|Kuratowski's Closure-Complement Cornucopia|
|There is a large and scattered literature on Kuratowski's
theorem, most of which focuses on topological spaces; an admirable survey is the paper of Gardner
[2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.].|
|—Janusz Brzozowski, Elyot Grant, and Jeffrey Shallit
[2009 BGS‑a AJanusz Brzozowski, Elyot Grant, Jeffrey Shallit, Closures in Formal Languages and Kuratowski's Theorem, arXiv:0901.3761 [cs.CC], arXiv.org, 2009, 12 pp.]|
After finishing his Ph.D. at the University of Warsaw in 1921,
Kazimierz Kuratowski published the first part of
his dissertation in the Polish research periodical
[1922 Kuratowski AKazimierz Kuratowski, Sur l'Opération Ā de l'Analysis Situs (On the Topological Closure Operation), Fund. Math., v. 3, 1922, pp. 182‑199, in French. (in French)],
[2012 Kuratowski AKazimierz Kuratowski, Sur l'Opération Ā de l'Analysis Situs (On the Topological Closure Operation), English translation by Mark Bowron, Math Transit.com, 2012, 11 pp. (English)].
In it Kuratowski observed that whenever one
subset in a topological space has closure and complement applied to it repeatedly (in any order),
the number of distinct subsets generated is always less than or equal to 14.
This unusual result eventually became known as Kuratowski's
closure‑complement theorem (a.k.a. 14‑set theorem, closure-complement problem).
Math Transit chronicles the abundance of literature on this topic.
updated 20 May 2015