Kuratowski's Closure-Complement Problem

Theorem

Let $T = \struct {S, \tau}$ 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

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)

Proof

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

Complement

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

$\ds A'$

$=$

$\ds \hointl \gets 0$

Definition of Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 1$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \hointr 2 3$

Definition of Half-Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \hointl 3 4$

... adjacent to Half-Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \paren {\paren {\R \setminus \Q} \cap \closedint 4 5}$

Irrational Numbers from $4$ to $5$

$\ds $

$\, \ds \cup \, $

$\ds \hointr 5 \to$

Definition of Unbounded Closed Real Interval

Interior

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

$\ds A^\circ$

$=$

$\ds \openint 0 1 \cup \openint 1 2$

Union of Adjacent Open Intervals

Closure

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

$\ds A^-$

$=$

$\ds \closedint 0 2$

Definition of Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 3$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \closedint 4 5$

Definition of Closed Real Interval

Exterior

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

$\ds A^e$

$=$

$\ds \openint \gets 0$

Definition of Unbounded Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \openint 2 3 \cup \openint 3 4$

Definition of Union of Adjacent Open Intervals

$\ds $

$\, \ds \cup \, $

$\ds \openint 5 \to$

Definition of Unbounded Open Real Interval

Closure of Complement

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

$\ds A^{\prime \, -}$

$=$

$\ds \hointl \gets 0$

Definition of Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \set 1$

Definition of Singleton

$\ds $

$\, \ds \cup \, $

$\ds \hointr 2 \to$

Definition of Unbounded Closed Real Interval

Closure of Interior

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

$\ds A^{\circ \, -}$

$=$

$\ds \closedint 0 2$

Definition of Closed Real Interval

Interior of Closure

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

$\ds A^{- \, \circ}$

$=$

$\ds \openint 0 2$

Definition of Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \openint 4 5$

Definition of Open Real Interval

Interior of Closure of Interior

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

$\ds A^{\circ \, - \, \circ}$

$=$

$\ds \openint 0 2$

Definition of Open Real Interval

Interior of Complement of Interior

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

$\ds A^{\circ \, \prime \, \circ}$

$=$

$\ds \openint \gets 0$

Definition of Unbounded Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \openint 2 \to$

Definition of Unbounded Open Real Interval

Closure of Interior of Complement

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

$\ds A^{\prime \, \circ \, -}$

$=$

$\ds \hointl \gets 0$

Definition of Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \closedint 2 4$

Definition of Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \hointr 5 \to$

Definition of Unbounded Closed Real Interval

Closure of Interior of Closure

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

$\ds A^{- \, \circ \, -}$

$=$

$\ds \closedint 0 2$

Definition of Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \closedint 4 5$

Definition of Closed Real Interval

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:

$\ds A^{- \, \circ \, \prime \, \circ}$

$=$

$\ds \left({\gets \,.\,.\, 0}\right)$

Definition of Unbounded Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \left({2 \,.\,.\, 4}\right)$

Definition of Open Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \left({5 \,.\,.\, \to}\right)$

Definition of Unbounded Open Real Interval

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:

$\ds A^{\circ \, - \, \circ \, \prime}$

$=$

$\ds \left({\gets \,.\,.\, 0}\right]$

Definition of Unbounded Closed Real Interval

$\ds $

$\, \ds \cup \, $

$\ds \left[{2 \,.\,.\, \to}\right)$

Definition of Unbounded Closed Real Interval

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 $T = \struct {S, \tau}$.

To simplify the presentation:

let $a$ be used to denote the operation of taking the complement of $A$ relative to $S$: $\map a A = S \setminus A$

let $b$ be used to denote the operation of taking the closure of $A$ in $T$: $\map b A = A^-$

let $I$ be used to denote the identity operation on $A$, that is: $\map I A = A$.

let the parentheses and the reference to $A$ be removed, so as to present, for example:

$\map a {\map b {\map a A} }$

as:

$a b a$

From Relative Complement of Relative Complement:

$\map a {\map a A} = A$

or, using the compact notation defined above:

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

and from Closure of Topological Closure equals Closure:

$\map b {\map b A} = \map b A = A^-$

or, using the compact notation defined above:

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

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

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:

$b a b = b a b a \paren {b a b}$

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

1922: Kazimierz Kuratowski: Sur l'operation A de l'Analysis Situs (Fundamenta Mathematicae Vol. 3: pp. 182 – 199)

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors
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: $9$