|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 and Jackson [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 completing his Ph.D. at the University of Warsaw in 1921
Kazimierz Kuratowski published the first part of
his dissertation one year later in the Polish journal
[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)].
This oft‑cited paper taught us something new about the number 14:|
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.
|To become famous, it helps to have a name.|
Kuratowski's off-the-wall discovery got its lucky break when John L. Kelley took the highly unusual step of naming every problem in his classic textbook [1955 Kelley BJohn L. Kelley, General Topology, 1955, p. 57.]. Our present-day cornucopia would probably contain less fruit had this notable author not gone this extra mile. Kelley gave Kuratowski's result the fitting name Kuratowski Closure and Complementation Problem.
Kelley's choice (or minor variations of it) stuck for nearly a half century; it wasn't until the early 21st century that the numeral 14 (or word fourteen) began to displace the operations in some titles [2003 Brandsma AH. S. Brandsma, The Fourteen Subsets Problem: Interiors, Closures and Complements, Topology Explained, Topology Atlas, 2003, 4 pp.], [2005 Muhm DPhilip Muhm, Kuratowski's 14‑Set Theorem — A Modal Logic View, 2005, 58 pp.], [2007 Beckman MRyan T. Beckman, Basic Topology and the Kuratowski 14‑set Problem, 2007, 70 pp.], [2010 Sherman ADavid Sherman, Variations on Kuratowski's 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123.]. Sherman attempts to persuade us that Kuratowski's result “is known as” the 14‑set theorem without any support from his thirteen references (Kelley would have made a nice fourteenth). Six coauthors of one recent paper [2015 BCMPRS AT. Banakh, O. Chervak, T. Martynyuk, M. Pylypovych, A. Ravsky, M. Simkiv, Kuratowski monoids of n‑topological spaces, arXiv:1508.07703v1 [math.GN], arXiv.org, 2015, 20 pp.] staked a middle ground by calling it “the famous 14‑set closure‑complement Theorem of Kuratowski.” Perhaps they took a vote that resulted in a deadlock?
The numeral 14's debut within the name of Kuratowski's theorem occurs in an editorial note following the solution to Problem 10577 in the March 1998 issue of the Monthly [1997 BRR PMark Bowron, Stanley Rabinowitz, Closure, Complement, and Arbitrary Union, Problem 10577, Amer. Math. Monthly, v. 104 no. 2, 1997, p. 169, Solution: John Rickard, v. 105 no. 3, 1998, pp. 282‑283.]. After casually referring to “the result about 14 sets” the note informs us that “Previous Monthly problems related to the 14 sets problem include 5569 [1968, 199; 1971, 411], 5996 [1974, 1034; 1978, 283], and 6260 [1979, 226; 1980, 680].” (italics mine)
What might have prompted this choice?
From the late 1980s to the mid 1990s, the Monthly's lag time from proposal to published solution gradually inflated to nearly four years. Intent on slashing this during his 1996‑2002 tenure as lead editor, Daniel Ullman—truly “the Paul Volcker of problem editors”—routinely packed as many solutions into each issue as he possibly could. Thus he would have strongly favored abbreviations—especially during the earlier part of his tenure when the note was written.
However Ullman's thinking went, it's doubtful he intended to tinker with the established name.
In these pages Math Transit chronicles the abundance of literature related to—Kelley's—Kuratowski Closure‑Complement Theorem.
updated 24 May 2016
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