Skip to main content
SpringerLink
Log in

Consonance and Cantor set-selectors

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

It is shown that every metrizable consonant space is a Cantor set-selector. Some applications are derived from this fact, also the relationship is discussed in the framework of hyperspaces and Prohorov spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alleche B., Calbrix J., On the coincidence of the upper Kuratowski topology with the cocompact topology, Topology Appl., 1999, 93(3), 207–218

    Article  MathSciNet  MATH  Google Scholar 

  2. Banakh T.O., Topology of spaces of probability measures. I. The functors P τ and P̃, Mat. Stud., 1995, 5, 65–87 (in Russian)

    MathSciNet  MATH  Google Scholar 

  3. Borges C.J.R., A study of multivalued functions, Pacific J. Math., 1967, 23, 451–461

    MathSciNet  MATH  Google Scholar 

  4. Bouziad A., Borel measures in consonant spaces, Topology Appl., 1996, 70(2–3), 125–132

    Article  MathSciNet  MATH  Google Scholar 

  5. Bouziad A., A note on consonance of P δ subsets, Topology Appl., 1998, 87(1), 53–61

    Article  MathSciNet  MATH  Google Scholar 

  6. Bouziad A., Consonance and topological completeness in analytic spaces, Proc. Amer. Math. Soc., 1999, 127(12), 3733–3737

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouziad A., Filters, consonance and hereditary Baireness, Topology Appl., 2000, 104(1–3), 27–38

    Article  MathSciNet  MATH  Google Scholar 

  8. Choban M.M., Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc., 1970, 22, 258–280

    Google Scholar 

  9. Costantini C., Watson S., On the dissonance of some metrizable spaces, Topology Appl., 1998, 84(1–3), 259–268

    Article  MathSciNet  MATH  Google Scholar 

  10. Debs G., Espaces héréditairement de Baire, Fund. Math., 1988, 129(3), 199–206

    MathSciNet  MATH  Google Scholar 

  11. Dolecki S., Greco G.H., Lechicki A., Sur la topologie de la convergence supérieure de Kuratowski, C. R. Acad. Sci. Paris, 1991, 312(12), 923–926

    MathSciNet  MATH  Google Scholar 

  12. Dolecki S., Greco G.H., Lechicki A., When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc., 1995, 347(8), 2869–2884

    Article  MathSciNet  MATH  Google Scholar 

  13. El’kin A.G., A-sets in complete metric spaces, Dokl. Akad. Nauk SSSR, 1967, 175, 517–520

    MathSciNet  Google Scholar 

  14. Gutev V., Selections and approximations in finite-dimensional spaces, Topology Appl., 2005, 146–147, 353–383

    Article  MathSciNet  Google Scholar 

  15. Gutev V., Completeness, sections and selections, Set-Valued Anal., 2007, 15(3), 275–295

    Article  MathSciNet  MATH  Google Scholar 

  16. Gutev V., Nedev S., Pelant J., Valov V., Cantor set selectors, Topology Appl., 1992, 44(1–3), 163–166

    Article  MathSciNet  MATH  Google Scholar 

  17. Gutev V., Valov V., Sections, selections and Prohorov’s theorem, J. Math. Anal. Appl., 2009, 360(2), 377–379

    Article  MathSciNet  MATH  Google Scholar 

  18. Koumoullis G., Cantor sets in Prohorov spaces, Fund. Math., 1984, 124(2), 155–161

    MathSciNet  MATH  Google Scholar 

  19. Kuratowski K., Topology. I, Academic Press, New York-London; PWN, Warsaw, 1966

  20. Michael E., Continuous selections. I, Ann. of Math., 1956, 63, 361–382

    Article  MathSciNet  MATH  Google Scholar 

  21. Michael E., Continuous selections. II, Ann. of Math., 1956, 64, 562–580

    Article  MathSciNet  MATH  Google Scholar 

  22. Michael E., A theorem on semi-continuous set-valued functions, Duke Math. J., 1959, 26, 647–651

    Article  MathSciNet  MATH  Google Scholar 

  23. van Mill J., Pelant J., Pol R., Selections that characterize topological completeness, Fund. Math., 1996, 149(2), 127–141

    MathSciNet  MATH  Google Scholar 

  24. Nedev S.J., Valov V.M., On metrizability of selectors, C. R. Acad. Bulgare Sci., 1983, 36(11), 1363–1366

    MathSciNet  MATH  Google Scholar 

  25. Nogura T., Shakhmatov D., When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?, Topology Appl., 1996, 70(2–3), 213–243

    Article  MathSciNet  MATH  Google Scholar 

  26. Preiss D., Metric spaces in which Prohorov’s theorem is not valid, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1973, 27, 109–116

    Article  MathSciNet  MATH  Google Scholar 

  27. Prokhorov Yu.V., Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl., 1956, 1(2), 157–214

    Article  Google Scholar 

  28. Przymusinski T., Collectionwise normality and absolute retracts, Fund. Math., 1978, 98(1), 61–73

    MathSciNet  MATH  Google Scholar 

  29. Scott D., Continuous lattices, In: Toposes, Algebraic Geometry and Logic, Halifax, January 16–19, 1971, Lecture Notes in Math., 274, Springer, Berlin, 1972, 97–136

    Chapter  Google Scholar 

  30. Stone A.H., On σ-discreteness and Borel isomorphism, Amer. J. Math., 1963, 85, 655–666

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Malta, Msida, MSD 2080, Malta

    Valentin Gutev

Authors
  1. Valentin Gutev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valentin Gutev.

About this article

Cite this article

Gutev, V. Consonance and Cantor set-selectors. centr.eur.j.math. 11, 341–348 (2013). https://doi.org/10.2478/s11533-012-0137-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-012-0137-0

MSC

Keywords

Search

Navigation