On Borel mappings and Baire functions
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- by R. W. Hansell
- Trans. Amer. Math. Soc. 194 (1974), 195-211
- DOI: https://doi.org/10.1090/S0002-9947-1974-0362270-7
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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
- Richard Arens, Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11–22. MR 49543, DOI 10.2140/pjm.1952.2.11 S. Banach, Über 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 10.4153/cjm-1951-022-3
- 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 10.1090/S0002-9947-1971-0288228-1 —, 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; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], 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 métriques non-séparables, Fund. Math. 25 (1935), 545. —, Sur le prolongement de l’homéomorphie, Comptes Rendus Hebdomadaires des Séances de l’Académie 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) Washington State University, Department of Mathematics, Pi Mu Epsilon, Pullman, Wash., 1970, pp. 20–33. MR 0268848
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 194 (1974), 195-211
- MSC: Primary 54H05; Secondary 54C50
- DOI: https://doi.org/10.1090/S0002-9947-1974-0362270-7
- MathSciNet review: 0362270