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Stability of the fixed point property and universal maps


Author: José M. R. Sanjurjo
Journal: Proc. Amer. Math. Soc. 105 (1989), 221-230
MSC: Primary 54H25; Secondary 54C08, 54F43
DOI: https://doi.org/10.1090/S0002-9939-1989-0931746-X
MathSciNet review: 931746
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Abstract: We give a stability condition for the fixed point property in terms of the fundamental metric and the metric of continuity introduced by K. Borsuk. This condition is equivalent to that originally given by V. Klee but reflects richer properties. We introduce the notion of a proximately universal map and study many of its properties. Relationships among proximately universal maps and some generalized refinable maps introduced by E. E. Grace are studied. In particular we prove that every weakly refinable map $ r:X \to Y$ is proximately universal whenever $ X$ has the proximate fixed point property. This generalizes a result of Grace.


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  • [1] K. Borsuk, Sur un probleme de MM. Kuratowski et Ulam, Fund. Math. 31 (1938), 154-559.
  • [2] Karol Borsuk, On a metrization of the hyperspace of a metric space, Fund. Math. 94 (1977), no. 3, 191–207. MR 0433397
  • [3] K. Borsuk, On nearly extendable maps, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 7, 753–760 (English, with Russian summary). MR 0394544
  • [4] K. Borsuk, On the Lefschetz-Hopf fixed point theorem for nearly extendable maps, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 12, 1273–1279 (English, with Russian summary). MR 0405406
  • [5] Z. Čerin and A. P. Šostak, Some remarks on Borsuk's fundamental metric, Colloq. Math. Soc. Janos Bolyai, Budapest, 1978, pp. 233-252.
  • [6] Michael H. Clapp, On a generalization of absolute neighborhood retracts, Fund. Math. 70 (1971), no. 2, 117–130. MR 0286081
  • [7] Jo Ford and J. W. Rogers Jr., Refinable maps, Colloq. Math. 39 (1978), no. 2, 263–269. MR 522365
  • [8] E. E. Grace, Refinable maps and the proximate fixed point property, Proceedings of the 1985 topology conference (Tallahassee, Fla., 1985), 1985, pp. 293–303. MR 876900
  • [9] E. E. Grace, Generalized refinable maps, Proc. Amer. Math. Soc. 98 (1986), no. 2, 329–335. MR 854042, https://doi.org/10.1090/S0002-9939-1986-0854042-5
  • [10] Chung Wu Ho, On a stability theorem for the fixed-point property, Fund. Math. 111 (1981), no. 2, 169–177. MR 609433
  • [11] W. Holsztyński, Une généralisation du théorème de Brouwer sur les points invariants, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 603–606. MR 0174041
  • [12] W. Holsztyński, On the composition and products of universal mappings, Fund. Math. 64 (1969), 181–188. MR 0243491
  • [13] V. Klee, Stability of the fixed-point property, Colloq. Math. 8 (1961), 43–46. MR 0126261
  • [14] Victor Klee and André Yandl, Some proximate concepts in topology, Symposia Mathematica, Vol. XVI (Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973) Academic Press, London, 1975, pp. 21–39. MR 0397656
  • [15] K. Kuratowski, Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR 0259835
  • [16] C. W. Saalfrank, Neighborhood retraction generalized for compact Hausdorff spaces, Portugal. Math. 20 (1961), 11–16. MR 0126830

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0931746-X
Keywords: Proximate fixed point property, refinable map, weakly refinable map, proximately universal map
Article copyright: © Copyright 1989 American Mathematical Society