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Kuratowski's Closure-Complement Problem

Let X be an arbitrary topological space. Denote the set closure of a subset A of X by A^- and the complement of A by A^'. Then at most 14 different sets can be derived from A by repeated application of closure and complementation (Berman and Jordan 1975, Fife 1991). The problem was first proved by Kuratowski (1922) and popularized by Kelley (1955).

See also

Complement Set, Kuratowski Reduction Theorem, Set Closure, Subset

Portions of this entry contributed by Mark Bowron

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References

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.

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Kuratowski's Closure-Complement Problem

Cite this as:

Bowron, Mark and Weisstein, Eric W. "Kuratowski's Closure-Complement Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KuratowskisClosure-ComplementProblem.html

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