On some properties of the space of upper semicontinuous functions
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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 phrasessequentially separable function space continuous function upper semicontinuous function Baire function of class 1
Mathematics Subject Classification54C35 54C30 54A20 54H05
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The authors are grateful to Sergey V. Medvedev and the anonymous referee for making several suggestions which improved this paper.
- 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
- 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.C. A. Rogers, J. E. Jayne et al., Analytic Sets, Academic Press (1980)Google Scholar
- 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