How many sets can you get by taking closures, complements, and intersections? 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?
 A: 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.
