# Kuratowski's Closure-Complement Problem

## Contents

- 1 Theorem
- 2 Proof
- 2.1 Complement
- 2.2 Interior
- 2.3 Closure
- 2.4 Exterior
- 2.5 Closure of Complement
- 2.6 Closure of Interior
- 2.7 Interior of Closure
- 2.8 Interior of Closure of Interior
- 2.9 Interior of Complement of Interior
- 2.10 Closure of Interior of Complement
- 2.11 Closure of Interior of Closure
- 2.12 Interior of Complement of Interior of Closure
- 2.13 Complement of Interior of Closure of Interior
- 2.14 Proof of Maximum

- 3 Also known as
- 4 Source of Name
- 5 Sources

## 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

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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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 |

### 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

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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Closures and Interiors

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 32: \ 9$