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The Kuratowski Closure-Complement Problem
The following result was introduced in 1922 by Kazimierz Kuratowski:
The highest number of distinct sets that can be generated from one set in a topological space by repeatedly applying closure and complement in any order is 14.
The proof breaks into two parts. First one must show that 14 is the maximum possible number. This follows from the identity kckckck = kck where k is closure and c is complement. Then a set that actually generates 14 sets must be found. Such sets are called Kuratowski 14-sets.
Readers are invited to construct a Kuratowski 14-set in the interactive diagram by following the link below.
This Supplement accompanies the article "Variations on Kuratowski's 14-set Theorem," by David Sherman, American Mathematical Monthly, February 2010 (vol. 117, no. 2), pp. 113-123.
Closures in Formal Languages and Kuratowski's Theorem, Janusz Brzozowski, Elyot Grant, and Jeffrey Shallit, Developments in Language Theory, pp. 125-144, Springer, 2009.
The Kuratowski Closure-Complement Theorem, B. J. Gardner and Marcel Jackson, New Zealand Journal of Mathemaics, 2008, pp. 9-44.
Sur l'operation A de l'analysis situs (English translation), Kazimierz Kuratowski, Fund. Math. 3, 182-199, 1922.
Variations on Kuratowski's 14-Set Theorem, David Sherman, American Mathematical Monthly, February 2010 (vol. 117, no. 2), pp. 113-123.
Bowron, Mark, "The Kuratowski Closure-Complement Problem," Loci (October 2009), DOI: 10.4169/loci003343