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The Kuratowski closure-complement problem yields 14 sets which can be formed by taking the closure and the complement of a single set. But if I want to also include such sets as the frontier or boundary, I also need to be able to take intersections between previously generated elements. In this case, how many different sets can you get?

Mario Carneiro
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    Could you explain your example of such a set? – user134824 Apr 14 '14 at 02:54
  • @user134824 I may have to retract that statement; my hope was to express $S'={x:x\in\overline{S\setminus{x}}}$ in terms of these operations, and then use $\omega^\omega$ under the order topology. But I don't think limit points can be written in terms of these operations. – Mario Carneiro Apr 14 '14 at 03:03
  • (Note that applying the limit point operation to $\omega^\omega$ $n$ times yields the set of all ordinals whose lowest exponent in cantor normal form is at least $n$.) – Mario Carneiro Apr 14 '14 at 03:08
  • @MarioCarneiro yes Kuratowski proved that limit points cannot be expressed in terms of the three operations. The proof appears at the very end of his 1922 paper, an English translation of which can be found at http://www.mathtransit.com/1922_kuratowski_english.pdf. – mathematrucker Jun 11 '17 at 01:18

1 Answers1

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This question has a long history in the literature.

1. (1922) Kuratowski proves that infinitely many distinct sets can be generated from one set under closure, complement and intersection. His proof appears on the middle of page 197 in his paper Sur l'Opération $\bar A$ de l'Analysis Situs (in French). Here it is in English:

Consider the operation $\varphi(A)=A\cap\overline{\overline A\setminus A}$. Let $B$ be a totally ordered set of order type $\omega^\omega$. Let $a\in B$ and define $A=B\setminus\{a+\omega$, $a+\omega^3$, $a+\omega^5,\ldots\}$.

It is easy to see that, under the order topology, $\varphi(A)$ consists of the elements of $A$ of the form $a+\omega^n$ with $n\geq2$, $\varphi(\varphi(A))$ those of the form $a+\omega^n$ with $n\geq4$, and so on. The operation $\varphi(A)$ thus leads to an infinite number of distinct sets.

(This was taken from my full translation of Kuratowski’s paper.)

2. (1944) Citing reference 1, McKinsey and Tarski reproduce Kuratowski’s construction on page 169 in their paper The Algebra of Topology.

3. (1966) The last part of Problem 5349 (Amer. Math. Monthly, proposed 1965 p. 1136, solved 1966 pp. 1132-1134) essentially asks readers to find a subset of the closed unit interval that generates infinitely many distinct sets under the three operations. J. C. Morgan II cites both 1 and 2 in his published solution.

4. (1977) W. J. Blok presents two examples on page 363 of his paper The Free Closure Algebra on Finitely Many Generators, one of which shows that infinitely many distinct sets can be generated starting with a finite seed set. References 1 and 2 are cited.

5. (1998) John Rickard presents the first published example of a set of reals (under the usual topology) that satisfies the condition in his solution to Problem 10577 (Amer. Math. Monthly, proposed 1997 p. 169, solved 1998 pp. 282-283). No previous solutions are cited.

6. (2004) David Sherman cites reference 1 then gives a proof similar to Kuratowski’s on page 9 of the 2004 arXiv preprint of his 2010 Monthly paper Variations on Kuratowski’s 14‑Set Theorem. The same proof appears in the final paper, except there Sherman adds that while his seed set distinguishes infinitely many operations (in the monoid generated by the three operations), it fails to distinguish all of them. This serves as a nice lead-in to the section on closure algebras that follows.

7. (2006) Without citing any references, Bruce Burdick’s published solution of Problem 11059 (Amer. Math. Monthly, proposed 2004 p. 64, solved 2006 p. 83) reproduces the finite seed set example from reference 4.

8. (2008) After citing 1, 2 and 4 on page 34 of their paper The Kuratowski Closure-Complement Theorem, Gardner and Jackson give an example showing it is possible to generate an infinite family by repeatedly applying just the two set operations of closure and set difference to a single seed set.

9. (2010) The question was asked here.

mathematrucker
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