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Borel isomorphisms at the first level—I

Part of: Connections with other structures, applications

Published online by Cambridge University Press:  26 February 2010

J. E. Jayne  and
C. A. Rogers
J. E. Jayne
Affiliation:
The Department of Mathematics, University College, London, WC1E 6BT.
C. A. Rogers
Affiliation:
The Department of Mathematics, University College, London, WC1E 6BT.
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Extract

A Borel isomorphism that, together with its inverse, maps ℱσ-sets to ℱσ-sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism, for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

MSC classification

Secondary: 54H99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 26 , Issue 1 , June 1979 , pp. 125 - 156
Copyright
Copyright © University College London 1979

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References

1Bade, W. G.. Complementation problems for the Baire classes, Pacific J. Math., 45 (1973), 1–11.CrossRefGoogle Scholar
2Engelking, R.. General topology. (Warsaw: Polish Scientific Publishers 1977).Google Scholar
3Fleissner, W. G.. An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc., 251 (1979), 309–328.CrossRefGoogle Scholar
4Freiwald, R. C.. Images of B(k), Fund. Math., 99 (1978), 73–78.CrossRefGoogle Scholar
5Frolik, Z.. On analytic spaces, Bull. Acad. Polon. Sc, 9 (1961), 529–631.Google Scholar
6Hansell, R. W.. Borel measurable mappings for nonseparable metric spaces, Trans. Amer. Math. Soc, 161 (1971), 145–169.CrossRefGoogle Scholar
7Hansell, R. W.. On Borel mappings and Baire functions, Trans. Amer. Math. Soc, 194 (1974), 195–211.CrossRefGoogle Scholar
8Harrington, L.. Analytic determinacy and 0 (preprint).Google Scholar
9Hausdorff, F.. Die schlichten stetigen Bilder des Nullraums, Fund. Math., 29 (1937), 151–158.CrossRefGoogle Scholar
10Hurewicz, W.. Relativ Perfekte Teile von Punktmengen und Mengen (A), Fund. Math., 12 (1928),78–109.CrossRefGoogle Scholar
11Hurewicz, W. and Wallman, H.. Dimension theory (Princeton: (Princeton Univ. Press, 1948).Google Scholar
12Jayne, J. E.. The space of class a Baire functions, Bull. Amer. Math. Soc, 80 (1974), 1151–1156.CrossRefGoogle Scholar
13Kondô, M.. Sur la representation parametrique des ensembles, J. Fac. Sci. Hokkaido Imp. Univ., 8 (1939–1940), 173–220.Google Scholar
14Kuratowski, K.. Sur une generalisation de la notion d'homeomorphie, Fund. Math., 22 (1934), 206–220.CrossRefGoogle Scholar
15Kuratowski, K.. Topology, Vol. I (New York: Academic Press, 1966).Google Scholar
16Kuratowski, K.. Topology, Vol. II (New York: Academic Press, 1968).Google Scholar
17Lefschetz, S.. On compact spaces, Ann. Math., 32 (1931), 521–538.CrossRefGoogle Scholar
18Menger, K.. Über umfassendste n-dimensionale Mengen, Proc. Akad. Wetensch. Amst., 29 (1926), 1125–1128.Google Scholar
19Nöbeling, G.. Über eine n-dimensionale Universalmenge im R2n + 1, Math. Ann., 104 (1930), 71–80.CrossRefGoogle Scholar
20Ostrovskii, A. V.. On nonseparable τ-A-sets and their mappings, Dokl. Akad. Nauk SSSR, 226 (1976), 269–272 (Soviet Math. DM., 17 (1976), 99–103).Google Scholar
21Preiss, D.. Completely additive disjoint system of Baire sets is of bounded class, Comment. Math. Univ. Carolinae, 15 (1974), 341–344.Google Scholar
22Rogers, C. A.. Universal properties of certain analytic sets, J. London Math. Soc.,(2), 16 (1977), 177–183.CrossRefGoogle Scholar
23Rogers, C. A.. The topological classification of open co-countable subsets of S2, J. London Math. Soc, (2), 18 (1978), 546–552.CrossRefGoogle Scholar
24Saint-Raymond, J.. Approximation des sous-ensembles analytiques par l'intérieur, C. R. Acad. Sc. Paris Ser. A, 281 (1975), 85–87.Google Scholar
25Sierpiński, W.. L'homéomorphisme des espaces rationnels (in Polish), Wektor, 4 (1915), 215–221.Google Scholar
26Sierpiński, W.. Sur une propriété topologique des ensembles denses en soi, Fund. Math., 1 (1920), 11–16.CrossRefGoogle Scholar
27Sierpiński, W.. Sur une propriété des ensembles Gδ non denombrables. Fund. Math., 21 (1933), 66–72.CrossRefGoogle Scholar
28Sierpiński, W.. Sur les images continues et biunivoques de l'ensemble de tous les numbres irrationels, Mathematica, 1 (1929), 18–21.Google Scholar
29Sierpiński, W.. Sur une problème concernant les fonctions projectives, Fund. Math., 30 (1938), 59–60.CrossRefGoogle Scholar
30Steel, J.. Thesis (Univ. of California, Berkeley 1976).Google Scholar
31Stone, A. H.. Absolute Fσ, spaces, Proc. Amer. Math. Soc, 13 (1962), 495–499.Google Scholar
32Stone, A. H.. Non-separable Borel sets, Rozprawy Mat. ( = Dissertations Math.), 28 (1962), 1–40.Google Scholar
33Stone, A. H.. On σ-discreteness and Borel isomorphisms, Amer. J. Math., 85 (1963), 655–666.CrossRefGoogle Scholar
34Stone, A. H.. Non-separable Borel sets II, Gen. Top. and its Appl., 2 (1972), 249–270.CrossRefGoogle Scholar
35Telgársky, R.. Spaces defined by topological games, Fund. Math., 88 (1975), 195–223.CrossRefGoogle Scholar