|Kuratowski's Closure-Complement Cornucopia|
|For info on the 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.] and
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.].
One of the most prolific authors to expand on Kuratowski’s result, and the first to do so, was Ukrainian mathematician Miron Zarycki (1889‑1961). 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)]
considers the four cases where exterior, interior, frontier (boundary), and border replace closure. 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. (in German)], [1928 Zarycki‑a EMiron Zarycki, Allgemeine Eigenschaften der Cantorschen Kohärenzen (General Properties of Cantor's Coherences), English translation by Mark Bowron, Math Transit.com, 1928, 7 pp. (English translation)],
[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. (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. (in German)],
[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. (in Ukrainian)], [1947 Zarycki EMiron Zarycki, Some Properties of the Derived Set Operation in Abstract Spaces, English translation by Mark Bowron, Math Transit.com, 1947, 11 pp. (English translation)],
[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. (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. (in Ukrainian)].
Breaking the closure-complement theorem down into its constituent parts will provide us with a handy framework for classifying existing results and constructing new questions.
Starting with some space (a specific one such as the reals, or a general space type), the definition of which usually involves a system of subsets, certain seed operations on the power set of the space are repeatedly applied to an initial family of seed sets (usually an arbitrary singleton) until the generated family of distinct subsets is closed under the seed operations. This procedure recursively defines a monoid of operations under composition.
For example, consider the following (open) question:
How small can the topology (system of subsets) be in a topological space (space type) that contains a “14‑set” (single seed set that maximizes the generated family under the operations of closure of complement)?
It is characterized within our framework by this 7-tuple:
|Note that we use the suffix “‑al” instead of “‑ize” to avoid ambiguity about precedence.
Also, the blank seed condition is to be interpreted as the usual “singleton”.
If we weaken the space type in this question to closure space by removing the topological closure axiom ∅ = ∅ and weakening additivity to A ⋃ B ⊇ A ⋃ B, 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.]). Soltan presents a 6‑point closure space that minimizes (under the “maximal” family condition) not only the system of closed sets (14 sets), but the space as well.
Since every topological space is a closure space, it follows that 14 is a lower bound for the answer in the topological case. Herda and Metzler [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.] show that minimal (7‑point) topological spaces (under the “maximal” family condition) cannot have a topology with less than 19 sets. They present such a 7‑point space that has a 19‑set topology, so 19 is an upper bound for the answer in the topological case.
If as with closure spaces the system and space can both be minimized (under the “maximal” family condition) within a single topological space, then the answer to our question above is 19. This question is currently open.
Here is another example:
|Solutions 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 BB PMark Bowron, More on Kuratowski 14‑Sets, Problem 1898, Math. Mag., v. 85 no. 3, 2012, p. 228, Solution: Bruce S. Burdick, v. 86 no. 3, 2013, p. 231.].
21 Dec 2018