# How many sets can we get by taking interiors and closures?

I'm having following problem. I'm looking for the maximum number of different sets that we can generate by one set $B \subseteq \mathbb{R}$ by taking a finite amount of closures and interiors. For example $\{0\}$ generates the sets $\{0\}$ and $\emptyset$. At first I thought the answer was 3 (we can only generate B, the closure of B and the interior of B) because I thought $\overline{B^\circ}=\overline{B}$ and $\overline{B}^\circ=B^\circ$. But then I looked at the example $B=\mathbb{Q}$ and found that $\overline{\mathbb{Q}}^\circ=\mathbb{R}^\circ=\mathbb{R}\neq\emptyset=\mathbb{Q}^\circ$.
This is not what I would expect and I clearly need another method te search that maximum.

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Note that both these operations are idempotent, so the only way to achieve actual results is to intertwine them. – Asaf Karagila May 29 '13 at 21:52
This isn't exactly what you're looking for, but it is related (and interesting): Kuratowski's Problem – Jared May 29 '13 at 21:55
As suggested by @Jared's link, have a look at closure and interior repeatedly applied to $B=(0,1)\cup(1,2)\cup \{3\}\cup([4,5]\cap \mathbb Q)$. – Hagen von Eitzen May 29 '13 at 21:59
@Jared: indeed related to my question and very interessenting. I'm suprised that the answer is 14 (I would absolutely not guess that answer). – mvcouwen May 29 '13 at 22:03
Note that as the Wikipedia article on the 14 set problem says, that $S^\circ = \overline{S^c}^c$ (or $iS = ckcS$), so it seems that the problem reduces to the closure-complement question. – kahen May 29 '13 at 22:32

## 1 Answer

Trying with the set $B=(0,1)\cup(1,2)\cup\{3\}\cup([4,5]\cap\mathbb{Q}$ (as suggested by Hagen von Eitzen) gives me that the answer has to be at least 8.
Taking first the closure followed by the interior followed by the closure and so on, gives me 4 extra possibilities.
Starting with the interior instead gives me 3 more possibilities.
Including the set I have started with, I have 8 possibilities. The only question that is left: is it the maximum? I guess so, but to prove it I think a need to find certain relations between the closures and interior so that I can relate them to the two sequences I found with this example. More precisely (if we denote taking an interior by I and taking a closure by C) we have to proof that:
CICIC = CIC and ICIC = IC
Any suggestions for proofing this?

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at.yorku.ca/p/a/c/a/24.pdf has proofs for these statements. – Henno Brandsma May 30 '13 at 17:55