png


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)


Proof

That there can be as many as $14$ will be demonstrated by example.

Complement

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

\(\displaystyle A'\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right]\) 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


Kuratowski-Closure-Complement-Theorem-Comp.png


Interior

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

\(\displaystyle A^\circ\) \(=\) \(\displaystyle \left({0 \,.\,.\, 1}\right) \cup \left({1 \,.\,.\, 2}\right)\) Union of Adjacent Open Intervals


Kuratowski-Closure-Complement-Theorem-Int.png


Closure

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


Exterior

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

\(\displaystyle A^e\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({2 \,.\,.\, 3}\right) \cup \left({3 \,.\,.\, 4}\right)\) Definition of Union of Adjacent Open Intervals
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({5 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-Ext.png


Closure of Complement

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


Closure of Interior

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

\(\displaystyle A^{\circ \, -}\) \(=\) \(\displaystyle \left[{0 \,.\,.\, 2}\right]\) Definition of Closed Real Interval
Kuratowski-Closure-Complement-Theorem-ClosInt.png


Interior of Closure

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

\(\displaystyle A^{- \, \circ}\) \(=\) \(\displaystyle \left({0 \,.\,.\, 2}\right)\) Definition of Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({4 \,.\,.\, 5}\right)\) Definition of Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntClos.png


Interior of Closure of Interior

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

\(\displaystyle A^{\circ \, - \, \circ}\) \(=\) \(\displaystyle \left({0 \,.\,.\, 2}\right)\) Definition of Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntClosInt.png


Interior of Complement of Interior

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

\(\displaystyle A^{\circ \, \prime \, \circ}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({2 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntCompInt.png


Closure of Interior of Complement

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

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


Kuratowski-Closure-Complement-Theorem-ClosIntComp.png


Closure of Interior of Closure

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

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


Kuratowski-Closure-Complement-Theorem-ClosIntClos.png


Interior of Complement of Interior of Closure

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

\(\displaystyle A^{- \, \circ \, \prime \, \circ}\) \(=\) \(\displaystyle \left({\gets \,.\,.\, 0}\right)\) Definition of Unbounded Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({2 \,.\,.\, 4}\right)\) Definition of Open Real Interval
\(\displaystyle \) \(\) \(\, \displaystyle \cup \, \) \(\displaystyle \left({5 \,.\,.\, \to}\right)\) Definition of Unbounded Open Real Interval


Kuratowski-Closure-Complement-Theorem-IntCompIntClos.png


Complement of Interior of Closure of Interior

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

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


Kuratowski-Closure-Complement-Theorem-CompIntClosInt.png



Proof of Maximum

It remains to be shown that there can be no more than $14$.


Consider an arbitrary subset $A$ of a topological space $T = \left({S, \tau}\right)$.

To simplify the presentation:

let $a$ be used to denote the operation of taking the complement of $A$ relative to $S$: $a \left({A}\right) = S \setminus A$
let $b$ be used to denote the operation of taking the closure of $A$ in $T$: $b \left({A}\right) = A^-$
let $I$ be used to denote the identity operation on $A$, that is: $I \left({A}\right) = A$.
let the parentheses and the reference to $A$ be removed, so as to present, for example:
$a \left({b \left({a \left({A}\right)}\right)}\right)$
as:
$a b a$


From Relative Complement of Relative Complement:

$a \left({a \left({A}\right)}\right) = A$

or, using the compact notation defined above:

$(1): \quad a a = I$

and from Closure of Topological Closure equals Closure:

$b \left({b \left({A}\right)}\right) = b \left({A}\right) = A^-$

or, using the compact notation defined above:

$(2): \quad b b = b$


Let $s$ be a finite sequence of elements of $\left\{{a, b}\right\}$.

By successive applications of $(1)$ and $(2)$, it is possible to eliminate all multiple consecutive instances of $a$ and $b$ in $s$, and so reduce $s$ to one of the following forms:

$\text{a)}: \quad a b a b \ldots a$
$\text{b)}: \quad b a b a \ldots a$
$\text{c)}: \quad a b a b \ldots b$
$\text{d)}: \quad b a b a \ldots b$


From Closure of Complement of Closure is Regular Closed:

$b a b$ is regular closed.

By Interior equals Complement of Closure of Complement, the interior of $A$ is:

$a b a$

Recall the definition of regular closed:

a set $A$ is regular closed if and only if it equals the closure of its interior.

And so as $b a b$ is regular closed:

$b a b = b a b a \left({b a b}\right)$


So, adding an extra $b$ to either of $a b a b a b a$ or $b a b a b a$ will generate a string containing $b a b a b a b$ which can be reduced immediately to $b a b$.


It follows that the possible different subsets of $S$ that can be obtained from $A$ by applying $a$ and $b$ can be generated by none other than:

$I$
$a$
$a b$
$a b a$
$a b a b$
$a b a b a$
$a b a b a b$
$a b a b a b a$
$b$
$b a$
$b a b$
$b a b a$
$b a b a b$
$b a b a b a$

... a total of $14$.

Hence the result.

$\blacksquare$


Also known as

This result is also known as Kuratowski's Closure-Complement Theorem.


Source of Name

This entry was named for Kazimierz Kuratowski.


Sources