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Homotopy Properties of the Space If(X) of Idempotent Probability Measures

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Abstract

A subspace If(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then If(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and If(X) are equal. It is also proved that, given a compact space X, the compact space If(X) is an absolute neighborhood retract if and only if so is X.

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References

  1. V. P. Maslov and V. N. Kolokoltsov, Idempotent Analysis and Its Application in Optimal Control (Nauka, Moscow, 1994; Kluwer, Dordrecht, 1997).

    Google Scholar 

  2. Idempotent Mathematics and Mathematical Physics, in Contemp. Math., Ed. by G. L. Litvinov and V. P. Maslov (Amer. Math. Soc, Providence, R., 2005), Vol. 377.

  3. G. L. Litvinov, “Maslov dequantization, idempotent and tropical mathematics: A brief introduction,” in Zap. Nauchn. Sem. POMI, Vol. 326: Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods. XIII (POMI, St. Petersburg, 2005), pp. 145–182 [J. Math. Sci. (New York) 140 (3), 426–444 (2007)].

    MATH  Google Scholar 

  4. P. Bernhard, “Max-plus algebra and mathematical fear in dynamic optimization,” Set-Valued Anal. 8 (1–2), 71–84 (2000).

    Article  MathSciNet  Google Scholar 

  5. J. P. Aubin and O. Dordan, “Fuzzy systems, viability theory and toll sets,” in Fuzzy Systems, Handb. Fuzzy Sets Ser. (Kluwer Acad. Publ., Boston, M., 1998), Vol. 2, pp. 461–488.

    MathSciNet  MATH  Google Scholar 

  6. J.-P. Aubin, Dynamic Economic Theory. A Viability Approach, in Stud. Econom. Theory (Springer-Verlag, Berlin, 1997), Vol. 5.

  7. V. V. Fedorchuk, “Probability measures in topology,” Uspekhi Mat. Nauk 46 (1(277)), 41–80 (1991) [Russian Math. Surveys 46 (1), 45–93 (1991)].

    MathSciNet  MATH  Google Scholar 

  8. M. M. Zarichnyi, “Spaces and maps of idempotent measures,” Izv Ross. Akad. Nauk Ser. Mat. 74 (3), 45–64 (2010) [Izv. Math. 74 (3), 481–499 (2010)].

    Article  MathSciNet  Google Scholar 

  9. E. V. Shchepin, “Functors and uncountable powers of compacta,” Uspekhi Mat. Nauk 36 (3(219)), 3–62 (1981) [Russian Math. Surveys 36 (3), 1–71 (1981)].

    MathSciNet  MATH  Google Scholar 

  10. V. V. Fedorchuk, “Fully closed mappings and their applications,” Fundam. Prikl. Mat. 9 (4), 105–235 (2003) [J. Math. Sci. (New York) 136 (5), 4201–4292 (2006)].

    MATH  Google Scholar 

  11. A. A. Zaitov, “Geometrical and topological properties of the subspace Pf(X) of probability measures,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 28–37 (2019).

    Google Scholar 

  12. K. Borsuk, Theory of Shape (PWN, Warsaw, 1975; Mir, Moscow, 1976).

    MATH  Google Scholar 

  13. K. Borsuk, Theory of Retracts (PWN, Warsaw, 1967; Mir, Moscow, 1971).

    MATH  Google Scholar 

  14. T. A. Chapman, Lectures on Hilbert cube manifolds (Am. Math. Soc, Providence, R., 1976; Mir, Moscow, 1981).

    MATH  Google Scholar 

  15. T. Radul, Idempotent Measures: Absolute Retracts and Soft Maps, arXiv: 1810.09140(2018).

    Google Scholar 

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Acknowledgments

The authors express deep gratitude to the referee for critical comments, suggestions, and useful advice.

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Authors and Affiliations

  1. Tashkent Institute of Architecture and Civil Engineering, Tashkent, 100011, Uzbekistan

    A. A. Zaitov & A. Ya. Ishmetov

Authors
  1. A. A. Zaitov
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  2. A. Ya. Ishmetov
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Correspondence to A. A. Zaitov or A. Ya. Ishmetov.

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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 4, pp. 531–542.

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Zaitov, A.A., Ishmetov, A.Y. Homotopy Properties of the Space If(X) of Idempotent Probability Measures. Math Notes 106, 562–571 (2019). https://doi.org/10.1134/S0001434619090244

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  • DOI: https://doi.org/10.1134/S0001434619090244

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