Kuratowski closurecomplement 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 closurecomplement 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.
${A}^{ccc}={A}^{c}$.
Proof.
For any set $A$ in a topological space $X$, ${A}^{}$ is closed, so that ${A}^{c}$ is regular closed. This means that ${A}^{c}={A}^{ccc}$. ∎
This means that $\sigma $ can be reduced to one of the following cases:
$$1{,}^{}{,}^{c}{,}^{c}{,}^{cc}{,}^{cc}{,}^{ccc}{,}^{c}{,}^{c}{,}^{cc}{,}^{cc}{,}^{ccc}{,}^{ccc}{,}^{cccc},$$ 
where $1{=}^{cc}$ is the identity operator. As there are a total of 14 combinations^{}, proving the closurecomplement 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.
${A}^{c}=(\mathrm{\infty},0)\cup (1,2)\cup (2,3)\cup (4,5)\cup (7,\mathrm{\infty})$,

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

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

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

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

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

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

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

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

13.
${A}^{cccc}=(0,1)\cup (5,7)$,
together with $A$, are $14$ pairwise distinct sets that can be generated by ${}^{}$ and ${}^{c}$.
Title  Kuratowski closurecomplement theorem 

Canonical name  KuratowskiClosurecomplementTheorem 
Date of creation  20130322 17:59:28 
Last modified on  20130322 17:59:28 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 54A99 
Classification  msc 54A05 