|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 introduced the following well-known theorem. While the theorem itself gives an upper bound, there appears to be no such bound on the number of questions and results related to it.|
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 became known as Kuratowski's
closure‑complement theorem (aka 14‑set theorem, closure-complement problem).
In these pages Math Transit chronicles the abundance of literature related to this topic.
If you know of any missing references, please send them to [my nickname below]@gmail.com so they can be added to the list.
updated 1 Dec 2015