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2019 Santiago AJosé Hernández Santiago, The Group‑Theoretic Analog of Kuratowski’s Closure‑Complement Theorem, Amer. Math. Monthly, v. 126 no. 6, 2019, pp. 519‑526.
2018 BCMPRS ATaras Banakh, Ostap Chervak, Tetyana Martynyuk, Maksym Pylypovych, Alex Ravsky, Markiyan Simkiv, Kuratowski Monoids of n‑Topological Spaces, Topol. Algebra Appl., v. 6, 2018, pp. 1‑25.
2018 Shum AKar-Ping Shum, The 14 Set Problem and Its Related Topics, ResearchGate, 2018, 21 pp.
2017 Berghammer ARudolf Berghammer, Tool-Based Relational Investigation of Closure-Interior Relatives for Finite Topological Spaces, Relational and Algebraic Methods in Computer Science, Lecture Notes in Computer Science, v. 10226, edited by P. Höfner, D. Pous, G. Struth, 2017, pp. 60‑76.
2017 GS AAnkit Gupta, Ratna Dev Sarma, A Note on Some Generalized Closure and Interior Operators in a Topological Space, Math. Appl., v. 6, 2017, pp. 11‑20.
2017 JPS AJozef Jirásek, Matúš Palmovský, Juraj Šebej, Kuratowski Algebras Generated by Factor‑, Subword‑, and Suffix‑Free Languages, International Conference on Descriptional Complexity of Formal Systems, Lecture Notes in Computer Science, v. 10316, edited by Cezar Câmpeanu, Giovanni Pighizzini, 2017, pp. 189‑201.
2017 Schwiebert ARyan C. Schwiebert, The Radical-Annihilator Monoid of a Ring, Comm. Algebra, v. 45 no. 4, 2017, pp. 1601‑1617.
2017 Shum AKar-Ping Shum, A Note on Kuratowski’s Theorem and Its Related Topics, v. 7, 2017, pp. 383‑406.
2016 JS AJozef Jirásek, Juraj Šebej, Kuratowski Algebras Generated by Prefix‑Free Languages, Implementation and Application of Automata, CIAA 2016, Lecture Notes in Computer Science, v. 9705, edited by Yo‑Sub Han, Kai Salomaa, pp. 150‑162.
2015 BCMPRS ATaras Banakh, Ostap Chervak, Tetyana Martynyuk, Maksym Pylypovych, Alex Ravsky, Markiyan Simkiv, Kuratowski Monoids of n‑Topological Spaces, arXiv:1508.07703 [math.GN], arXiv.org, 2015, 20 pp.
2014 BJZ AJanusz Brzozowski, Galina Jirásková, Chenglong Zou, Quotient Complexity of Closed Languages, Theory Comput. Syst., v. 54 no. 2, 2014, pp. 277‑292.
2014 SG ARatna Dev Sarma, Ankit Gupta, On the k‑Numbers of Some Generalized Kuratowski Operators in Topology, APMSCSET‑2014, pp. 50‑54.
2013 PW ASzymon Plewik, Marta Walczyńska, The Monoid Consisting of Kuratowski Operations, J. Math., v. 2013, 2013, pp. 1‑9.
2012 AMSS‑a AOsama Al-Hassani, Quratul‑ain Mahesar, Claudio Sacerdoti Coen, Volker Sorge, A Term Rewriting System for Kuratowski’s Closure-Complement Problem, Proceedings of the 23rd International Conference on Rewriting Techniques and Applications, RTA 2012, Nagoya, Japan, Monday, May 28, 2012 to Saturday, June 2, 2012, Leibniz International Proceedings in Informatics (LIPIcs), v. 15, edited by Ashish Tiwari, pp. 38‑52.
2012 AMSS‑b AOsama Al-Hassani, Quratul‑ain Mahesar, Claudio Sacerdoti Coen, Volker Sorge, Solving Kuratowski Problems by Term Rewriting, Proceedings of the 19th Automated Reasoning Workshop, 2‑4 April 2012, Manchester UK, edited by Fabio Papacchini, Renate A. Schmidt, pp. 51‑52.
2012 CDHS AÉmilie Charlier, Mike Domaratzki, Tero Harju, Jeffrey Shallit, Composition and Orbits of Language Operations: Finiteness and Upper Bounds, Intl. J. Comp. Math., 2012, pp. 1‑26.
2012 Cato ASusumu Cato, Complements and Consistent Closures, Discrete Math., v. 312, 2012, pp. 1218‑1221.
2012 PW ASzymon Plewik, Marta Walczyńska, The Monoid Consisting of Kuratowski Operations, arXiv:1205.3391 [math.GN], arXiv.org, 2012, 18 pp.
2011 BGS AJanusz Brzozowski, Elyot Grant, Jeffrey Shallit, Closures in Formal Languages and Kuratowski’s Theorem, Intl. J. Found. Comp. Sci., v. 22, 2011, pp. 301‑321.
2011 CDHS AÉmilie Charlier, Mike Domaratzki, Tero Harju, Jeffrey Shallit, Finite Orbits of Language Operations, Language and Automata Theory and Applications, Proceedings of the 5th International Conference, LATA 2011, Tarragona, Spain, May 26‑31, 2011, Lecture Notes in Computer Science, v. 6638, edited by A. H. Dediu, S. Inenaga, C. Martín-Vide, pp. 204‑215.
2011 HK AŠtěpán Holub, Juha Kortelainen, On Partitions Separating Words, Intl. J. Algebra Comp., v. 21 no. 8, 2011, pp. 1305‑1316.
2011 Kudlek AManfred Kudlek, A Relation Between Modal Logic and Language Closure Operators, Fund. Inform., v. 109, 2011, pp. 297‑304.
2011 SW AJeffrey Shallit, Ross Willard, Kuratowski’s Theorem for Two Closure Operators, arXiv:1109.1227 [math.GN], arXiv.org, 2011, 6 pp.
2010 BJZ AJanusz Brzozowski, Galina Jirásková, Chenglong Zou, Quotient Complexity of Closed Languages, Computer Science — Theory and Applications, Proceedings of the 5th International Computer Science Symposium in Russia, CSR 2010, Kazan, Russia, June 16‑20, 2010, Lecture Notes in Computer Science, v. 6072, edited by Farid Ablayev, Ernst W. Mayr, pp. 84‑95.
2010 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123.
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.
2009 BGS‑b AJanusz Brzozowski, Elyot Grant, Jeffrey Shallit, Closures in Formal Languages and Kuratowski’s Theorem, Developments in Language Theory, 13th International Conference, DLT 2009, Lecture Notes in Computer Science, v. 5583, pp. 125‑144.
2009 BGS‑c AJanusz Brzozowski, Elyot Grant, Jeffrey Shallit, Closures in Formal Languages: Concatenation, Separation, and Algorithms, arXiv:0901.3763 [cs.CC], arXiv.org, Comp. Research Rep. — CORR, v. abs/0901.3, 2009, 12 pp.
2009 BJZ AJanusz Brzozowski, Galina Jirásková, Chenglong Zou, Quotient Complexity of Closed Languages, arXiv:0912.1034 [cs.FL], arXiv.org, 2009, 12 pp.
2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.
2006 GJ AAdam Grabowski, Magdalena Jastrzebska, Enhancing MML via Isomichi's Classification of Domains, Mech. Math. Appl., v. 5, 2006, pp. 11‑18.
2006 Grabowski AAdam Grabowski, Solving Two Problems in General Topology Via Types, Types for Proofs and Programs International Workshop, TYPES 2004, Jouy‑en‑Josas, France, December 15‑18, 2004, Revised Selected Papers, Lecture Notes in Computer Science, v. 3839, edited by Jean-Christophe Filliâtre, Christine Paulin-Mohring, Benjamin Werner, 2006, pp. 138‑153.
2005 Grabowski AAdam Grabowski, On the Computer-Checked Solution of the Kuratowski Closure-Complement Problem, Mech. Math. Appl., v. 4, 2005, pp. 25‑34.
2004 Farmer AFrank D. Farmer, Containment for c, s, t Operators on a Binary Relation, Algebra Universalis, v. 51, 2004, pp. 153‑175.
2004 JS AMarcel Jackson, Tim Stokes, Semilattice Pseudo-complements on Semigroups, Comm. Algebra, v. 32, 2004, pp. 2895‑2918.
2004 Jackson AMarcel Jackson, Semilattices with Closure, Algebra Universalis, v. 52 no. 1, 2004, pp. 1‑37.
2004 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, arXiv:math/0405401 [math.GN], arXiv.org, 2004, 11 pp.
2003 BG ALilla Krystyna Baginska, Adam Grabowski, On the Kuratowski Closure-Complement Problem, J. Formal. Math., v. 15, 2003, pp. 1‑6.
2003 Brandsma AH. S. Brandsma, The Fourteen Subsets Problem: Interiors, Closures and Complements, Topology Explained, Topology Atlas, 2003, 4 pp.
2002 Voutsadakis AGeorge Voutsadakis, On Some Operations on Classes of Algebras and Coalgebras from a Bialgebraic Viewpoint, Voutsadakis.com, 2002.
2001 MT ADragan Mašulović, Boža Tasić, Operators on Classes of Coalgebras, Theoret. Comput. Sci., v. 269, 2001, pp. 419‑431.
2001 Tasic ABoža Tasić, On the Partially Ordered Monoid Generated by the Operators H, S, P, Ps on Classes of Algebras, J. Algebra, v. 245, 2001, pp. 1‑19.
2000 WY AHe Wei, Zhang Yaoming, Interior and Boundary in a Locale, Adv. Math., v. 29, 2000, pp. 439‑443.
1999 GS AB. J. Gardner, Tim Stokes, Closure Rings, Comm. Math. Univ. Carolinae, v. 40 no. 3, 1999, pp. 413‑427.
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.
1996 Shum AKar‑Ping Shum, Closure Functions on the Set of Positive Integers, Sci. China Ser. A, v. 39 no. 4, 1996, pp. 337‑346.
1995 GO‑a AEmmanuelle Garel, Jean Pierre Olivier, On Closures Verifying that the Interior of a Closed Element is Closed, Comm. Algebra, v. 23, 1995, pp. 3715‑3728.
1995 GO‑b AEmmanuelle Garel, Jean Pierre Olivier, The Opoid Generated by Transitive Closure and Interior and Symmetric Closure and Interior: A Characterization Using Generators and Relations, Rend. Sem. Mat. Univ. Poi. Torino, v. 53, 1995, pp. 71‑88.
1995 MT AM. S. Moslehian, N. Tavallaii, A Generalization of the Kuratowski Closure-Complement Problem, Punjab Univ. J. Math., v. 28, 1995, pp. 1‑9.
1995 Pfaltz AJohn L. Pfaltz, Evaluating the Binary Partition Function When N = 2^n, Cong. Num., v. 109, 1995, pp. 3‑12.
1993 GO AEmmanuelle Garel, Jean Pierre Olivier, About the Finiteness of Monoids Generated by Closures and Interiors, Semigroup Forum, v. 47, 1993, pp. 341‑352.
1993 Shum AKar‑Ping Shum, The Amalgamation of Closure and Boundary Functions on Semigroups and Partially Ordered Sets, Proceedings of the Conference on Ordered Structures and Algebra of Computer Languages: Hong Kong, 26‑29 June 1991, 1993, pp. 232‑243.
1992 Aull RC. E. Aull, Corrections and Additions for Classification of Topological Spaces II, Bull. Pol. Acad. Sci. Math., v. 40 no. 2, 1992, p. 175.
1992 KS AS. M. Kam, Kar‑Ping Shum, On a Problem of P. C. Hammer, Southeast Asian Bull. of Math., v. 16, 1992, pp. 123‑127.
1991 Fife AJames H. Fife, The Kuratowski Closure-Complement Problem, Math. Mag., v. 64 no. 3, 1991, pp. 180‑182.
1988 Aull AC. E. Aull, Classification of Topological Spaces II, Bull. Pol. Acad. Sci. Math., v. 36 no. 11-12, 1988, pp. 649‑654.
1988 Kuratowski AKazimierz Kuratowski, Selected Papers of Kazimierz Kuratowski, Sur l'Opération Ā de l'Analysis Situs (On the Topological Closure Operation), edited by Karol Borsuk, Ryszard Engelking, 1988, pp. 95‑112, in French.
1986 Wronski AStanislaw Wronski, On the Operations Definable in Terms of the Complementation and the Closure, Bull. Sect. Logic, Pol. Acad. Sci., v. 15, 1986, pp. 117‑121.
1984 Peleg AD. Peleg, A Generalized Closure and Complement Phenomenon, Discrete Math., v. 50, 1984, pp. 285‑293.
1984 Shum AKar‑Ping Shum, On the Boundary of Kuratowski 14‑Sets in Connected Spaces, Glas. Mat. Ser. III, v. 19, 1984, pp. 293‑296.
1982 Chagrov AA. V. Chagrov, Kuratowski Numbers, Appl. Funct. Anal. Approx. Th., 1982, pp. 186‑190, in Russian.
1982 Chagrov EA. V. Chagrov, Kuratowski Numbers, English translation by Mark Bowron, Math Transit.com, 6 pp.
1982 Soltan AV. P. Soltan, Problems of Kuratowski Type, Mat. Issled., v. 65, 1982, pp. 121‑131, in Russian.
1982 Soltan EV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 18 pp.
1980 FP AM. J. Fischer, M. S. Paterson, The Fast Skew-Closure Algorithm, L'Enseign. Math., v. 26, 1980, pp. 345‑360.
1980 Radulescu AFlorin Radulescu, On the Frontier Operator, Bull. Math. de la Soc. Sci. Math de la R.S. de Roumanie, v. 24 no. 4, 1980, pp. 401‑406.
1980 Soltan AV. P. Soltan, On Kuratowski’s Problem, Bull. Acad. Pol. Sci., Sér. Sci. Math., v. 28, 1980, pp. 369‑375, in Russian.
1980 Soltan EV. P. Soltan, On Kuratowski’s Problem, English translation by Mark Bowron, Math Transit.com, 11 pp.
1979 Soltan AV. P. Soltan, The Kuratowski Number of a Closure Space, IV Tiraspol Symposium on General Topology and its Applications, 1979, pp. 137‑139, in Russian.
1979 Soltan EV. P. Soltan, The Kuratowski Number of a Closure Space, English translation by Mark Bowron, Math Transit.com, 1 p.
1978 Fishburn APeter C. Fishburn, Operations on Binary Relations, Discrete Math., v. 21, 1978, pp. 7‑22.
1977 Blok AW. J. Blok, The Free Closure Algebra on Finitely Many Generators, Ind. Math., v. 80, 1977, pp. 362‑379.
1977 Kleiner AĬ. Z. Kleĭner, Closure and Boundary Operators in Topological Spaces, Ukr. Math. J. (English), v. 29 no. 3, 1977, pp. 295‑296.
1977 Moser ALouise E. Moser, Closure, Interior, and Union in Finite Topological Spaces, Colloq. Math., v. 38, 1977, pp. 41‑51.
1975 BJ AJoel Berman, Steven L. Jordan, The Kuratowski Closure-Complement Problem, Amer. Math. Monthly, v. 82 no. 8, 1975, pp. 841‑842.
1975 YS AKai-wing Yip, Kar‑ping Shum, On the Structure of Kuratowski Sets, J. Chin. Univ. Hong Kong, v. 3, 1975, pp. 429‑439.
1974 GE AYu. R. Gaida, A. É. Eremenko, On the Frontier Operator in Boolean Algebras with a Closure, Ukr. Math. J. (English), v. 26 no. 6, 1974, pp. 662‑664.
1974 Goldberg AA. A. Gol'dberg, On Boolean Algebras with a Derivation, Ukr. Math. J. (English), v. 26, 1974, pp. 365‑369.
1972 CJ AS. Comer, J. Johnson, The Standard Semigroup of Operators of a Variety, Algebra Universalis, v. 2, 1972, pp. 77‑79.
1972 GKM AR. L. Graham, D. E. Knuth, T. S. Motzkin, Complements and Transitive Closures, Discrete Math., v. 2, 1972, pp. 17‑29.
1972 Pigozzi AD. Pigozzi, On Some Operations on Classes of Algebras, Algebra Universalis, v. 2, 1972, pp. 346‑353.
1971 AS AJ. Anusiak, Kar‑Ping Shum, Remarks on Finite Topological Spaces, Colloq. Math., v. 23 no. 2, 1971, pp. 217‑223.
1971 Isomichi AYoshinori Isomichi, New Concepts in the Theory of Topological Space — Supercondensed Set, Subcondensed Set, and Condensed Set, Pacific J. Math., v. 38 no. 3, 1971, pp. 657‑668.
1971 Langford AEric Langford, Characterization of Kuratowski 14‑Sets, Amer. Math. Monthly, v. 78 no. 4, 1971, pp. 362‑367.
1970 Abian AAlexander Abian, The Derived Set Axioms for Topology, Mathematica (Cluj), v. 12 no. 2, 1970, pp. 213‑215.
1968 Aull RC. E. Aull, Corrigendum: Classification of Topological Spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., v. 16 no. 6, 1968.
1968 Staley ADavid H. Staley, On the Commutativity of the Boundary and Interior Operators in a Topological Space, Ohio J. Sci., v. 68 no. 2, 1968, p. 84.
1967 Aull AC. E. Aull, Classification of Topological Spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., v. 15, 1967, pp. 773‑778.
1967 Coleman APatrick J. Coleman, On the Composition Semigroup Generated by Complement and an Expansive Function, Nieuw Arch. Wisk., v. 15, 1967, pp. 207‑210.
1967 Nelson AEvelyn Nelson, Finiteness of Semigroups of Operators in Universal Algebra, Canad. J. Math., v. 19, 1967, pp. 764‑768.
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.
1966 Koenen AWilliam Koenen, The Kuratowski Closure Problem in the Topology of Convexity, Amer. Math. Monthly, v. 73 no. 7, 1966, pp. 704‑708.
1965 Hintikka AJ. Hintikka, A Closure and Complement Result for Nested Topologies, Fund. Math., v. 57 no. 1, 1965, pp. 97‑106.
1963 Harvey AFrank Reese Harvey, The Derived Set Operator, Amer. Math. Monthly, v. 70 no. 10, 1963, pp. 1085‑1086.
1962 Chapman‑a AThomas A. Chapman, An Extension of the Kuratowski Closure and Complementation Problem, Math. Mag., v. 35 no. 1, 1962, pp. 31‑35.
1962 Chapman‑b AThomas A. Chapman, A Further Note on Closure and Interior Operators, Amer. Math. Monthly, v. 69 no. 6, 1962, pp. 524‑529.
1962 Hammer AP. C. Hammer, Extended Topology: Set-Valued Set-Functions, Nieuw Arch. Wisk., v. 10, 1962, pp. 55‑77.
1961 CM AHisashi Choda, Katuyosi Matoba, On a Theorem of Levine, Proc. Japan Acad., v. 37, 1961, pp. 462‑463.
1961 Levine ANorman Levine, On the Commutativity of the Closure and Interior Operators in Topological Spaces, Amer. Math. Monthly, v. 68 no. 5, 1961, pp. 474‑477.
1960 Hammer AP. C. Hammer, Kuratowski’s Closure Theorem, Nieuw Arch. Wisk., v. 8 no. 2, 1960, pp. 74‑80.
1956 Gravett AK. A. H. Gravett, A Characterization of Frontier, Math. Proc. Camb. Philos. Soc., v. 52 no. 1, 1956, pp. 152‑153.
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.
1954 Halfar AEdwin Halfar, The Isolated Points of a Set, Port. Math., v. 13 no. 3, 1954, pp. 125‑128.
1951 Halfar AEdwin Halfar, A Note on Point-Set Operators, Port. Math., v. 10 no. 2, 1951, pp. 103‑104.
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 Derived Set Operation in Abstract Spaces, Nauk. Zap. Ser. Fiz.‑Mat., v. 5 no. 1, 1947, pp. 22‑33, in Ukrainian.
1947 Zarycki EMiron Zarycki, Some Properties of the Derived Set Operation in Abstract Spaces, English translation by Mark Bowron, Math Transit.com, 11 pp.
1944 MT AJ. C. C. McKinsey, Alfred Tarski, The Algebra of Topology, Ann. of Math., v. 45, 1944, pp. 141‑191.
1943 Hewitt AEdwin Hewitt, A Problem of Set-Theoretic Topology, Duke Math. J., v. 10 no. 2, 1943, pp. 309‑333.
1941 Chittenden‑a AE. W. Chittenden, A New Proof of a Formula of Kuratowski, Proc. Iowa Acad. Sci., v. 48, 1941, pp. 299‑300.
1941 Chittenden‑b AE. W. Chittenden, On the Reduction of Topological Functions, Lectures in Topology: The University of Michigan Conference of 1940, edited by William L. Ayres, Raymond L. Wilder, 1941, pp. 267‑285.
1941 Haight AFrank Haight, On the Independence of Operators on a Lattice, Proc. Iowa Acad. Sci., v. 48, 1941, pp. 301‑304.
1941 McKinsey AJ. C. C. McKinsey, A Solution of the Decision Problem for the Lewis Systems S2 and S4, With an Application to Topology, J. Symbol. Logic, v. 6 no. 4, 1941, pp. 117‑134.
1939 Stopher AE. C. Stopher Jr., Point Set Operators and Their Interrelations, Bull. Amer. Math. Soc., v. 46, 1939, pp. 758‑762.
1937 Sanders ASamuel T. Sanders Jr., Functional Equations Defining the Complementary Operations, Natl. Math. Mag., v. 12 no. 3, 1937, pp. 115‑117.
1937 Stopher AE. C. Stopher Jr., Cyclic Relations in Point Set Theory, Bull. Amer. Math. Soc., v. 43, 1937, pp. 686‑694.
1936 Sanders ASamuel T. Sanders Jr., Derived Sets and Their Complements, Bull. Amer. Math. Soc., v. 42, 1936, pp. 577‑584.
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.
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‑a EMiron Zarycki, Allgemeine Eigenschaften der Cantorschen Kohärenzen (General Properties of Cantor's Coherences), English translation by Mark Bowron, Math Transit.com, 7 pp.
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.
1927 Zarycki AMiron Zarycki, Quelques Notions Fondamentales de l'Analysis Situs au Point du Vue de l'Algèbre de la Logique (Some Fundamental Concepts of Topology in Terms of the Algebra of Logic), Fund. Math., v. 9, 1927, pp. 3‑15, 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, 8 pp.
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.
1922 Kuratowski EKazimierz Kuratowski, Sur l'Opération Ā de l'Analysis Situs (On the Topological Closure Operation), English translation by Mark Bowron, Math Transit.com, 11 pp.
DISSERTATIONS
2014 Mahesar DQuratul‑ain Mahesar, Computing Relatively Large Algebraic Structures by Automated Theory Exploration, 2014, pp. 103‑186.
2005 Frasson DMiguel V. S. Frasson, Stellingen (Propositions) in: Large Time Behaviour of Neutral Delay Systems, 2005, 2 pp.
2005 Muhm DPhilip Muhm, Kuratowski’s 14‑Set Theorem — A Modal Logic View, 2005, 58 pp.
1963 Staley DDavid H. Staley, Commutativity of Operators in a Topological Space with Applications, 1963, 65 pp.
1947 Halfar DEdwin Halfar, Reduction of Topological Operators and Their Canonical Representation, 1947, 48 pp.
1937 Stopher DE. C. Stopher Jr., Interrelations of a Family of Operators on Point Sets and Their Canonical Representation, 1937, 50 pp.
1935 Sanders DSamuel T. Sanders Jr., Derived Sets and Their Complements: The Characterization of Certain Set-Valued Set-Functions, 1935, 74 pp.
THESES
2015 Matta MNoopur Matta, A Study on Kuratowski’s Closure Complement Theorem, 2015.
2010 Schmidt MAndrin Schmidt, A Theory of Subsets of Topological Spaces, 2010, 21 pp.
2009 Strabel MGreg Strabel, The Kuratowski Closure-Complement Theorem, Math 4530 — Introduction to Topology, Lecturer: Tim Riley, TA: Andrew Marshall, Math Transit.com, 2009, 11 pp.
2007 Beckman MRyan T. Beckman, Basic Topology and the Kuratowski 14‑Set Problem, 2007, 70 pp.
2006 Nguyen MVan‑Trinh Nguyen, Concerning a Problem of K. Kuratowski, 2006, 30 pp.
1974 Grannan MPatrick James Grannan, The Kuratowski Closure and Complement Problem, 1974, 22 pp.
1970 Palmer MBobra Jane Pitman Palmer, The Kuratowski Closure Problem and Some Generalizations, 1970, 48 pp.
1941 Haight MFrank Haight, Independence of Topological Operators on a Lattice, 1941, 30 pp.
TALK SLIDES
2013 Ziemann SBen Ziemann, Kuratowski Numbers of Finite Topological Spaces, 2013, 24 slides.
2012 Awang SJenni Awang, Stars, Bars, and Languages: Kuratowski’s Closure‑Complement Theorem for Formal Languages, 2012, 30 slides.
2012 Harju STero Harju, Kuratowski Closure Results On Languages, 2012, 51 slides.
2012 PW SSzymon Plewik, Marta Walczyńska, Kuratowski Operations Revisited, 2012 Winter School in Abstract Analysis: Set Theory and Topology, Hejnice, Czech Republic, 23 slides.
2011 Moot SRichard Moot, Some Remarks on Logic and Topology, Linguistic Signs, Grammar and Meaning: Computational Logic of Language, 2011, 93 slides.
PROBLEMS
2012 AB PAlex Aguado, Closures and Complements, Problem 1888, Math. Mag., v. 85 no. 1, 2012, p. 62, Solution: Mark Bowron, v. 86 no. 1, 2013, pp. 67‑68.
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.
2004 BB PMark Bowron, Closure, Complement, and Union, Problem 11059, Amer. Math. Monthly, v. 111 no. 1, 2004, pp. 64‑65, Solution: Bruce Burdick, v. 113 no. 1, 2006, p. 83.
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.
1986 BF PEdwin Buchman, Generating Sets in a Topological Space, Problem E3144, Amer. Math. Monthly, v. 93 no. 4, 1986, p. 299, Solution: Jesús Ferrer, v. 95 no. 4, 1988, p. 353.
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.
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.
1969 Shum PKar‑Ping Shum, Bounds on the Kuratowski Family, Problem 153, Can. Math. Bull., v. 12 no. 2, 1969, p. 237, Solution: K. P. Shum, v. 14 no. 4, 1971, pp. 602‑603.
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.
1966 KM PWilliam Koenen, On Two Closure Problems, Problem E1941, Amer. Math. Monthly, v. 73 no. 10, 1966, p. 1123, Solution: L. F. Meyers, v. 75 no. 4, 1968, pp. 408‑409.
1965 LM PS. P. Lloyd, A Kuratowski Closure and Complement Problem, Problem 5349, Amer. Math. Monthly, v. 72 no. 10, 1965, p. 1136, Solution: J. C. Morgan II, v. 73 no. 10, 1966, pp. 1132‑1134.
Q&A THREADS
2017 SA QDistinct Sets in Topology?, studyAcer, 2017.
2016 AM QArkanj3l, mathematrucker, Which math books title every problem/exercise?, reddit, 2016.
2016 BDMSS Qbof, Did, Saaqib Mahmuud, sinbadh, Martin Sleziak, Prob. 21, Sec. 17 in Munkres’ TOPOLOGY, 2nd ed: Closure and Complementation Can Result in at Most 14 Different Sets, Mathematics Stack Exchange, 2016.
2016 RSSU Qrschwieb, Brian M. Scott, Slade, user43208, Moore Closure and Realizing the Kuratowski Monoid, Mathematics Stack Exchange, 2016.
2015 AGMU Qanakhronizein, Dejan Govc, mathematrucker, user210387, Does There Exist a Kuratowski Set Which is Uncountable?, Mathematics Stack Exchange, 2015.
2015 AMNPS QArthur, mathcounterexamples.net, J.‑E. Pin, SamM, Playing With Closure and Interior to Get the Maximum Number of Sets, Mathematics Stack Exchange, 2015.
2015 Anonymous QAnonymous, Kuratowski’s Closure Complement Problem, warosu.org, 2015.
2015 DKPST QDontknowanything, Arpit Kansal, phaiakia, Brian M. Scott, Tom‑Tom, Need a Counterexample Related to Interior and Closure, Mathematics Stack Exchange, 2015.
2015 MSU Qmathematrucker, Martin Sleziak, User23, Variations of Kuratowski Closure Complement Theorem, Mathematics Stack Exchange, 2015.
2015 MU Qmathematrucker, User23, Classifying Topological Spaces by Kuratowski Monoid, Mathematics Stack Exchange, 2015.
2015 OEWM Qoctatoan, endymion32, whirligig231, mathematrucker, The Kuratowski closure‑complement problem (very fun and not really topological), reddit, 2015.
2014 AACMMSU QMichael Albanese, arjafi, Andrés E. Caicedo, Saaqib Mahmuud, mathematrucker, Martin Sleziak, user45765, A Question on the Proof of 14 Distinct Sets Can Be Formed by Complementation and Closure, Mathematics Stack Exchange, 2014.
2014 ABDEJMNP Qabx, Kevin Beanland, Alex Degtyarev, Nate Eldredge, Will Jagy, mathematrucker, Norbert, Gerhard Paseman, Is Grothendieck Classification of Tensor Norms and Kuratowski’s 14 Sets Theorem Somehow Related?, MathOverflow, 2014.
2014 CMU QMario Carneiro, mathematrucker, user134824, How Many Sets Can You Get by Taking Closures, Complements, and Intersections?, Mathematics Stack Exchange, 2014.
2014 Campin QJack Campin, The Meaning of Life, S3 and the Kuratowski Closure‑Complement Theorem, sci.logic (at tech-archive.net), 2014.
2014 EMPS Qepimorphic, mathematrucker, Abhishek Parab, shamisen, A Variation of Kuratowski Closure-Complement Problem Using Dual Cones, Mathematics Stack Exchange, 2014.
2014 HM Qhengxin, mathematrucker, Applications of the Kuratowski Closure‑Complement Theorem, Mathematics Stack Exchange, 2014.
2013 ACFM QAshot, Haskell Curry, Arthur Fischer, mathematrucker, How to Prove That int(cl(A)) = cl(int(A))?, 2013.
2013 BC QB11, Haskell Curry, Example of a Set That Satisfies the Upper Bound in the Kuratowski Closure‑Complement Theorem, Mathematics Stack Exchange, 2013.
2013 BCGJKKMMUV QHenno Brandsma, Andrés E. Caicedo, Glougloubarbaki, Jared, kahen, Asaf Karagila, mathematrucker, mvcouwen, user64494, How Many Sets Can We Get by Taking Interiors and Closures?, Mathematics Stack Exchange, 2013.
2013 HJVV QJoel David Hamkins, Emil Jeřábek, Joseph Van Name, vincenzoml, Name of the Concept “Topological Boundary of A Intersected With A”, MathOverflow, 2013.
2012 ACGJ Qarjafi, Community, Dejan Govc, John, Does the Interior of a Kuratowski 14‑Set in a Finite Space Always Have Cardinality 1?, Mathematics Stack Exchange, 2012.
2012 DMMV QJason DeVito, Arturo Magidin, David Mitra, Voldemort, Is the Interior of a Set Always Equal to the Interior of its Closure?, Mathematics Stack Exchange, 2012.
2012 FGJMM QArthur Fischer, Dejan Govc, John, mathematrucker, Gerry Myerson, What is the Smallest Cardinality of a Kuratowski 14‑Set?, Mathematics Stack Exchange, 2012.
2012 MSSU QDylan Moreland, Brian M. Scott, Sigur, user31899, Closure of the Interior of Another Closure, Mathematics Stack Exchange, 2012.
2012 RU Qricky, user02138, What are Some Examples of a Mathematical Result Being Counterintuitive?, Mathematics Stack Exchange, 2012.
2011 AM QAdelle, mars, Distinct Sets in the Usual Topology of R, Topology Q+A Board, Ask a Topologist 2011.
2011 BL QBeni Bogosel, Zhen Lin, Why Some People Don't Like Proofs by Contradiction, Mathematics Stack Exchange, 2011.
2011 MTa Qmbowron, tylerc1991, Kuratowski’s Closure-Complement (Topology), Physics Forums, 2011.
2011 MTb Qmathematrucker, talisman, Kuratowski’s Closure-Complement Theorem, My Math Forum, 2011.
2010 AM Qanonymous, Austin Mohr, Funny Identities, Mathematics Stack Exchange, 2010.
2010 CCMY QS. Carnahan, Zev Chonoles, mathematrucker, Qiaochu Yuan, Kuratowski Closure-Complement Problem for Other Mathematical Objects?, MathOverflow, 2010.
2010 DHMM QFrançois G. Dorais, Joel David Hamkins, mathematrucker, Greg Muller, The Closure-Complement-Intersection Problem, MathOverflow, 2010.
2010 MMST Qmathematrucker, me@home, statham, t0rajir0u, Closure; Closed Set and Open Set, Art of Problem Solving, 2010.
2008 Brandsma QH. S. Brandsma, Topology, Kuratowski Problem, Topology Q+A Board, Ask a Topologist 2008.
2005 BGMS QRussell Blackadar, George Greene, mitch, Jim Sprigs, "Separate Entity" and Individuation in Mathematics, The Math Forum, 2005.
2005 BS QClaudio Buffara, Paul Santa Rita, Problema do Kuratowski (Kuratowski’s Problem), 2005, in Portuguese.
2005 CHOR QGeorge Cox, Stephen J. Herschkorn, Michael Orion, Keith Ramsay, Why 14 is an Interesting Number (Topology), The Math Forum, 2005.
2005 EM QWilliam Elliot, mitch, 14 Sets, 14 Modalities, Epistemic Logic, Cantor Spaces and an Euler Tour, The Math Forum, 2005.
2004 Mitch Qmitch, The Anticlassicalist }{ : Mitchism for Daleks, The Math Forum, 2004.
2002 LRT QMoufang Loop, Dave Rusin, Bill Taylor, Operators and Saltus, The Math Forum, 2002.
2001 KKR QKathleen, Andres Koropecki, Dave L. Renfro, Looking for a Subset of R that .... , The Math Forum, 2001.
1999 Bowron QMark Bowron, New Kuratowski Closure-Complement (and Union) Results?, The Math Forum, 1999.
1998 DGM QJulian Dontchev, Fred Galvin, David McIntyre, How Many Sets are Generated by Closure, Union and Complement?, The Math Forum, 1998.
1997 BHHPV QH. S. Brandsma, Kjetil Helstrup, Jeffrey C. Humpherys, Barry Paul, Valerio Venturi, Closure and Complement Give Only 14 Different Sets, The Math Forum, 1997.
1997 ENS QFred Erickson, J. H. Nieto, Brian M. Scott, Kuratowski — Distinct Sets from Closure and Complement, The Math Forum, 1997.
WEB PAGES
2015 Blogdemaths WLe Théorème de Kuratowski (The Theorem of Kuratowski), Blogdemaths, 2015, in French.
2015 PL WPrime.mover, Lord Farin, Kuratowski’s Closure‑Complement Problem, ProofWiki, 2015.
2015 Trimble WTodd Trimble, Closed Subspaces, nLab, 2015.
2015 Whitty WRobin Whitty, Kuratowski’s 14-Set Theorem, Theorem of the Day, 2015.
2012 Mathematrucker Wmathematrucker, Kuratowski’s Closure-Complement Cornucopia, Math Transit, 2012.
2012 Medina WMiguel A. M. Medina, El Teorema Clausura-Complemento de Kuratowski (Kuratowski’s Closure-Complement Theorem), Gaussian, 2012, in Spanish.
2012 Pfaltz WJohn L. Pfaltz, Closure Spaces: A Brief Overview, 2012.
2011 FB WKuratowski Closure-Complement Theorem, Facebook, 2011.
2010 CNPT Wchandrasekhar, nishantchandgotia, Abhishek Parab, Raghu Teja, Kuratowski’s Closure-Complement Problem, Concrete Nonsense, 2010.
2010 ElJj WEl Jj, Fermeture ou Complémentaire? (Closure or Complement?), Canalblog, 2010, in French.
2010 TFW WKuratowski’s Closure-Complement Problem: Wikis, The Full Wiki, 2010.
2009 Bowron WMark Bowron, The Kuratowski Closure-Complement Problem, Loci, v. 1, 2009.
2008 Richeson WDave Richeson, Kuratowski’s Closure-Complement Theorem, Division by Zero, 2008.
2007 Factobot Wfactobot, Kuratowski’s Closure-Complement Problem, Factolex, 2007.
2007 Mathematrucker Wmathematrucker, Kuratowski’s Closure-Complement Problem, Wikipedia, 2007.
2007 Nadish WNadish, Combining Interior, Complement, and Closure Operators, The Knot Atlas, 2007.
2003 FK WAlex Fink, Egil Kvaleberg et al., 14 (number), Wikipedia, 2003.
2000 BW WMark Bowron, Eric W. Weisstein, Kuratowski’s Closure-Complement Problem, MathWorld — A Wolfram Web Resource, 2000.
1999 Weisstein WEric W. Weisstein, Kuratowski’s Closure-Complement Problem, WolframAlpha, 1999.
1994 DR WJohn C. D. Diamantopoulos, Dave Rusin, Math 550 HELP !!!, The Mathematical Atlas: A Gateway to Modern Mathematics, 1994.
1972 BGS WM. Beeler, R. W. Gosper, R. Schroeppel, HAKMEM, Item 105, AI Memo 239, 1972, p. 45.
BOOKS
2016 Yan BMin Yan, Introduction to Topology, 2016, pp. 88‑89.
2015 Chamberland BMarc Chamberland, Single Digits: In Praise of Small Numbers, 2015, pp. 179‑181.
2009 GH BSteven Givant, Paul Halmos, Introduction to Boolean Algebras, 2009, p. 65.
2009 Patty BC. Wayne Patty, Foundations of Topology, 2009, p. 37.
2008 BBT BAndrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson, Elementary Real Analysis (second edition), 2008, p. 637.
2008 IKNV BO. A. Ivanov, V. M. Kharlamov, N. Yu. Netsvetaev, O. Ya. Viro, Elementary Topology: Problem Textbook, 2008, pp. 31‑32.
2003 Weisstein BEric W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd Edition, 2003, 3242 pp.
2000 Munkres BJames R. Munkres, Topology, Second Edition, 2000, p. 102.
1992 JJ BGlenn James, Robert C. James, Mathematics Dictionary, Fifth Edition, 1992, p. 240.
1976 KM BKazimierz Kuratowski, Andrzej Mostowski, Set Theory: With an Introduction to Descriptive Set Theory, 2nd Edition, Completely Revised, Studies in Logic and the Foundations of Mathematics, v. 86, 1976, pp. 27‑30.
1974 Munkres BJames R. Munkres, Topology: A First Course, 1974, p. 101.
1970 SS BLynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, 1970, pp. 61‑62.
1968 KM BKazimierz Kuratowski, Andrzej Mostowski, Set Theory, English translation by Maciej Mączyński, 1968, pp. 26‑30.
1968 Kelley BJohn L. Kelley, General Topology (with a foreword by P. S. Alexandrov), Russian translation by Mihail Âkovlevič Antonovskij, 1968, in Russian.
1966 Bourbaki BNicolas Bourbaki, Elements of Mathematics: General Topology Part 1, Exercise I.1.3, 1966, pp. 117‑118.
1966 Kuratowski‑a BKazimierz Kuratowski, Topology. Volume I. New Edition, Revised and Augmented., English translation by Jan Jaworowski, 1966, pp. 41‑43.
1966 Kuratowski‑b BKazimierz Kuratowski, Topology. Volume 1. With a foreword by P. S. Alexandrov., Russian translation by Mihail Âkovlevič Antonovskij, 1966, in Russian.
1963 Halmos BPaul Halmos, Lectures on Boolean Algebras, Van Nostrand Mathematical Studies, v. 1, 1963, p. 17.
1958 Kuratowski BKazimierz Kuratowski, Topologie. Volume 1, 4th Edition, Monografie Matematyczne, v. 20, 1958, in French.
1955 Kelley BJohn L. Kelley, General Topology, 1955, p. 57.
1952 Kuratowski BKazimierz Kuratowski, Teoria Mnogości (Set Theory), 1952, in Polish.
1948 Kuratowski BKazimierz Kuratowski, Topologie. I. Espaces Metrisables, Espaces Complets. 2nd ed., Revue et Augmentee (Topologie. I. 2nd edition, Revised and Augmented), Mathematical Monographs, v. 20, 1948, in French.
1933 Kuratowski BKazimierz Kuratowski, Topologie. I. Espaces Métrisables, Espaces Complets (Topology. I. Metric Spaces, Complete Spaces), 1933, pp. 48‑49, in French.
COURSEWORK
2017 Khatchatourian CIvan Khatchatourian, MAT327 Big List, 2017, p. 11.
2016 Kwasnicki CMateusz Kwaśnicki, Math 2001: Real Analysis Class Test, Oct 25, 2016, 2016, p. 4.
2014 Clark CDuncan Clark, What is Kuratowski’s 14 Set Theorem?, "What is ... ?" Seminar, Ohio State University, 2014, 5 pp.
2012 SM CT. Ø. Sørensen, A. Michelangeli, Functional Analysis, Problem in Class - Week 2, 2012, 2 pp.
2011 DL CEranda Dragoti‑Çela, Franz Lehner, Pflichtfach (Compulsory Subject), 2011, 5 pp, in German.
2009 Mohr CAustin Mohr, General Topology I: Math 730 Homework, Problem 3C, 2009, pp. 18‑19.
2006 Holm CTara Suzanne Holm, Problem Set 1, 2006, 1 p.
2004 MP CJesper Michael Møller, Morten Poulsen, Solutions to Exercises in Munkres, 2004, p. 3.
2002 McTague CCarl McTague, A Fun, Relatively Well-Known Problem, mctague.org, 2002, 8 pp.
LETTERS
1991 Robertson LJohn P. Robertson, News and Letters, Math. Mag., v. 64 no. 5, 1991, p. 361.
TALKS
2016 Jiaying TShen Jiaying, 14 Set Problem and Its Related Topics, 2016, in Chinese.
2014 PS TM. Pylypovych, M. Simkiv, Generalizing Kuratowski Theorem About 14 Sets, 2014.
2012 Pfeiffer TMarkus Pfeiffer, A Surprising Theorem, 2012.
2012 SC TClaudio Sacerdoti Coen, Kuratowski’s Closure-Complement Problem in Formal Topology, 2012.
2012 Walczynska TMarta Walczyńska, The Monoids of Kuratowski Operations, International Conference Dedicated to the 120th Anniversary of Stefan Banach, September 17‑21, 2012, Lviv, Ukraine.
2008 Sherman TDavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, UVA Topology Seminar, 2008.
2005 Lindsey TKathryn A. Lindsey, The Kuratowski Closure-Complement Problem, 2005.
2004 Aguado TAlex Aguado, Kuratowski’s Closure-Complement Problem, 2004.
2004 Jackson TMarcel Jackson, The Kuratowski Closure-Complement Problem, Analysis & Geometry Seminar, 2004.
2004 Sherman TDavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, Geometric Potpourri Seminar, 2004.
2003 Jackson TMarcel Jackson, The Kuratowski Closure-Complement Problem, 2003.
2000 Langford TEric Langford, Math Colloquium, 2000.
1975 Langford TEric Langford, Problems of Kuratowski Type Involving Unions and Intersections, Notices Amer. Math. Soc., v. 22, 1975, p. A-485.
1928 Zarycki TMiron Zarycki, General Properties of Cantorian Coherences, The February Meeting in New York, Bull. Amer. Math. Soc., v. 34 no. 3, 1928, pp. 262‑263.
VIDEOS
2013 Mathematrucker Vmathematrucker, Kuratowski’s Closure‑Complement Problem, YouTube, 2013.
SOFTWARE
2012 Bowron XMark Bowron, C Program for Applying Various Operations in Various Spaces, Math Transit.com, 2012.
HOME ABOUT NOTES AUTHORS
Kuratowski Closure-Complement Cornucopia
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 in the Polish journal Fundamenta Mathematicae the following year [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)], [1922 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.]. In this paper, he proved the following result:

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.

In his classic textbook which titled every problem [1955 Kelley BJohn L. Kelley, General Topology, 1955, p. 57.], John L. Kelley named this the Kuratowski Closure and Complement Problem.

kelley graph

Kelley’s General Topology spurred more papers on the subject.

Somewhat like PageRank, the number of citing sources within this website serves to rank the importance of the literature on the subject.

101:  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.
37:  2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.
35:  1960 Hammer AP. C. Hammer, Kuratowski’s Closure Theorem, Nieuw Arch. Wisk., v. 8 no. 2, 1960, pp. 74‑80.
29:  1955 Kelley BJohn L. Kelley, General Topology, 1955, p. 57.
22:  1971 Langford AEric Langford, Characterization of Kuratowski 14‑Sets, Amer. Math. Monthly, v. 78 no. 4, 1971, pp. 362‑367., 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.
21:  2010 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123., 1984 Peleg AD. Peleg, A Generalized Closure and Complement Phenomenon, Discrete Math., v. 50, 1984, pp. 285‑293.
19:  1962 Chapman‑b AThomas A. Chapman, A Further Note on Closure and Interior Operators, Amer. Math. Monthly, v. 69 no. 6, 1962, pp. 524‑529.
18:  1971 AS AJ. Anusiak, Kar‑Ping Shum, Remarks on Finite Topological Spaces, Colloq. Math., v. 23 no. 2, 1971, pp. 217‑223.
17:  1972 GKM AR. L. Graham, D. E. Knuth, T. S. Motzkin, Complements and Transitive Closures, Discrete Math., v. 2, 1972, pp. 17‑29., 1967 Aull AC. E. Aull, Classification of Topological Spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., v. 15, 1967, pp. 773‑778.
16:  1991 Fife AJames H. Fife, The Kuratowski Closure-Complement Problem, Math. Mag., v. 64 no. 3, 1991, pp. 180‑182.
15:  1966 Koenen AWilliam Koenen, The Kuratowski Closure Problem in the Topology of Convexity, Amer. Math. Monthly, v. 73 no. 7, 1966, pp. 704‑708., 1944 MT AJ. C. C. McKinsey, Alfred Tarski, The Algebra of Topology, Ann. of Math., v. 45, 1944, pp. 141‑191.
14:  1927 Zarycki AMiron Zarycki, Quelques Notions Fondamentales de l'Analysis Situs au Point du Vue de l'Algèbre de la Logique (Some Fundamental Concepts of Topology in Terms of the Algebra of Logic), Fund. Math., v. 9, 1927, pp. 3‑15, in French.
13:  2000 Munkres BJames R. Munkres, Topology, Second Edition, 2000, p. 102., 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.
12:  1982 Chagrov AA. V. Chagrov, Kuratowski Numbers, Appl. Funct. Anal. Approx. Th., 1982, pp. 186‑190, in Russian., 1970 SS BLynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, 1970, pp. 61‑62., 1961 Levine ANorman Levine, On the Commutativity of the Closure and Interior Operators in Topological Spaces, Amer. Math. Monthly, v. 68 no. 5, 1961, pp. 474‑477.
11:  2009 BGS‑b AJanusz Brzozowski, Elyot Grant, Jeffrey Shallit, Closures in Formal Languages and Kuratowski’s Theorem, Developments in Language Theory, 13th International Conference, DLT 2009, Lecture Notes in Computer Science, v. 5583, pp. 125‑144.
10:  1978 Fishburn APeter C. Fishburn, Operations on Binary Relations, Discrete Math., v. 21, 1978, pp. 7‑22., 1966 Kuratowski‑a BKazimierz Kuratowski, Topology. Volume I. New Edition, Revised and Augmented., English translation by Jan Jaworowski, 1966, pp. 41‑43.
9:  1975 BJ AJoel Berman, Steven L. Jordan, The Kuratowski Closure-Complement Problem, Amer. Math. Monthly, v. 82 no. 8, 1975, pp. 841‑842., 1974 Munkres BJames R. Munkres, Topology: A First Course, 1974, p. 101.
8:  1936 Sanders ASamuel T. Sanders Jr., Derived Sets and Their Complements, Bull. Amer. Math. Soc., v. 42, 1936, pp. 577‑584.
7:  1977 Moser ALouise E. Moser, Closure, Interior, and Union in Finite Topological Spaces, Colloq. Math., v. 38, 1977, pp. 41‑51., 1972 Pigozzi AD. Pigozzi, On Some Operations on Classes of Algebras, Algebra Universalis, v. 2, 1972, pp. 346‑353., 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., 1966 Kuratowski‑b BKazimierz Kuratowski, Topology. Volume 1. With a foreword by P. S. Alexandrov., Russian translation by Mihail Âkovlevič Antonovskij, 1966, in Russian.
6:  2011 SW AJeffrey Shallit, Ross Willard, Kuratowski’s Theorem for Two Closure Operators, arXiv:1109.1227 [math.GN], arXiv.org, 2011, 6 pp., 1986 BF PEdwin Buchman, Generating Sets in a Topological Space, Problem E3144, Amer. Math. Monthly, v. 93 no. 4, 1986, p. 299, Solution: Jesús Ferrer, v. 95 no. 4, 1988, p. 353., 1984 Shum AKar‑Ping Shum, On the Boundary of Kuratowski 14‑Sets in Connected Spaces, Glas. Mat. Ser. III, v. 19, 1984, pp. 293‑296., 1982 Soltan AV. P. Soltan, Problems of Kuratowski Type, Mat. Issled., v. 65, 1982, pp. 121‑131, in Russian., 1974 GE AYu. R. Gaida, A. É. Eremenko, On the Frontier Operator in Boolean Algebras with a Closure, Ukr. Math. J. (English), v. 26 no. 6, 1974, pp. 662‑664.
5:  1996 Shum AKar‑Ping Shum, Closure Functions on the Set of Positive Integers, Sci. China Ser. A, v. 39 no. 4, 1996, pp. 337‑346., 1980 Soltan AV. P. Soltan, On Kuratowski’s Problem, Bull. Acad. Pol. Sci., Sér. Sci. Math., v. 28, 1980, pp. 369‑375, in Russian., 1966 Bourbaki BNicolas Bourbaki, Elements of Mathematics: General Topology Part 1, Exercise I.1.3, 1966, pp. 117‑118.
4:  2003 Brandsma AH. S. Brandsma, The Fourteen Subsets Problem: Interiors, Closures and Complements, Topology Explained, Topology Atlas, 2003, 4 pp., 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., 1995 MT AM. S. Moslehian, N. Tavallaii, A Generalization of the Kuratowski Closure-Complement Problem, Punjab Univ. J. Math., v. 28, 1995, pp. 1‑9., 1995 GO‑a AEmmanuelle Garel, Jean Pierre Olivier, On Closures Verifying that the Interior of a Closed Element is Closed, Comm. Algebra, v. 23, 1995, pp. 3715‑3728., 1993 Shum AKar‑Ping Shum, The Amalgamation of Closure and Boundary Functions on Semigroups and Partially Ordered Sets, Proceedings of the Conference on Ordered Structures and Algebra of Computer Languages: Hong Kong, 26‑29 June 1991, 1993, pp. 232‑243., 1980 FP AM. J. Fischer, M. S. Paterson, The Fast Skew-Closure Algorithm, L'Enseign. Math., v. 26, 1980, pp. 345‑360., 1977 Blok AW. J. Blok, The Free Closure Algebra on Finitely Many Generators, Ind. Math., v. 80, 1977, pp. 362‑379., 1975 YS AKai-Wing Yip, Kar‑Ping Shum, On the Structure of Kuratowski Sets, J. Chin. Univ. Hong Kong, v. 3, 1975, pp. 429‑439., 1972 CJ AS. Comer, J. Johnson, The Standard Semigroup of Operators of a Variety, Algebra Universalis, v. 2, 1972, pp. 77‑79., 1967 Nelson AEvelyn Nelson, Finiteness of Semigroups of Operators in Universal Algebra, Canad. J. Math., v. 19, 1967, pp. 764‑768., 1963 Halmos BPaul Halmos, Lectures on Boolean Algebras, Van Nostrand Mathematical Studies, v. 1, 1963, p. 17., 1962 Chapman‑a AThomas A. Chapman, An Extension of the Kuratowski Closure and Complementation Problem, Math. Mag., v. 35 no. 1, 1962, pp. 31‑35., 1937 Stopher AE. C. Stopher Jr., Cyclic Relations in Point Set Theory, Bull. Amer. Math. Soc., v. 43, 1937, pp. 686‑694., 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.

Kelley’s name stuck for the rest of the 20th century: the subset count did not appear in the title of a paper until 2003, in Henno Brandsma’s Fourteen Subsets Problem [2003 Brandsma AH. S. Brandsma, The Fourteen Subsets Problem: Interiors, Closures and Complements, Topology Explained, Topology Atlas, 2003, 4 pp.]. Just one year later, David Sherman claimed that Kuratowski’s result is “known as” the 14‑set theorem [2004 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, arXiv:math/0405401 [math.GN], arXiv.org, 2004, 11 pp.] [2010 Sherman ADavid Sherman, Variations on Kuratowski’s 14‑Set Theorem, Amer. Math. Monthly, v. 117 no. 2, 2010, pp. 113‑123.].* Other authors followed [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.]. The count first appeared in the theorem’s name in 1997 [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.] and again in 1998 [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.].” 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.] combined the names: “the famous 14‑set closure‑complement Theorem of Kuratowski.”

In these pages we chronicle the abundance of literature related to the Kuratowski closure‑complement theorem.

updated 23 Jun 2019
by mathematrucker
 mathematrucker smiley
[portmanteau above]@gmail.com

*Edwin Hewitt criticizes Kelley for similar infractions in a 1956 book review of his textbook:

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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