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.

#### Further reading:

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.