REFERENCES
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2012 Aguado PAlex Aguado, On the Cardinality of the Kuratowski Family, Problem 1888, Math. Mag., v. 85 no. 1, 2012, p. 62.
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 Awang SJenni Awang, Stars, Bars, and Languages: Kuratowski's Closure‑Complement Theorem for Formal Languages, 2012, 30 slides.
2012 Bowron PMark Bowron, How Small Can a Kuratowski 14‑Set Be?, Problem 1898, Math. Mag., v. 85 no. 3, 2012, p. 228.
2012 Cato ASusumu Cato, Complements and Consistent Closures, Discrete Math., v. 312, 2012, pp. 1218‑1221.
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 Chagrov AA. V. Chagrov, Kuratowski Numbers, English translation by Mark Bowron, Math Transit.com, 2012, 6 pp.
2012 Harju STero Harju, Kuratowski Closure Results On Languages, 2012, 51 slides.
2012 Kuratowski AKazimierz 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.
2012 Pfeiffer TMarkus Pfeiffer, A Surprising Theorem, 2012.
2012 PW SSzymon Plewik, Marta Walczyńska, Kuratowski Operations Revisited, 2012 Winter School in Abstract Analysis: Set Theory and Topology, Hejnice, Czech Republic, 23 slides.
2012 PW‑a ASzymon Plewik, Marta Walczyńska, Kuratowski Operations Revisited, arXiv:1205.3391 [math.GN], arXiv.org, 2012, 18 pp.
2012 PW‑b ASzymon Plewik, Marta Walczyńska, The Monoid Consisting of Kuratowski Operations, arXiv:1205.3391v2 [math.GN], arXiv.org, 2012, 20 pp.
2012 Sacerdoti Coen TClaudio Sacerdoti Coen, Kuratowski's Closure-Complement Problem in Formal Topology, 2012.
2012 Soltan‑a AV. P. Soltan, The Kuratowski Number of a Closure Space, English translation by Mark Bowron, Math Transit.com, 2012, 1 p.
2012 Soltan‑b AV. P. Soltan, On Kuratowski's Problem, English translation by Mark Bowron, Math Transit.com, 2012, 11 pp.
2012 Soltan‑c AV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 2012, 18 pp.
2012 Walczyńska TMarta Walczyńska, The Monoids of Kuratowski Operations, International Conference Dedicated to the 120th Anniversary of Stefan Banach, September 17‑21, 2012, Lviv, Ukraine.
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.
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 Moot SRichard Moot, Some Remarks on Logic and Topology, Linguistic Signs, Grammar and Meaning: Computational Logic of Language, 2011, 93 slides.
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 Schmidt MAndrin Schmidt, A Theory of Subsets of Topological Spaces, 2010, 21 pp.
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 GH BSteven Givant, Paul Halmos, Introduction to Boolean Algebras, 2009, p. 65.
2009 Strabel MGreg Strabel, The Kuratowski Closure-Complement Theorem, Math 4530 — Introduction to Topology, Lecturer: Tim Riley, TA: Andrew Marshall, 2009, 11 pp.
2008 BBT BAndrew M. Bruckner, Judith B. Bruckner, Brian S. Thomson, Elementary Real Analysis (second edition), 2008, p. 637.
2008 GJ AB. J. Gardner, Marcel Jackson, The Kuratowski Closure-Complement Theorem, New Zealand J. Math., v. 38, 2008, pp. 9‑44.
2008 IKNV BO. A. Ivanov, V. M. Kharlamov, N. Yu. Netsvetaev, O. Ya. Viro, Elementary Topology: Problem Textbook, 2008, pp. 31‑32.
2008 Sherman TDavid Sherman, Variations on Kuratowski's 14‑Set Theorem, UVA Topology Seminar, 2008.
2007 Beckman MRyan T. Beckman, Basic Topology and the Kuratowski 14‑set Problem, 2007, 70 pp.
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.
2006 Nguyen MVan‑Trinh Nguyen, Concerning a Problem of K. Kuratowski, 2006, 30 pp.
2005 Frasson DMiguel V. S. Frasson, Stellingen (Propositions) in: Large Time Behaviour of Neutral Delay Systems, 2005, 2 pp.
2005 Grabowski AAdam Grabowski, On the Computer-Checked Solution of the Kuratowski Closure-Complement Problem, Mech. Math. Appl., v. 4, 2005, pp. 25‑34.
2005 Lindsey TKathryn A. Lindsey, The Kuratowski Closure-Complement Problem, 2005.
2005 Muhm DPhilip Muhm, Kuratowski's 14‑Set Theorem — A Modal Logic View, 2005, 58 pp.
2004 Aguado TAlex Aguado, Kuratowski's Closure-Complement Problem, 2004.
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.
2004 Farmer AFrank D. Farmer, Containment for c, s, t Operators on a Binary Relation, Algebra Universalis, v. 51, 2004, pp. 153‑175.
2004 Jackson‑a AMarcel Jackson, Semilattices with Closure, Algebra Universalis, v. 52 no. 1, 2004, pp. 1‑37.
2004 Jackson‑b TMarcel Jackson, The Kuratowski Closure-Complement Problem, Analysis & Geometry Seminar, 2004.
2004 JS AMarcel Jackson, Tim Stokes, Semilattice Pseudo-complements on Semigroups, Comm. Algebra, v. 32, 2004, pp. 2895‑2918.
2004 Sherman TDavid Sherman, Variations on Kuratowski's 14‑Set Theorem, Geometric Potpourri Seminar, 2004.
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.
2003 Jackson TMarcel Jackson, The Kuratowski Closure-Complement Problem, 2003.
2003 Weisstein BEric W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd Edition, 2003, 3242 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 Tasić 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 Langford TEric Langford, Math Colloquium, 2000.
2000 Munkres BJames R. Munkres, Topology, Second Edition, 2000, p. 102.
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 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.
1996 Pfaltz AJohn L. Pfaltz, Closure Lattices, Discrete Math., v. 154, 1996, pp. 217‑236.
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 EC. E. Aull, Corrections and Additions for Classification of Topological Spaces II, Bull. Pol. Acad. Sci. Math., v. 40 no. 2, 1992, p. 175.
1992 JJ BGlenn James, Robert C. James, Mathematics Dictionary, Fifth Edition, 1992, p. 240.
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.
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.
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 Soltan AV. P. Soltan, Problems of Kuratowski Type, Mat. Issled., v. 65, 1982, pp. 121‑131, in Russian.
1981 Soltan AV. P. Soltan, On Kuratowski's Problem, Bull. Acad. Pol. Sci., Sér. Sci. Math., v. 28, 1981, pp. 369‑375, in Russian.
1980 FP AM. J. Fischer, M. S. Paterson, The Fast Skew-Closure Algorithm, L'Enseign. Math., v. 26, 1980, pp. 345‑360.
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.
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.
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 AI. Z. Kleiner, 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.
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.
1975 BJ AJoel Berman, Steven L. Jordan, The Kuratowski Closure-Complement Problem, Amer. Math. Monthly, v. 82 no. 8, 1975, pp. 841‑842.
1975 Langford TEric Langford, Problems of Kuratowski Type Involving Unions and Intersections, Notices Amer. Math. Soc., v. 22, 1975, p. A-485.
1975 SY AKar‑Ping Shum, K. W. Yip, 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 Gol'dberg AA. A. Gol'dberg, On Boolean Algebras with a Derivation, Ukr. Math. J. (English), v. 26, 1974, pp. 365‑369.
1974 Grannan MPatrick James Grannan, The Kuratowski Closure and Complement Problem, 1974, 22 pp.
1974 Munkres BJames R. Munkres, Topology: A First Course, 1974, p. 101.
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.
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 Palmer MBobra Jane Pitman Palmer, The Kuratowski Closure Problem and Some Generalizations, 1970, 48 pp.
1970 SS BLynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, 1970, pp. 61‑62.
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 Aull EC. E. Aull, Corrigendum: Classification of Topological Spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., v. 16 no. 6, 1968.
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.
1968 Kelley BJohn L. Kelley, General Topology (with a foreword by P. S. Alexandrov), Russian translation by Mihail Âkovlevič Antonovskij, 1968, in Russian.
1968 KM BKazimierz Kuratowski, Andrzej Mostowski, Set Theory, English translation by Maciej Mączyński, 1968, pp. 26‑30.
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 Bourbaki BNicolas Bourbaki, Elements of Mathematics: General Topology Part 1, Exercise I.1.3, 1966, pp. 117‑118.
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 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.
1966 Koenen AWilliam Koenen, The Kuratowski Closure Problem in the Topology of Convexity, Amer. Math. Monthly, v. 73 no. 7, 1966, pp. 704‑708.
1966 Kuratowski BKazimierz Kuratowski, Topology. Volume I. New Edition, Revised and Augmented., English translation by Jan Jaworowski, 1966, pp. 41‑43.
1966 Kuratowski BKazimierz Kuratowski, Topology. Volume 1. With a foreword by P. S. Alexandrov., Russian translation by Mihail Âkovlevič Antonovskij, 1966, in Russian.
1965 Hintikka AJ. Hintikka, A Closure and Complement Result for Nested Topologies, Fund. Math., v. 57 no. 1, 1965, pp. 97‑106.
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.
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.
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.
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.
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.
1952 Kuratowski BKazimierz Kuratowski, Teoria Mnogości (Set Theory), 1952, in Polish.
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.
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.
1947 Zarycki AMiron Zarycki, The Role of Cardinality in the Notion of the Derivation of a Set in Abstract Spaces, Nauk. Zap. Ser. Fiz.‑Mat., v. 5 no. 1, 1947, in Ukrainian.
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.
1937 Stopher DE. C. Stopher Jr., Interrelations of a Family of Operators on Point Sets and Their Canonical Representation, 1937, 50 pp.
1936 Sanders ASamuel T. Sanders Jr., Derived Sets and Their Complements, Bull. Amer. Math. Soc., v. 42, 1936, pp. 577‑584.
1935 Sanders DSamuel T. Sanders Jr., Derived Sets and Their Complements: The Characterization of Certain Set-Valued Set-Functions, 1935, 74 pp.
1933 Kuratowski BKazimierz Kuratowski, Topologie. I. Espaces Métrisables, Espaces Complets (Topology. I. Metric Spaces, Complete Spaces), 1933, pp. 48‑49, in French.
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 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.
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.
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.
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.

OTHER
 =  coursework
 =  letter
 =  Q&A thread
 =  web page
 =  software

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, Stack Exchange, 2013.
2012 Bowron XMark Bowron, C Program for Applying Various Operations in Various Spaces, Math Transit.com, 2012.
2012 DMMV QJason DeVito, Arturo Magidin, David Mitra, Voldemort, Is the Interior of a Set Always Equal to the Interior of its Closure?, Stack Exchange, 2012.
2012 Dranishnikov CAlexander Dranishnikov, Homework and Extra Credit Problems, MTG4302/MTG5316, 2012, 2 pp.
2012 FGJMM QArthur Fischer, Dejan Govc, John, mathematrucker, Gerry Myerson, What is the Smallest Cardinality of a Kuratowski 14‑Set?, Stack Exchange, 2012.
2012 Medina WMiguel A. M. Medina, El Teorema Clausura-Complemento de Kuratowski (Kuratowski's Closure-Complement Theorem), Gaussian, 2012, in Spanish.
2012 MSSU QDylan Moreland, Brian M. Scott, Sigur, user31899, Closure of the Interior of Another Closure, Stack Exchange, 2012.
2012 Pfaltz WJohn L. Pfaltz, Closure Spaces: A Brief Overview, 2012.
2012 RU Qricky, user02138, What are Some Examples of a Mathematical Result Being Counterintuitive?, 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, Stack Exchange, 2011.
2011 DL CEranda Dragoti‑Çela, Franz Lehner, Pflichtfach (Compulsory Subject), 2011, 5 pp, in German.
2011 FB WKuratowski Closure-Complement Theorem, Facebook, 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, Stack Exchange, 2010.
2010 CCMY QS. Carnahan, Zev Chonoles, mathematrucker, Qiaochu Yuan, Kuratowski Closure-Complement Problem for Other Mathematical Objects?, MathOverflow, 2010.
2010 CNPT Wchandrasekhar, nishantchandgotia, Abhishek Parab, Raghu Teja, Kuratowski's Closure-Complement Problem, Concrete Nonsense, 2010.
2010 DHMM QFrançois G. Dorais, Joel David Hamkins, mathematrucker, Greg Muller, The Closure-Complement-Intersection Problem, MathOverflow, 2010.
2010 El Jj WEl Jj, Fermeture ou Complémentaire? (Closure or Complement?), Canalblog, 2010, in French.
2010 Robertson WStewart Robertson, Kuratowski's Closure-Complement Problem, Musings of a Maths Student: A Public Log of My Studies in Mathematics, 2010.
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.
2009 Mohr CAustin Mohr, General Topology I: Math 730 Homework, Problem 3C, 2009, pp. 18‑19.
2009 Rickey WFrederick Rickey, Kazimierz Kuratowski (1896-1980), 2009.
2008 Brandsma QH. S. Brandsma, Topology, Kuratowski Problem, Topology Q+A Board, Ask a Topologist 2008.
2008 Richeson WDave Richeson, Kuratowski's Closure-Complement Theorem, Division by Zero, 2008.
2008 Woo WChi Woo, Kuratowski Closure-Complement Theorem, Planet Math, 2008.
2007 Bowron WMark Bowron et al., Kuratowski's Closure-Complement Problem, Wikipedia, 2007.
2007 Factobot Wfactobot, Kuratowski's Closure-Complement Problem, Factolex, 2007.
2007 Nadish WNadish, Combining Interior, Complement, and Closure Operators, The Knot Atlas, 2007.
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 Litherland CRichard A. Litherland, Kuratowski's Closure-Complement Problem, Math 7510 Fall 2004, 2 pp.
2004 Mitch Qmitch, The Anticlassicalist }{ : Mitchism for Daleks, The Math Forum, 2004.
2004 MP CJesper Michael Møller, Morten Poulsen, Solutions to Exercises in Munkres, 2004, p. 3.
2003 FK WAlex Fink, Egil Kvaleberg et al., 14 (number), Wikipedia, 2003.
2002 LRT QMoufang Loop, Dave Rusin, Bill Taylor, Operators and Saltus, The Math Forum, 2002.
2002 McTague CCarl McTague, A Fun, Relatively Well-Known Problem, mctague.org, 2002, 8 pp.
2001 KKR QKathleen, Andres Koropecki, Dave L. Renfro, Looking for a Subset of R that .... , The Math Forum, 2001.
2000 BW WMark Bowron, Eric W. Weisstein, Kuratowski's Closure-Complement Problem, MathWorld — A Wolfram Web Resource, 2000.
1999 Bowron QMark Bowron, New Kuratowski Closure-Complement (and Union) Results?, The Math Forum, 1999.
1999 Weisstein WEric W. Weisstein, Kuratowski's Closure-Complement Problem, WolframAlpha, 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.
1994 DR WJohn C. D. Diamantopoulos, Dave Rusin, Math 550 HELP !!!, The Mathematical Atlas: A Gateway to Modern Mathematics, 1994.
1991 Robertson LJohn P. Robertson, News and Letters, Math. Mag., v. 64 no. 5, 1991, p. 361.
1972 BGS WM. Beeler, R. W. Gosper, R. Schroeppel, HAKMEM, Item 105, AI Memo 239, 1972, p. 45.

Kuratowski's Closure-Complement Cornucopia
by Mark Bowron
Fans of mathematics have much to be thankful for, including the works of the great Polish mathematician Kazimierz Kuratowski (1896‑1980), After finishing his Ph.D. at the University of Warsaw in 1921, the first part of his dissertation was published in the Polish research periodical Fundamenta Mathematicae [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 AKazimierz 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)]. In it Kuratowski observed that 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. This unusual result eventually became known as Kuratowski's closure‑complement theorem (also 14‑set theorem, or problem). It has inspired a veritable cornucopia's worth of related results over time, some surprisingly deep.

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.].

The purpose of the present article is to harvest this literature and serve it up as a mathematical feast. On the appetizer plate are some delicious English translations: [2012 Chagrov AA. V. Chagrov, Kuratowski Numbers, English translation by Mark Bowron, Math Transit.com, 2012, 6 pp.], [2012 Kuratowski AKazimierz 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.], [2012 Soltan‑a AV. P. Soltan, The Kuratowski Number of a Closure Space, English translation by Mark Bowron, Math Transit.com, 2012, 1 p.], [2012 Soltan‑b AV. P. Soltan, On Kuratowski's Problem, English translation by Mark Bowron, Math Transit.com, 2012, 11 pp.], [2012 Soltan‑c AV. P. Soltan, Problems of Kuratowski Type, English translation by Mark Bowron, Math Transit.com, 2012, 18 pp.], [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.]. The main dish consists of a hearty soup of references. For dessert, a light and fluffy C program capable of applying various operations in various spaces is served: [2012 Bowron XMark Bowron, C Program for Applying Various Operations in Various Spaces, Math Transit.com, 2012.].

Bon appétit!

Last updated 23 Jan 2014