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.

                                                —Brzozowski, Grant, and Shallit

This page lists and assigns a broad subject area to papers in the literature above. Link colors denote free, hoops, paywall.

2025F. Ciraulo, Kuratowski’s problem in constructive topology. Journal of Logic & Analysis, 17:FDS1 1–27, 1 February 2025. JLA Wayback Scholar Semantic
2024L. Staiger and K. W. Wagner, The forty nine Kuratowski lattices in the Cantor space. Research Report 573, April 2024. CDMTCS Wayback Scholar Semantic ResearchGate
2024P. Caron, J.-G. Luque, B. Patrou, Presentation of monoids generated by a projection and an involution. 26 pp., 2 February 2024. arXiv Wayback Scholar Semantic
2024M. Bowron, Boundary-border extensions of the Kuratowski monoid. Topology and its Applications, 341(108703):1–35, 1 January 2024. Elsevier Scholar Semantic MR ZB arXiv Wayback
2023P. Caron, A. Durand, B. Patrou, A study of monoids generated by three unary regular operations: reversal, root and star. Elsevier preprint, 16 May 2023. SSRN Wayback Scholar
2022J. Dassow, The orbit of closure-involution operations: the case of Boolean functions. Contributions to Algebra and Geometry, 63(2):321–334, 1 June 2022. Springer Wayback Scholar Semantic MR ZB
2021J. Dassow, Some remarks on the orbit of closure-involution operations on languages. Information and Computation, 281(104811), 8 pp., December 2021. Elsevier ResearchGate Scholar Semantic MR ZB
2021S. Canilang, M. P. Cohen, N. Graese, I. Seong, The closure-complement-frontier problem in saturated polytopological spaces. New Zealand Journal of Mathematics, 51:3–27, 6 August 2021. NZMS Wayback Scholar Semantic MR ZB arXiv Wayback
2020M. P. Cohen, T. Johnson, A. Kral, A. Li, J. Soll, A Kuratowski closure-complement variant whose solution is independent of ZF. arXiv preprint, 26 May 2020. arXiv Wayback Scholar Semantic
2020E. Przemska, The lattices of families of regular sets in topological spaces. Mathematica Slovaca, 70(2):477–488, 2020. De Gruyter ResearchGate Wayback Scholar Semantic MR ZB
2020I. Cohen, G. Wences, J. R. Contreras, A. R. Linero, R. S. S. Anillo, A linear version of the Kuratowski problem (in Spanish). Journal of the Foundations of Science, 5(1):63–71, January 2020. Wayback Scholar Semantic English translation by Google Translate
2019J. Jirásek, M. Palmovský, J. Šebej, Kuratowski Algebras Generated by Prefix-, Suffix-, Factor-, and Subword-Free Languages Under Star and Complementation. International Journal of Foundations of Computer Science, 30(6–7):1091–1115, September 2019. World Scientific Scholar Semantic MR ZB Springer HAL Wayback Scholar Semantic ZB
2019J. Dassow, On the orbit of closure-involution operations - the case of formal languages. Theoretical Computer Science, 777:192–203, 19 July 2019. Elsevier Scholar Semantic MR ZB
2019J. H. Santiago, The group‑theoretic analog of Kuratowski’s closure‑complement theorem. American Mathematical Monthly, 126(6)519–526, June 2019. Taylor & Francis JSTOR UNAM Wayback Scholar Semantic MR ZB
2019B. Nikolić, Degenerated topological spaces. Novi Sad Journal of Mathematics, 49(2)95–107, 31 March 2019. University of Novi Sad Wayback Scholar Semantic ZB
2019M. V. Volkov, The identities of the free product of a pair of two-element monoids. Algebra Universalis, 80(14), 6 pp., 22 February 2019. Springer Scholar Semantic MR ZB arXiv Wayback
2018T. Banakh, O. Chervak, T. Martynyuk, M. Pylypovych, A. Ravsky, M. Simkiv, Kuratowski monoids of n‑topological spaces. Topological Algebra and its Applications, 6:1–25, 28 March 2018. De Gruyter Wayback Scholar Semantic MR ZB arXiv Wayback
2018A. P. Farajzadeh and A. Shafie, On Kuratowski’s problem in vector spaces. Bulletin of L. N. Gumilyov Eurasian National University (Series: Mathematics, Computer Science, Mechanics), 125(4):117-119, 2018. Wayback Scholar Semantic arXiv Wayback
2017P. Gammie and G. Gioiosa, The Kuratowski Closure-Complement Theorem. Archive of Formal Proofs, 27 October 2017. Isabelle-AFP Wayback Scholar Semantic
2017H. Wang, Y. Wang, D. Tao, Y. Peng, Kuratowski monoids of various orders on lattices. Journal of Suzhou University of Science and Technology (Natural Science Edition), 34(3):15-18, September 2017. CAOD FX ZB
2017K. P. Shum, A note on Kuratowski’s theorem and its related topics. Advances in Pure Mathematics 7(8):383–406, August 2017. SCIRP Wayback Scholar Semantic ResearchGate Wayback
2017R. Berghammer, Tool-based relational investigation of closure-interior relatives for finite topological spaces. In P. Höfner, D. Pous, and G. Struth, editors, Relational and Algebraic Methods in Computer Science, RAMICS 2017, Lyon, France, volume 10226 of Lecture Notes in Computer Science (LNICS), pp. 60–76, 2017. Springer ResearchGate Scholar Semantic MR ZB
2017A. Gupta and R. D. Sarma, A note on some generalized closure and interior operators in a topological space. Mathematics for Applications, 6:11–20, 2017. Brno University of Technology Wayback Scholar Semantic MR ZB
2017R. C. Schwiebert, The radical-annihilator monoid of a ring. Communications in Algebra, 45(4):1601–1617, 2017. Taylor & Francis Scholar Semantic MR ZB arXiv Wayback
2016C. Câmpeanu, N. Moreira, R. Reis, Distinguishability operations and closures. Fundamenta Informaticae, 148(3–4):243–266, 24 December 2016. IOS Press University of Porto Wayback Scholar Semantic MR ZB arXiv Wayback
2016B. Tasić, Operator properties of congruence permutable varieties with strongly definable principal congruences. Algebra Universalis, 75:61–74, 1 February 2016. Springer ResearchGate Scholar Semantic MR ZB
2016J. Jirásek and J. Šebej, Kuratowski algebras generated by prefix‑free languages. In Y. S. Han and K. Salomaa, editors, Implementation and Application of Automata, 21st International Conference, CIAA 2016, Seoul, South Korea, July 19-22, 2016, Proceedings, volume 9705 of Lecture Notes in Computer Science (LNICS), pp. 150–162, 2016. Springer Scholar Semantic MR ZB
2015J. Jirásek and G. Jirásková, On the boundary of regular languages. Theoretical Computer Science, 578:42–57, 3 May 2015. Elsevier Scholar Semantic MR ZB Springer St. Mary’s University Wayback
2014R. D. Sarma and A. Gupta, On the k‑numbers of some generalized Kuratowski operators in topology. In 3rd International Conference On Innovative Approach in Applied Physical, Mathematical/Statistical, Chemical Sciences and Emerging Energy Technology for Sustainable Development, APMSCSET 2014, New Delhi, India, September 27-28, 2014, Proceedings, pp. 50–54, 2014. ResearchGate Krishi Sanskriti Wayback Scholar
2014E. A. Ok and G. Riella, Topological closure of translation invariant preorders. Mathematics of Operations Research, 39(3):737–745, August 2014. INFORMS JSTOR Scholar Semantic MR ZB EPRG Wayback
2014A. Grabowski, Topological interpretation of rough sets. Formalized Mathematics, 22(1):89–97, March 2014. De Gruyter Scholar Semantic ZB
2014J. Brzozowski, G. Jirásková, C. Zou, Quotient complexity of closed languages. Theory of Computing Systems, 54:277–292, February 2014. Springer Scholar Semantic MR ZB Springer arXiv Wayback
2013É. Charlier, M. Domaratzki, T. Harju, J. Shallit, Composition and orbits of language operations: finiteness and upper bounds. International Journal of Computer Mathematics, 90(6):1171–1196, 2013. Taylor & Francis Scholar Semantic MR ZB Springer arXiv Wayback
2013S. Plewik and M. Walczyńska, The monoid consisting of Kuratowski operations. Journal of Mathematics, 2013(289854), 9 pp., 6 March 2013. Wiley Wayback Scholar Semantic MR ZB arXiv Wayback
2013M. Bowron (proposer), B. S. Burdick (solver), Problem 1898: More on Kuratowski 14-sets. Mathematics Magazine, 85(3):228, March 2012 (proposal), 86(3):231, March 2013 (solution). Taylor & Francis JSTOR ResearchGate Wayback
2013A. Aguado (proposer), M. Bowron (solver), Problem 1888: Closures and complements. Mathematics Magazine, 85(1):62 January 2012 (proposal), 86(1):67–68, February 2013 (solution). Taylor & Francis JSTOR ResearchGate Wayback
2012G. Jirásková and J. Shallit, The state complexity of star-complement-star. In H. C. Yen, O. H. Ibarra (editors), Developments in Language Theory (DLT 2012), Taipei, Taiwan, August 14-17, 2012, volume 7410 of Lecture Notes in Computer Science (LNICS), Springer, Berlin, Heidelberg, pp. 380–391, 2012. Springer Scholar Semantic MR ZB arXiv Wayback
2012O. Al-Hassani, Q. Mahesar, C. Sacerdoti Coen, V. Sorge, A term rewriting system for Kuratowski’s closure-complement problem. In 23rd International Conference on Rewriting Techniques and Applications (RTA ’12), Nagoya, Japan, May 28 to June 2, 2012, Leibniz International Proceedings in Informatics (LIPIcs), 15:38–52, 2012. Dagstuhl Wayback Scholar Semantic MR ZB ARW 2012 Wayback UBIRA Wayback Scholar Semantic
2012S. Cato, Complements and consistent closures. Discrete Mathematics, 312(6):1218–1221, 28 March 2012. Elsevier Scholar Semantic MR ZB
2011Š. Holub and J. Kortelainen, On partitions separating words. International Journal of Algebra and Computation, 21(8):1305–1316, December 2011. World Scientific Charles University Wayback Scholar Semantic MR ZB
2011J. Shallit and R. Willard, Kuratowski’s theorem for two closure operators. 6 pp., 7 September 2011. arXiv Wayback Scholar Semantic
2011J. Brzozowski, E. Grant, J. Shallit, Closures in formal languages and Kuratowski’s theorem. International Journal of Foundations of Computer Science, 22(2):301–321, 2011. World Scientific Scholar Semantic MR ZB Springer Wayback ZB arXiv Wayback
2011M. Kudlek, A relation between modal logic and language closure operators. Fundamenta Informaticae, 109(3):297–304, 2011. IOS Press Scholar Semantic MR ZB
2010D. Sherman, Variations on Kuratowski’s 14‑set theorem. American Mathematical Monthly, 117(2):113–123, February 2010. Taylor & Francis JSTOR Wayback Scholar Semantic MR ZB arXiv Wayback
2008B. J. Gardner and M. Jackson, The Kuratowski closure-complement theorem. New Zealand Journal of Mathematics, 38:9–44, 2008. University of Tasmania Wayback Scholar Semantic MR ZB
2007A. Christie, Q. Wang, S. L. Wismath, The inflation class operator. Algebra Universalis, 56:107–118, February 2007. Springer Scholar Semantic MR ZB
2007A. E. McCluskey, D. W. McIntyre, W. S. Watson, The propositional theory of closure. Experimental Mathematics, 16(4):501–512, 2007. Taylor & Francis Project Euclid Wayback Scholar Semantic MR ZB
2006A. Grabowski and M. Jastrzębska, Enhancing MML via Isomichi's classification of domains. Mechanized Mathematics and Its Applications, 5(2):11–18, December 2006. ResearchGate Wayback Scholar Semantic
2006M. Bowron (proposer), B. S. Burdick (solver), Problem 11059: Closure, complement, and union. American Mathematical Monthly, 111(1):64–65, January 2004 (proposal), 113(1):83, January 2006 (solution). Taylor & Francis JSTOR ResearchGate Wayback Semantic
2006J. Lawrence and B. Tasić, HSP ≠ SHPS for commutative rings with identity. Proceedings of the American Mathematical Society, 134(4):943–948, 2006. AMS ResearchGate Wayback Scholar Semantic MR ZB
2006A. Grabowski, Solving two problems in general topology via types. In J. C. Filliâtre, C. Paulin-Mohring, and B. Werner, editors, Types for Proofs and Programs International Workshop (TYPES 2004), Jouy-en-Josas, France, December 15-18, 2004, Revised Selected Papers, volume 3839 of Lecture Notes in Computer Science (LNICS), pp. 138–153, 2006. Springer Scholar Semantic MR ZB
2005B. Tasić, Partially ordered monoids of operators on classes of algebras. PhD thesis, University of Waterloo, 97 pp., 2005. LAC Scholar MR
2005P. Muhm, Kuratowski’s 14‑set theorem — a modal logic view. PhD thesis, University of Houston, 58 pp., December 2005. Proquest Scholar MR
2005A. Grabowski, On the computer-checked solution of the Kuratowski closure-complement problem. Mechanized Mathematics and Its Applications, 4(1):25–34, March 2005. ResearchGate Wayback Scholar Semantic
2004M. Jackson, Semilattices with closure. Algebra Universalis, 52(1):1–37, November 2004. Springer ResearchGate Wayback Scholar Semantic MR ZB
2004F. D. Farmer, Containment for c, s, t operators on a binary relation. Algebra Universalis, 51:153–175, August 2004. Springer Scholar Semantic MR ZB
2004M. Jackson and T. Stokes, Semilattice pseudo-complements on semigroups. Communications in Algebra, 32(8):2895–2918, 2004. Taylor & Francis ResearchGate Wayback Scholar Semantic MR ZB
2004B. Tasić, A note on homomorphic images, subalgebras and various products. Algebra Universalis, 52(4):431–438, 2004. Springer ResearchGate Scholar Semantic MR ZB
2003G. Sambin, Some points in formal topology. Theoretical Computer Science, 305(1–3):347–408, 18 August 2003. Elsevier CORE Scholar Semantic ZB
2003L. K. Bagińska and A. Grabowski, On the Kuratowski closure-complement problem. Journal of Formalized Mathematics, 11, 6 pp., 2003. Mizar Wayback Scholar Semantic
2002J. A. Pomykala, Similarity in relational databases and in information systems theory. Demonstratio Mathematica, 35(4):927–953, 1 October 2002. De Gruyter Wayback Scholar Semantic MR ZB University of Amsterdam Wayback
2002G. Voutsadakis, On some operations on classes of algebras and coalgebras from a bialgebraic viewpoint. 17 pp., 2002. Wayback
2001B. Tasić, On the partially ordered monoid generated by the operators H, S, P, Ps on classes of algebras. Journal of Algebra, 245(1):1–19, 1 November 2001. Elsevier Scholar Semantic MR ZB
2001D. Mašulović and B. Tasić, Operators on classes of coalgebras. Theoretical Computer Science, 269(1):419–431, 28 October 2001. Elsevier Scholar Semantic MR ZB
2001B. Tasić, Partially Ordered Monoids Generated by Operators H,S,P, and by H,S,Pf Are Isomorphic. Semigroup Forum, 62(3):485–490, April 2001. Springer ResearchGate Scholar Semantic MR ZB
2001R. Sz. Madarász and B. Tasić, On the partially ordered semigroup generated by the class operators I,R,H,S,P. Order, 18(1):49–60, March 2001. Springer ResearchGate Scholar Semantic MR ZB
2000W. He and Y. Zhang, Interior and boundary in a locale. Advances in Mathematics (China), 29(5):439–443, October 2000. CAOD Scholar MR ZB
1999B. J. Gardner and T. Stokes, Closure rings. Commentationes Mathematicae Universitatis Carolinae, 40(3):413–427, 1999. DML-CZ Wayback Scholar Semantic MR ZB
1998M. Bowron and S. Rabinowitz (proposers), J. Rickard (solver), Problem 10577: Closure, complement, and arbitrary union. American Mathematical Monthly, 104(2):169, February 1997 (proposal), 105(3):282–283, March 1998 (solution). Taylor & Francis JSTOR ResearchGate Wayback Semantic MR
1997E. Q. F. Rezende, R. A. Bacci, R. C. Bassanezi, Semi-group generated by K-limits. Proyecciones Journal of Mathematics, 16(1):67–82, 1997. Catholic University of the North Wayback Scholar Semantic MR ZB
1996K. P. Shum, Closure functions on the set of positive integers. Science in China Series A: Mathematics, 39(4):337–346, 1996. ResearchGate Scholar MR ZB
1995E. Garel and J. P. Olivier, On closures verifying that the interior of a closed element is closed. Communications in Algebra, 23(10):3715–3728, 1995. Taylor & Francis Scholar Semantic MR ZB
1995E. Garel and J. P. Olivier, The opoid generated by transitive closure and interior and symmetric closure and interior: a characterization using generators and relations. Rendiconti del Seminario Matematico Università e Politecnico di Torino, 53(2):71–88, 1995. Wayback Scholar MR ZB
1995M. S. Moslehian and N. Tavallaii, A generalization of the Kuratowski closure-complement problem. Punjab University Journal of Mathematics, 28:1–9, 1995. University of the Punjab Wayback Scholar MR ZB
1994G. M. Bergman, Partially ordered sets, and minimal systems of counterexamples. Algebra Universalis, 32:13–30, 1 March 1994. Springer Scholar Semantic MR ZB
1993E. Garel and J. P. Olivier, About the finiteness of monoids generated by closures and interiors. Semigroup Forum, 47(1):341–352, December 1993. EuDML Wayback Scholar Semantic MR ZB
1993K. P. Shum, The amalgamation of closure and boundary functions on semigroups and partially ordered sets. In K. P. Shum and P. C. Yuen, editors, Proceedings of the Conference on Ordered Structures and Algebra of Computer Languages: Hong Kong, 26–29 June 1991, pp. 232–243, 1993. Worldcat Scholar MR ZB
1992S. M. Kam and K. P. Shum, On a problem of P. C. Hammer. Southeast Asian Bulletin of Mathematics, 16(2):123–127, 1992. Worldcat Scholar MR ZB
1991J. H. Fife, The Kuratowski closure-complement problem. Mathematics Magazine, 64(3):180–182, June 1991. Taylor & Francis JSTOR Scholar Semantic MR ZB
1988E. Buchman (proposer), J. Ferrer (solver), Problem E3144: Generating sets in a topological space. American Mathematical Monthly, 93(4):299, April 1986 (proposal), 95(4):353, April 1988 (solution). Taylor & Francis JSTOR MR
1988C. E. Aull, Classification of topological spaces II. Bulletin of the Polish Academy of Sciences, Mathematics, 36(11):649–654, 1988 (errata ibid., 40(2):175, 1992). Worldcat MR ZB
1986S. Wroński, On the operations definable in terms of the complementation and the closure. Bulletin of the Section of Logic, Polish Academy of Sciences, 15(3):117–122, 1986. Wayback Scholar MR ZB
1984D. Peleg, A generalized closure and complement phenomenon. Discrete Mathematics, 50:285–293, 1984. Elsevier Scholar Semantic MR ZB
1984K. P. Shum, On the boundary of Kuratowski 14‑sets in connected spaces. Glasnik Matematicki Series III, 19(39):293–296, 1984. Google Books Scholar MR ZB
1982A. Capelo and L. Sanchez, On certain operators expressible in a topological space (in Portuguese). Boletim Sociedade Portuguesa De Matematica, 5:37–48, November 1982. Wayback
1982A. V. Chagrov, Kuratowski numbers (in Russian). Application of Functional Analysis in Approximation Theory, pp. 186–190, 1982. Scholar MR ZB English translation by M. Bowron
1982V. P. Soltan, Problems of Kuratowski type (in Russian). Matematicheskie Issledovaniya, 65:121–131, 1982. Scholar MR ZB English translation by M. Bowron
1981H. Mortazavian and M. Boutillier, On a theorem of Kuratowski. Working Report, INRIA, Rocquencourt, France.
1981H. Mortazavian, Patterns arising from a theorem of Kuratowski in topology with applications. Working Report, INRIA, Rocquencourt, France.
1980J. Anderson, Closure operators on languages and binary relations. Semigroup Forum, 21(1):91–92, December 1980. Springer EuDML Wayback Scholar MR ZB
1980E. Langford (proposer), W. Myers (solver), Problem 6260: Sets formed by iterated closure, interior, and union. American Mathematical Monthly, 86(3):226–226, March 1979 (proposal), 87(8):680–681, October 1980 (solution). Taylor & Francis JSTOR MR
1980V. P. Soltan, On Kuratowski’s problem (in Russian). Bulletin of the Polish Academy of Sciences, Mathematics, 28:369–375, 1980. Scholar MR English translation by M. Bowron
1980M. J. Fischer and M. S. Paterson, The fast skew-closure algorithm. L'Enseignement Mathématique, 26(3):345–360, 1980. ETH Zurich Scholar MR ZB
1979V. P. Soltan, The Kuratowski number of a closure space (in Russian). IV Tiraspol Symposium on General Topology and its Applications, pp. 137–139, 1979. Scholar English translation by Mark Bowron
1978A. Smith (proposer), C. Y. Yu (solver), Problem 5996: Kuratowski sets. American Mathematical Monthly, 81(9):1034, November 1974 (proposal), 85(4):283–284, April 1978 (solution). Taylor & Francis JSTOR Scholar MR
1978P. C. Fishburn, Operations on binary relations. Discrete Mathematics, 21:7–22, 1978. Elsevier Scholar Semantic MR ZB
1977W. J. Blok, The free closure algebra on finitely many generators. Indagationes Mathematicae (Proceedings), 80:362–379, 1977. Elsevier Wayback Scholar Semantic MR ZB
1977I. Z. Kleiner, Closure and boundary operators in topological spaces. Ukrainian Mathematical Journal (English edition), 29(3):295–296, 1977. Springer Scholar Semantic MR ZB
1977L. E. Moser, Closure, interior, and union in finite topological spaces. Colloquium Mathematicum, 38(1):41–51, 1977. Library of Science Wayback Scholar Semantic MR ZB
1976J. C. Oxtoby, The kernel operation on subsets of a T1-space. Fundamenta Mathematicae, 90(3):275–284, 1976. IMPAN Wayback Scholar Semantic MR ZB
1975J. Berman and S. L. Jordan, The Kuratowski closure-complement problem. American Mathematical Monthly, 82(8):841–842, October 1975. Taylor & Francis JSTOR Scholar Semantic MR ZB
1975E. Langford, Problems of Kuratowski type involving unions and intersections. Notices of the American Mathematical Society, 22(4):A–485, June 1975. AMS Wayback Scholar
1975K. W. Yip and K. P. Shum, On the structure of Kuratowski sets. Journal of the Chinese University of Hong Kong, 3:429–439, 1975. ResearchGate Scholar
1974Y. R. Gaida and A. É. Eremenko, On the frontier operator in Boolean algebras with a closure. Ukrainian Mathematical Journal (English edition), 26(6):662–664, 1974. Springer Purdue Wayback Scholar Semantic MR ZB unpublished diagram Wayback
1974A. A. Gol'dberg, On Boolean algebras with a derivation. Ukrainian Mathematical Journal (English edition), 26(4):365–369, 1974. Springer Scholar Semantic MR ZB
1972R. L. Graham, D. E. Knuth, T. S. Motzkin, Complements and transitive closures. Discrete Mathematics, 2(1):17–29, March 1972. Elsevier Scholar Semantic MR ZB
1972S. Comer and J. Johnson, The standard semigroup of operators of a variety. Algebra Universalis, 2(1):77–79, 1972. Springer Homepage Wayback Scholar Semantic MR ZB
1972D. Pigozzi, On some operations on classes of algebras. Algebra Universalis, 2(1):346–353, 1972. Springer Academia Wayback Scholar Semantic MR ZB
1971K. P. Shum (proposer and solver), Problem 153: Bounds on the Kuratowski family. Canadian Mathematical Bulletin, 12(2):237, April 1969 (proposal), 14(4):602–603, December 1971 (solution). Wayback
1971S. Baron (proposer), E. Langford (solver), Problem 5569: Kuratowski’s 14‑sets. American Mathematical Monthly, 75(2):199, February 1968 (proposal), 78(4):411, April 1971 (solution). Taylor & Francis JSTOR MR
1971E. Langford, Characterization of Kuratowski 14‑sets. American Mathematical Monthly, 78(4):362–367, April 1971. Taylor & Francis JSTOR Scholar Semantic MR ZB
1971J. Anusiak and K. P. Shum, Remarks on finite topological spaces. Colloquium Mathematicum, 23(2):217–223, 1971. IMPAN ResearchGate Wayback Scholar Semantic MR ZB
1971Y. Isomichi, New concepts in the theory of topological space — supercondensed set, subcondensed set, and condensed set. Pacific Journal of Mathematics, 38(3):657–668, 1971. Project Euclid MSP Wayback Scholar Semantic MR ZB
1968W. Koenen (proposer), L. F. Meyers (solver), Problem E1941: On two closure problems. American Mathematical Monthly, 73(10):1123, December 1966 (proposal), 75(4):408–409, April 1968 (solution). Taylor & Francis JSTOR MR
1968D. H. Staley, On the commutativity of the boundary and interior operators in a topological space. Ohio Journal of Science, 68(2):84, March 1968. OSU Knowledge Bank Wayback Scholar Semantic OhioLINK Wayback Scholar Semantic
1967C. E. Aull, Classification of topological spaces. Bulletin of the Polish Academy of Sciences, Mathematics, Astronomy, and Physics, 15(11):773–778, 1967 (errata ibid., 16(6), 1968). Worldcat Scholar MR ZB
1967P. J. Coleman, On the composition semigroup generated by complement and an expansive function. Nieuw Archief voor Wiskunde, 15(2):207–210, 1967. Worldcat Scholar MR ZB
1967E. Nelson, Finiteness of semigroups of operators in universal algebra. Canadian Journal of Mathematics, 19:764–768, 1967. Cambridge Wayback Scholar Semantic MR ZB
1966S. P. Lloyd (proposer), J. C. Morgan II (solver), Problem 5349: A Kuratowski closure and complement problem. American Mathematical Monthly, 72(10):1136, December 1965 (proposal), 73(10):1132–1134, December 1966 (solution). Taylor & Francis JSTOR MR
1966W. Koenen, The Kuratowski closure problem in the topology of convexity. American Mathematical Monthly, 73(7):704–708, August 1966. Taylor & Francis JSTOR Scholar Semantic MR ZB
1966H. H. Herda and R. C. Metzler, Closure and interior in finite topological spaces. Colloquium Mathematicum, 15(2):211–216, 1966. Library of Science Wayback EuDML Scholar Semantic MR ZB
1965J. Hintikka, A closure and complement result for nested topologies. Fundamenta Mathematicae, 57(1):97–106, 1965. Library of Science Wayback Scholar Semantic MR ZB
1964T. Tamura, Operations on binary relations and their applications. Bulletin of the American Mathematical Society, 70:113–120, 1964. AMS Wayback Scholar Semantic MR ZB
1962T. A. Chapman, A further note on closure and interior operators. American Mathematical Monthly, 69(6):524–529, June 1962. Taylor & Francis JSTOR Scholar Semantic MR ZB
1962T. A. Chapman, An extension of the Kuratowski closure and complementation problem. Mathematics Magazine, 35(1):31–35, January 1962. Taylor & Francis JSTOR Scholar Semantic MR ZB
1961N. Levine, On the commutativity of the closure and interior operators in topological spaces. American Mathematical Monthly, 68(5):474–477, May 1961. Taylor & Francis JSTOR Scholar Semantic MR ZB
1961H. Choda and K. Matoba, On a theorem of Levine. Proceedings of the Japan Academy, Series A (Mathematical Sciences), 37(8):462–463, 1961. J-Stage Project Euclid Scholar Semantic MR ZB
1960P. C. Hammer, Kuratowski’s closure theorem. Nieuw Archief voor Wiskunde, 8(2):74–80, 1960. Worldcat MR ZB
1957L. Rieger, A note on the so-called free closure algebras (in Russian). Czechoslovak Mathematical Journal, 7(1):16–20, 1957. DML-CZ Scholar MR ZB
1955M. Zarycki, Boolean algebras with a closure and Boolean algebras with a derivation (in Ukrainian). Dopovidi Akademiї Nauk Ukrains'koї RSR (Reports of the Academy of Sciences of the Ukrainian SSR), 1:3–6, 1955. Worldcat
1954E. Halfar, The isolated points of a set. Portugaliae Mathematica, 13(3):125–128, 1954. National Library of Portugal Wayback Scholar MR ZB
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< 1920?Did anyone alternately add limit points and take complements before Kuratowski?
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