Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Borel mappings and Baire functions

Author: R. W. Hansell
Journal: Trans. Amer. Math. Soc. 194 (1974), 195-211
MSC: Primary 54H05; Secondary 54C50
MathSciNet review: 0362270
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies conditions under which classes of Borel mappings (i.e., mappings such that the inverse image of open sets are Borel sets) coincide with certain classes of Baire functions (i.e., functions which belong to the smallest family containing the continuous functions and closed with respect to pointwise limits). Generalizations of the classical Lebesgue-Hausdorff and Banach theorems are obtained for the class of mappings which we call “$\sigma$-discrete". These results are then applied to the problem of extending Borel mappings over Borel sets, and generalizations of the theorems of Lavrentiev and Kuratowski are obtained.

References [Enhancements On Off] (What's this?)

  • Richard Arens, Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11–22. MR 49543
  • S. Banach, Ăśber analytisch darstellbare Operationen in abstrakten Räumen, Fund. Math. 17 (1931), 283-295.
  • R. H. Bing, Metrization of topological spaces, Canad. J. Math. 3 (1951), 175–186. MR 43449, DOI
  • Gustave Choquet, Lectures on analysis. Vol. I: Integration and topological vector spaces, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Edited by J. Marsden, T. Lance and S. Gelbart. MR 0250011
  • R. W. Hansell, Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc. 161 (1971), 145–169. MR 288228, DOI
  • ---, Borel measurable mappings for nonseparable metric spaces, Dissertation, University of Rochester, Rochester, N. Y., 1969. F. Hausdorff, Mengenlehre, 3rd ed., de Gruyter, Berlin, 1937; English transl., Chelsea, New York, 1957. MR 19, 111.
  • Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
  • K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; PaĹ„stwowe Wydawnictwo Naukowe, Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
  • ---, Topologie. Vol. 2, 3rd ed., Monografie Mat., Tom 21, PWN, Warsaw, 1961; English transl., Academic Press, New York; PWN, Warsaw, 1968. MR 24 #A2958; 41 #4467. ---, Quelques problèmes concernant les espaces mĂ©triques non-sĂ©parables, Fund. Math. 25 (1935), 545. ---, Sur le prolongement de l’homĂ©omorphie, Comptes Rendus Hebdomadaires des SĂ©ances de l’AcadĂ©mie des Sciences, 197 (1933), 1090.
  • A. H. Stone, Non-separable Borel sets, Rozprawy Mat. 28 (1962), 41. MR 152457
  • A. H. Stone, Borel and analytic metric spaces, Proc. Washington State Univ. Conf. on General Topology (Pullman, Wash., 1970), Pi Mu Epsilon, Dept. of Math., Washington State Univ., Pullman, Wash., 1970, pp. 20–33. MR 0268848

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H05, 54C50

Retrieve articles in all journals with MSC: 54H05, 54C50

Additional Information

Keywords: Borel measurable mappings, Baire functions, Borel classifications, Baire classifications, <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\sigma$">-discrete mappings
Article copyright: © Copyright 1974 American Mathematical Society