|Kuratowski's Closure-Complement Cornucopia|
|To learn about
Kuratowski's closure-complement theorem, see
Gardner and Jackson’s extensive paper
[2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.].
David Sherman’s informative 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.
An abundance of questions similar to the closure-complement problem can be formed by varying the inputs. This territory was first explored by Ukrainian mathematician Miron Zarycki (1889‑1961) in his 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)], [1927 Zarycki EMiron 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 translation)]. See also [1928 Zarycki‑a AMiron Zarycki, Allgemeine Eigenschaften der Cantorschen Kohärenzen (General Properties of Cantor's Coherences), Trans. Amer. Math. Soc., v. 30 no. 3, 1928, pp. 498‑506, in German.], [1928 Zarycki‑b AMiron Zarycki, Derivation and Coherence in Abstract Sets, Zb. Mat.‑Prir.‑Likar. Sekt. Nauk. Tov., v. 27, 1928, pp. 247‑259, in Ukrainian.], [1930 Zarycki AMiron Zarycki, Über den Kern einer Menge (On the Core of a Set), Jahr. Deutsch. Math. Ver., v. 39, 1930, pp. 154‑158, in German.], [1949 Zarycki AMiron Zarycki, On an Operation in the Theory of Point Sets, Nauk. Zap. Ser. Fiz.‑Mat., v. 12 no. 3, 1949, pp. 35‑43, in Ukrainian.], [1947 Zarycki AMiron Zarycki, Some Properties of the Notion of Derived Set in Abstract Spaces, Nauk. Zap. Ser. Fiz.‑Mat., v. 5 no. 1, 1947, pp. 22‑33, in Ukrainian.], [1955 Zarycki AMiron Zarycki, Boolean Algebras With a Closure and Boolean Algebras With a Derivation, Dop. Akad. Nauk. Ukrain., v. 1, 1955, pp. 3‑6, in Ukrainian.] (none of these papers are in English).
Many questions can be characterized using the following general framework. Starting with some (general or specific) space defined in terms of a system of subsets satisfying certain properties, apply certain seed operations repeatedly to an initial family of one (or more) seed set(s) until the generated family of distinct subsets is closed under the seed operations. The collection of distinct set operations (defined on the power set of the space) that are obtainable in this fashion (for a given space) forms a monoid under composition.
Here is just one example of such a question:
How small can the topology be in a topological space that contains a 14‑set?
(A seed set that generates 14 distinct sets under closure and complement in a topological space is called a 14‑set.) This question fits into our framework as follows:
|If we weaken the space type to closure spaces,
the answer is 14 (see Theorem 3 on page 6 of
[1982 Soltan EV. 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
family (14) 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 [1982 Soltan EV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 2012, 18 pp.] showed that a minimal system of closed sets (14) 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 there is currently no published proof.
By shifting the word “minimize” to a different input, we get the following question:
|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.] and
[2012 Bowron PMark Bowron, How Small Can a Kuratowski 14‑Set Be?, Problem 1898, Math. Mag., v. 85 no. 3, 2012, p. 228.].
27 Jun 2017