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

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

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

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

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

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

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

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

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

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 ${}^c$.