|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 EKazimierz 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)].|
Kuratowski’s 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 rare step of naming every problem in his classic textbook
[1955 Kelley BJohn L. Kelley, General Topology, 1955, p. 57.].
Our present-day “cornucopia” might contain far less fruit without Kelley’s contribution. He called Kuratowski’s result the Kuratowski Closure and Complementation Problem.|
Kelley’s choice (or minor variations of it) stuck for nearly half a century. The numeral 14, or word fourteen, did not appear in titles until the early twenty-first century [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.].
It did not appear in the theorem’s name until 1997, in a paper on topological molecular lattices [1997 FW ATai‑He Fan, Guo‑Jun Wang, On Some Gross Misunderstandings About the Theory of Topological Molecular Lattices, Fuzzy Set. Syst., v. 90, 1997, pp. 61‑67.]: “The Kuratowski 14 set theorem is true: Let A be an element in a symmetric TML, then, by using alternatively interior, pseudocomplement and closure to A can give at most 14 different sets.” (sic, except the italics are mine)
It next appeared in 1998, in an editor’s note following the published solution to Monthly Problem 10577 [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.]: “Previous Monthly problems related to the 14 sets problem include 5569 [1968 BL PStephen Baron, Kuratowski’s 14‑Sets, Problem 5569, Amer. Math. Monthly, v. 75 no. 2, 1968, p. 199, Solution: Eric Langford, v. 78 no. 4, 1971, p. 411.], 5996 [1974 SY PArthur Smith, Kuratowski Sets, Problem 5996, Amer. Math. Monthly, v. 81 no. 9, 1974, p. 1034, Solution: Chie Y. Yu, v. 85 no. 4, 1978, pp. 283‑284.], and 6260 [1979 LM PEric Langford, Sets Formed by Iterated Closure, Interior, and Union, Problem 6260, Amer. Math. Monthly, v. 86 no. 3, 1979, p. 226, Solution: William Myers, v. 87 no. 8, 1980, pp. 680‑681.].” (italics mine) During the 1980s and early 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 — the “Paul Volcker of problem editors” — routinely packed as many solutions as possible into each issue. Thus he would have strongly favored abbreviations, especially during the earlier part of his tenure when the note was written. However his thinking went, it is doubtful Ullman intended to tinker with the established name.
Those first two appearances of 14 set in place of closure-complement were not trendsetting. However, five years later, not only did the term fourteen subsets problem appear in a paper that was freely available on the web [2003 Brandsma AH. S. Brandsma, The Fourteen Subsets Problem: Interiors, Closures and Complements, Topology Explained, Topology Atlas, 2003, 4 pp.], it appeared in the title of that paper. This proved to be a game changer: the new name spread quickly.
The following year, David Sherman claimed — with no support from his thirteen references (Kelley would have made a nice fourteenth) — that Kuratowski’s result is known as the 14‑set theorem. Sherman’s misstatement appears in both his Monthly article [2010 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123.] and arXiv preprint [2004 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, arXiv:math/0405401 [math.GN], arXiv.org, 2004, 11 pp.].* Six coauthors of a more 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.] employed a shotgun approach, calling it “the famous 14‑set closure‑complement Theorem of Kuratowski.”
In these pages Math Transit chronicles the abundance of literature related to — Kelley’s — Kuratowski Closure‑Complement Theorem.
*Ironically Edwin Hewitt criticizes Kelley for similar naming infractions in a 1956 book review of General Topology:
No reasonable person would alter established terminology without grave reasons. While recognizing that the author undoubtedly has such reasons, the reviewer takes exception to the new definitions given for complete regularity (p. 117) and compacta (implicitly on p. 229) and to the introduction of the new term “Tychonoff space.”
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