设 
为任意拓扑空间。将 子集 
的集合闭包 
记为 
，补集 
记为 
。那么，通过重复应用闭包和补运算，最多可以从 
导出 14 个不同的集合 (Berman and Jordan 1975, Fife 1991)。这个问题最早由库拉托夫斯基 (Kuratowski) (1922) 证明，并由凯利 (Kelley) (1955) 推广。
库拉托夫斯基闭包-补集问题
另请参阅
补集, 库拉托夫斯基归约定理, 集合闭包, 子集本条目部分内容由 Mark Bowron 贡献
使用 Wolfram|Alpha 探索
参考文献
Anusiak, J. and Shum, K. P. "Remarks on Finite Topological Spaces." Colloq. Math. 23, 217-223, 1971.Aull, C. E. "Classification of Topological Spaces." Bull. de l'Acad. Pol. Sci. Math. Astron. Phys. 15, 773-778, 1967.Baron, S. Advanced Problem 5569. Amer. Math. Monthly 75, 199, 1968.Beeler et al. Item 105 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 45, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/group.html#item105.Berman, J. and Jordan, S. L. "The Kuratowski Closure-Complement Problem." Amer. Math. Monthly 82, 841-842, 1975.Bowron, M. and Rabinowitz, S. "Problem 10577." Amer. Math. Monthly 104, 169, 1997.Buchman, E. "Problem E 3144." Amer. Math. Monthly 93, 299, 1986.Chagrov, A. V. "Kuratowski Numbers, Application of Functional Analysis in Approximation Theory." Kalinin: Kalinin Gos. Univ., pp. 186-190, 1982.Chapman, T. A. "A Further Note on Closure and Interior Operators." Amer. Math. Monthly 69, 524-529, 1962.Fife, J. H. "The Kuratowski Closure-Complement Problem." Math. Mag. 64, 180-182, 1991.Fishburn, P. C. "Operations on Binary Relations." Discrete Math. 21, 7-22, 1978.Graham, R. L.; Knuth, D. E.; and Motzkin, T. S. "Complements and Transitive Closures." Discrete Math. 2, 17-29, 1972.Grannan, P. J. "The Kuratowski Closure and Complement Problem." Masters thesis. University of Virginia, Charlottesville, Virginia, 1974.Hammer, P. C. "Kuratowski's Closure Theorem." Nieuw Arch. Wisk. 8, 74-80, 1960.Herda, H. H. and Metzler, R. C. "Closure and Interior in Finite Topological Spaces." Colloq. Math. 15, 211-216, 1966.Kelley, J. L. General Topology. Princeton: Van Nostrand, p. 57, 1955.Koenen, W. "The Kuratowski Closure Problem in the Topology of Convexity." Amer. Math. Monthly 73, 704-708, 1966.Kuratowski, C. "Sur l'operation A de l'analysis situs." Fund. Math. 3, 182-199, 1922.Langford, E. "Characterization of Kuratowski 14-Sets." Amer. Math. Monthly 78, 362-367, 1971.Langford, E. "Problem 6260." Amer. Math. Monthly 86, 226, 1979.Levine, N. "On the Commutativity of the Closure and Interior Operators in Topological Spaces." Amer. Math. Monthly 68, 474-477, 1961.Moser, L. E. "Closure, Interior, and Union in Finite Topological Spaces." Colloq. Math. 38, 41-51, 1977.Moslehian, M. S. and Tavallaii, N. "A Generalization of the Kuratowski Closure-Complement Problem.' Punjab Univ. J. Math. 28, 1-9, 1995.Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.Palmer, B. P. "The Kuratowski Closure Problem and Some Generalizations." Masters thesis. Vanderbilt University, Nashville, Tennessee, 1970.Peleg, D. "A Generalized Closure and Complement Phenomenon." Discrete Math. 50, 285-293, 1984.Shum, K. P. "On the Boundary of Kuratowski 14-Sets in Connected Spaces." Glas. Mat. Ser. III 19, 293-296, 1984.Shum, K. P. "The Amalgamation of Closure and Boundary Functions on Semigroups and Partially Ordered Sets." In Proceedings of the Conference on Ordered Structures and Algebra of Computer Languages. Singapore: World Scientific, pp. 232-243, 1993.Smith, A. Advanced Problem 5996. Amer. Math. Monthly 81, 1034, 1974.Soltan, V. P. "On Kuratowski's Problem." Bull. Acad. Polon. Sci. Ser. Sci. Math. 28, 369-375, 1981.Soltan, V. P. "Problems of Kuratowski Type." Mat. Issled. 65, 121-131 and 155, 1982.Steen, L. A. and Seebach, J. A. Jr. Counterexamples in Topology. New York: Dover, 1996.在 Wolfram|Alpha 中被引用
库拉托夫斯基闭包-补集问题请引用为
鲍龙，马克 和 韦斯坦，埃里克·W.。“库拉托夫斯基闭包-补集问题”。来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/KuratowskisClosure-ComplementProblem.html