|Kuratowski's Closure-Complement Cornucopia|
|Anyone looking for a good place to start learning about
Kuratowski's closure-complement theorem need look no further than
B. J. Gardner and Marcel Jackson's extensive survey and research article
[2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.].
This 36-page paper stands out as the single most comprehensive study of Kuratowski's closure-complement theorem ever written.
David Sherman's delightful Monthly article
[2010 Sherman ADavid Sherman, Variations on Kuratowski's 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123.]
is also well worth reading.
Many interesting results are obtained by varying the inputs in the 14‑set theorem. The first such appeared five years after Kuratowski's 1922 paper, in yet another Ph.D. dissertation [1927 Zarycki AMiron Zarycki, Quelques Notions Fondamentales de l'Analysis Situs au Point du Vue de l'Algèbre de la Logique (Some Basic Topological Concepts in Terms of the Algebra of Logic), Fund. Math., v. 9, 1927, pp. 3‑15, in French. (in French)], [2012 Zarycki AMiron Zarycki, Quelques Notions Fondamentales de l'Analysis Situs au Point du Vue de l'Algèbre de la Logique (Some Basic Topological Concepts in Terms of the Algebra of Logic), English translation by Mark Bowron, Math Transit.com, 2012, 8 pp. (English)] published in Fundamenta Mathematicae, this one by the celebrated Ukrainian mathematician Miron Zarycki (1889‑1961).
The closure-complement theorem is based on the following general construction:
Starting with some general or specific space, usually definable in terms of a system of subsets satisfying certain properties, various given operators are applied repeatedly to a single seed subset until the resulting family of subsets becomes closed under the operators. These operators also generate a monoid (of operators) under composition.
Various max-min questions arise naturally. Here is one such question:
How small can the topology be when there exists a seed that generates a maximal (14-set) family under closure and complement in a topological space?
The following data summarize this question:
|If we loosen up the space requirement to include all closure spaces,
the answer is 14 (see Theorem 3 on page 6 of
[2012 Soltan‑c AV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 2012, 18 pp.]).
Since every topological space has equal numbers of open and closed sets, this gives a lower
bound of 14 for the topological case. In
[1966 HM AH. H. Herda, R. C. Metzler, Closure and Interior in Finite Topological Spaces, Colloq. Math., v. 15 no. 2, 1966, pp. 211‑216.]
it is shown that among all minimal (7-point) topological spaces capable of generating a maximal
(14-set) family under closure and complement, the topology always contains at least 19 open sets.
An example is given with a 19-set topology, so 19 is an upper bound for the topological case.
In the case of closure spaces, Soltan [2012 Soltan‑c AV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 2012, 18 pp.] showed that a minimal system of (14) closed sets is attainable within a minimal (6-point) space. If the analogous situation holds for topological spaces, then the answer to our question above is 19. This seems likely to be true, but we leave it for others to prove (there is currently no published proof).
By moving the word “minimize” to a different field we get the following question, which appears in [2012 Bowron PMark Bowron, How Small Can a Kuratowski 14‑Set Be?, Problem 1898, Math. Mag., v. 85 no. 3, 2012, p. 228.]:
|Multiple solutions to this problem can be found in
[2012 FGJMM QArthur Fischer, Dejan Govc, John, mathematrucker, Gerry Myerson, What is the Smallest Cardinality of a Kuratowski 14‑Set?, Stack Exchange, 2012.].
12 Mar 2013