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2020 | 34 (1) | 81--95
Tytuł artykułu

On the Borel Classes of Set-Valued Maps of Two Variables

Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
sing the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class α in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α + 1. Also other cases are considered.(original abstract)
Rocznik
Tom
Strony
81--95
Opis fizyczny
Twórcy
  • Academy of Sciences, Bratislava, Slovakia
  • University of Gdansk, Poland
Bibliografia
  • R. Brisac, Les classes de Baire des fonctions multiformes, C. R. Acad. Sci. Paris 224 (1947), 257-258.
  • J. Ewert, Multivalued Mappings and Bitopological Spaces (Polish), Pomeranian University in Słupsk, Słupsk, 1985.
  • J. Ewert and T. Lipski, Lower and upper quasicontinuous functions, Demonstratio Math. 16 (1983), no. 1, 85-93.
  • K.M. Garg, On the classification of set-valued functions, Real Anal. Exchange 9 (1983/84), no. 1, 86-93.
  • R.W. Hansell, Hereditarily additive families in descriptive set theory and Borel measurable multimaps, Trans. Amer. Math. Soc. 278 (1983), no. 2, 725-749.
  • S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, vol. I, Kluwer Academic Publishers, Dordrecht-Boston-London, 1997.
  • S. Kempisty, Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184-197.
  • K. Kuratowski, Sur la théorie des fonctions dans les espaces métriques, Fund. Math. 17 (1931), 275-282.
  • K. Kuratowski, On set-valued B-measurable mappings and a theorem of Hausdorff, in: G. Asser, J. Flachsmeyer, and W. Rinow (eds.), Theory of Sets and Topology (in Honour of Felix Hausdorff, 1868-1942), VEB Deutsh. Verlag Wissench., Berlin, 1972, pp. 355-362.
  • K. Kuratowski, Some remarks on the relation of classical set-valued mappings to the Baire classification, Colloq. Math. 42 (1979), 273-277.
  • G. Kwiecińska, On the Borel class of multivalued functions of two variables, Topology Proc. 25 (2000), 601-613.
  • G. Kwiecińska, B-measurability of multifunctions of two variables, Real Analysis Exchange, Summer Symposium 2011, 36-41.
  • T. Neubrunn, Quasi-continuity, Real Anal. Exchange 14 (1988), no. 2, 259-306.
  • W. Zygmunt, The Scorza-Dragoni Property (Polish), Thesis, M. Curie-Skłodowska University, Lublin, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ekon-element-000171605111

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