Advertisement

Acta Mathematica Hungarica

, Volume 157, Issue 2, pp 459–464 | Cite as

On some properties of the space of upper semicontinuous functions

  • A. V. OsipovEmail author
  • E. G. Pytkeev
Article
  • 59 Downloads

Abstract

For a Tychonoff space X, we will denote by USCp(X) (B1(X)) the set of all real-valued upper semicontinuous functions (the set of all Baire functions of class 1) defined on X endowed with the pointwise convergence topology.

In this paper we describe a class of Tychonoff spaces X for which the space USCp(X) is sequentially separable. Unexpectedly, it turns out that this class coincides with the class of spaces for which a stronger form of the sequential separability for the space B1(X) holds.

Key words and phrases

sequentially separable function space continuous function upper semicontinuous function Baire function of class 1 

Mathematics Subject Classification

54C35 54C30 54A20 54H05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to Sergey V. Medvedev and the anonymous referee for making several suggestions which improved this paper.

References

  1. 1.
    M. Kačena, L. Motto Ros and B. Semmes, Some observations on “A new proof of a theorem of Jayne and Rogers”, Real Anal. Exchange, 38 (2012/2013), 121–132Google Scholar
  2. 2.
    Noble, N.: The density character of functions spaces. Proc. Amer. Math. Soc. 42, 228–233 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Osipov, A.V.: Application of selection principles in the study of the properties of function spaces. Acta Math. Hungar. 154, 362–377 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Osipov, A.V., Pytkeev, E.G.: On sequential separability of functional spaces. Topology Appl. 221, 270–274 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. V. Pestriakov, Spaces of Baire functions, in: Investigations in the theory of convex sets and graphs (Issledovanii po teorii vypuklikh mnogestv i grafov), 82, Akad. Nauk SSSR, Ural. Nauchn Tsentr (Sverdlovsk, 1987), pp. 53–59 (in Russian)Google Scholar
  6. 6.
    C. A. Rogers, J. E. Jayne et al., Analytic Sets, Academic Press (1980)Google Scholar
  7. 7.
    G. Tironi and R. Isler, On some problems of local approximability in compact spaces, in: General Topology and its Relations to Modern Analysis and Algebra, III (Prague, August 30–September 3, 1971), Academia (Prague, 1972), pp. 443–446Google Scholar
  8. 8.
    Velichko, N.V.: On sequential separability. Math. Notes 78, 610–614 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wilansky, A.: How separable is a space? Amer. Math. Monthly 79, 764–765 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Federal UniversityEkaterinburgRussia
  2. 2.Ural State University of EconomicsEkaterinburgRussia

Personalised recommendations