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Topological intersection theorems


Author: Jürgen Kindler
Journal: Proc. Amer. Math. Soc. 117 (1993), 1003-1011
MSC: Primary 54C60; Secondary 49J35, 54D05
DOI: https://doi.org/10.1090/S0002-9939-1993-1127141-4
MathSciNet review: 1127141
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{ {C_x}:x \in X\} $ be a family of subsets of some set $ Y$. A purely topological condition is given that is both necessary and sufficient for $ \bigcap {\{ {C_x}:x \in X\} } $ to be nonvoid. Applications to minimax theorems are sketched.


References [Enhancements On Off] (What's this?)

  • [1] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310. MR 0131268 (24:A1120)
  • [2] -, A minimax inequality and applications, Inequalities. III, Academic Press, New York and London, 1972, pp. 103-113. MR 0341029 (49:5779)
  • [3] M. A. Geraghty and B.-L. Lin, Topological minimax theorems, Proc. Amer. Math. Soc. 91 (1984), 377-380. MR 744633 (86c:49009)
  • [4] Ch.-W. Ha, Minimax and fixed point theorems, Math. Ann. 248 (1980), 73-77. MR 569411 (81i:47058)
  • [5] Ch. Horvath, Quelques théorèmes en théorie des Mini-Max, C. R. Acad. Sci. Paris Ser. I 310 (1990), 269-272. MR 1042861 (91f:49008)
  • [6] I. Joé, A simple proof for von Neumann's minimax theorem, Acta Sci. Math. 42 (1980), 91-94. MR 576940 (81i:49008)
  • [7] -, On some convexities, Acta Math. Hungar. 54 (1989), 163-172. MR 1015786 (90i:52003)
  • [8] J. Kindler and R. Trost, Minimax theorems for interval spaces, Acta Math. Hungar 54 (1989), 39-49. MR 1015776 (91f:49009)
  • [9] E. Klein and A. C. Thomson, Theory of correspondences, Wiley, New York, 1984. MR 752692 (86a:90012)
  • [10] H. Komiya, Elementary proof for Sion's minimax theorem, Kodai Math. J. 11 (1988), 5-7. MR 930413 (89f:49017)
  • [11] -, On minimax theorems without linear structure, Hiyoshi Review of Natural Science 8 (1990), 74-78.
  • [12] V. Komornik, Minimax theorems for upper semicontinuous functions, Acta Math. Acad. Sci. Hungar. 40 (1982), 159-163. MR 686004 (84h:49023)
  • [13] H. König, A general minimax theorem based on connectedness, Arch. Math. (to appear).
  • [14] S. Simons, A flexible minimax theorem, Acta Math. Hungar. (to appear). MR 1260284 (95f:49005)
  • [15] M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176. MR 0097026 (20:3506)
  • [16] L. L. Stachó, Minimax theorems beyond topological vector spaces, Acta Sci. Math. 42 (1980), 157-164. MR 576949 (82a:49017)
  • [17] H. Tuy, On a general minimax theorem, Soviet Math. Dokl. 15 (1974), 1689-1693. MR 0425733 (54:13686)
  • [18] Wu Wen-Tsün, A remark on the fundamental theorem in the theory of games, Sci. Rec. (N.S.) 3 (1959), 229-233. MR 0122587 (22:13311)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1127141-4
Keywords: Intersecting sets, connectedness, upper semicontinuous correspondences, minimax
Article copyright: © Copyright 1993 American Mathematical Society

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