# Kuratowski closure-complement theorem

Problem. Let $X$ be a topological space and $A$ a subset of $X$. How many (distinct) sets can be obtained by iteratively applying the closure and complement operations to $A$?

Kuratowski studied this problem, and showed that at most $14$ sets that can be generated from a given set in an arbitrary topological space. This is known as the Kuratowski closure-complement theorem.

Let us examine this problem more closely. For convenience, let us denote ${}^{-}:X\to X$ be the closure operator:

 $A\mapsto A^{-},$

and ${}^{c}:X\to X$ the complementation operator:

 $A\mapsto A^{c}.$

A set that can be obtained from $A$ by iteratively applying ${}^{-}$ and ${}^{c}$ has the form $A^{\sigma}$, where $\sigma$ is an operator on $X$ that is the composition of finitely many ${}^{-}$ and ${}^{c}$. In other words, $\sigma$ is a word on the alphabet $\{^{-},^{c}\}$.

First, notice that $A^{--}=A^{-}$ and $A^{cc}=A$. This means that $\sigma$ can be reduced (or simplified) to a form such that ${}^{-}$ and ${}^{c}$ occurs alternately.

In addition, we have the following:

###### Proposition 1.

$A^{-c-c-c-}=A^{-c-}$.

###### Proof.

For any set $A$ in a topological space $X$, $A^{-}$ is closed, so that $A^{-c-}$ is regular closed. This means that $A^{-c-}=A^{-c-c-c-}$. ∎

This means that $\sigma$ can be reduced to one of the following cases:

 $1,^{-},^{-c},^{-c-},^{-c-c},^{-c-c-},^{-c-c-c},^{c},^{c-},^{c-c},^{c-c-},^{c-c% -c},^{c-c-c-},^{c-c-c-c},$

where $1=\,^{cc}$ is the identity operator. As there are a total of 14 combinations, proving the closure-complement theorem is to exhibit an example. To do this, pick $X=\mathbb{R}$, the real line. Let $A=(0,1)\cup\{2\}\cup((3,4)\cap\mathbb{Q})\cup((5,7)-\{6\})$. In other words, $A$ is the union of a real interval, a point, a rational interval, and a real interval with a point deleted. Then

1. 1.

$A^{-}=[0,1]\cup\{2\}\cup[3,4]\cup[5,7]$,

2. 2.

$A^{-c}=(-\infty,0)\cup(1,2)\cup(2,3)\cup(4,5)\cup(7,\infty)$,

3. 3.

$A^{-c-}=(-\infty,0]\cup[1,3]\cup[4,5]\cup[7,\infty)$,

4. 4.

$A^{-c-c}=(0,1)\cup(3,4)\cup(5,7)$,

5. 5.

$A^{-c-c-}=[0,1]\cup[3,4]\cup[5,7]$,

6. 6.

$A^{-c-c-c}=(-\infty,0)\cup(1,3)\cup(4,5)\cup(7,\infty)$,

7. 7.

$A^{c}=(-\infty,0]\cup[1,2)\cup(2,3]\cup((3,4)-\mathbb{Q})\cup[4,5]\cup\{6\}% \cup[7,\infty)$,

8. 8.

$A^{c-}=(-\infty,0]\cup[1,5]\cup\{6\}\cup[7,\infty)$,

9. 9.

$A^{c-c}=(0,1)\cup(5,6)\cup(6,7)$,

10. 10.

$A^{c-c-}=[0,1]\cup[5,7]$,

11. 11.

$A^{c-c-c}=(-\infty,0)\cup(1,5)\cup(7,\infty)$,

12. 12.

$A^{c-c-c-}=(-\infty,0]\cup[1,5]\cup[7,\infty)$,

13. 13.

$A^{c-c-c-c}=(0,1)\cup(5,7)$,

together with $A$, are $14$ pairwise distinct sets that can be generated by ${}^{-}$ and ${}^{c}$.

Title Kuratowski closure-complement theorem KuratowskiClosurecomplementTheorem 2013-03-22 17:59:28 2013-03-22 17:59:28 CWoo (3771) CWoo (3771) 9 CWoo (3771) Theorem msc 54A99 msc 54A05