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

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

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

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

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

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

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

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

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

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

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

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 ${}^{c}$.

## Mathematics Subject Classification

54A99*no label found*54A05

*no label found*

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