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Compact subsets of the first Baire class

Author: Stevo Todorcevic
Journal: J. Amer. Math. Soc. 12 (1999), 1179-1212
MSC (1991): Primary 26A21, 28A05, 28A20, 05D10, 03E05, 03E15, 54H05, 46B25, 46B45
Published electronically: July 6, 1999
MathSciNet review: 1685782
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Abstract: In this paper we present results about the structure of compact subsets of the first Baire class. For example, we give a complete description of characters of points in such compacta as well as a complete list of `critical' members of this class of compacta. Moreover, we describe the close relationship between this class of compacta and the class of compact metric spaces.

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  • [AU] P. Alexandroff, P. Urysohn, Mèmoire sur les espaces topologiques compacts, Verh. Akad. Wetensch. Amsterdam 14 (1929).
  • [AM] S. Argyros and S. Mercourakis, On weakly Lindelöf Banach spaces, Rocky Mountain J. Math. 23 (1993), 395-446. MR 94i:46016
  • [Ba] R. Baire, Sur les fonctions des variables rèelles, Ann. Mat. Pura Appl. 3 (1899), 16-30.
  • [Bl] A. Blass, A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82 (1981), 271-277. MR 83k:03063
  • [Bo] J. Bourgain, Some remarks on compact sets of first Baire class, Bull. Soc. Math. Belg. 30 (1978), 3-10. MR 80j:54008
  • [BFT] J. Bourgain, D.H. Fremlin and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), 845-886. MR 80b:54017
  • [CP] E. \v{C}ech, B. Pospi\v{s}il, Sur les espaces compacts, Publ. Fac. Sci. Univ. Masaryk Brno 258 (1938), 3-7.
  • [D] G. Debs, Effective properties in compact sets of Borel function, Mathematika 34 (1987), 64-68. MR 89b:03082
  • [F1] D. H. Fremlin, Consequences of Martin's axiom, Cambridge University Press, 1984. MR 86i:03001
  • [F] D.H. Fremlin, On compact sets carrying Radon measure of uncountable Maharam type, Fund. Math. 154 (1997), no. 3, 295-304. MR 99d:28019
  • [Go] G. Godefroy, Compacts de Rosenthal, Pacific J. Math. 91 (1980), 293-306. MR 82f:54030
  • [GL] G. Godefroy and A. Louveau, Axioms of determinacy and biorthogonal systems, Israel J. Math. 67 (1989), 109-116. MR 91k:03126
  • [GT] G. Godefroy and M. Talagrand, Espaces de Banach representables, Israel J. Math. 41 (1982), 321-330. MR 84g:46019
  • [Gr] G. Gruenhage, Perfectly normal compacta, cosmic spaces, and some partition problems, in: Open Problems in Topology (J. van Mill and G.M. Reed, eds.), Elsevier Sci. Publ., Amsterdam, 1990. MR 92c:54001
  • [H] F. Hausdorff, Die Graduierung nach dem Endrerlauf, Abl. Königl. Sächs Gesell. Wiss Math.-Phys. Kl. 31 (1909), 296-334.
  • [J] R.C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743. MR 54:5811
  • [Ke] A.S. Kechris, Classical descriptive set theory, Springer-Verlag, New York 1995. MR 96e:03057
  • [Kr] A. Krawczyk, Rosenthal compacta and analytic sets, Proc. Amer. Math. Soc. 115 (1992), 1095-1100. MR 92j:54050
  • [LS] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $\ell^1$ and whose duals are non-separable, Studia Math. 54 (1975), 81-105. MR 52:11543
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, 1977. MR 58:17766
  • [LSV] A. Louveau, S. Shelah and B. Velickovic, Borel partitions of infinite subtrees of a prefect tree, Ann. Pure Appl. Logic 63 (1993), 271-281. MR 94g:04003
  • [Lo] S. Lojasiewicz, An introduction to the theory of real functions, Wiley-Interscience Publ., New York 1988. MR 89e:26001
  • [M1] W. Marciszewski, On a classification of pointwise compact sets of the first Baire class functions, Fund. Math. 133 (1989), 195-209. MR 91k:54026
  • [M2] W. Marciszewski, On properties of Rosenthal compacta, Proc. Amer. Math. Soc. 115 (1992), 797-805. MR 92i:54023
  • [MN] S. Mercourakis and S. Negrepontis, Banach spaces and Topology II, in: Recent Progress in Topology (M. Hu\v{s}ek and J. van Mill, eds.), Elsevier Sci. Publ., New York, 1992. MR 95g:54004
  • [Mi] A.W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic 41 (1989), 179-203. MR 90b:03070
  • [N] I.P. Nathanson, Theory of functions of real variable, Moscow, 1950.
  • [Na] I. Namioka, Separate continuity and joint continuity, Pacific J. Math. 51 (1974), 515-531. MR 51:6693
  • [OR] E. Odell and H.P. Rosenthal, A double-dual characterization of separable Banach spaces containing $\ell^1$, Israel J. Math. 20 (1975), 375-384. MR 51:13654
  • [Pa] J. Pawlikowski, Parametrized Ellentuck Theorem, Topology and its Appl. 37 (1990), 65-73. MR 91j:04002
  • [Po1] R. Pol, Note on the spaces $P(S)$ of regular probability measures whose topology is determined by countable subsets, Pacific J. Math. 100 (1982), 185-201. MR 83g:54024
  • [Po2] R. Pol, On pointwise and weak topology in function spaces, Warszaw University, 1984.
  • [Po3] R. Pol, Note on Pointwise convergence of sequences of Analytic sets, Mathematika 36 (1989), 290-300. MR 91j:54070
  • [R1] H.P. Rosenthal, A characterization of Banach spaces containing $\ell^1$, Proc. Nat. Acad. Sci. USA 71 (1974), 2411-2413. MR 50:10773
  • [R2] H.P. Rosenthal, Pointwise compact subsets of the first Baire class, Amer. J. Math. 99 (1977), 362-378. MR 55:11032
  • [R3] H.P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803-831. MR 80d:46023
  • [Ro] F. Rothberger, Sur la familles indenombrables de suites de nombres naturels et les problemes concernant la propriete $C$, Proc. Cambridge Phil. Soc. 37 (1941), 109-126. MR 2:352a
  • [S] D. Scott, A proof of the independence of the Continuum Hypothesis, Math. Systems Theory 1 (1966), 89-111. MR 36:1321
  • [St] C. Stegall, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223. MR 51:10581
  • [Ste] J. Stern, A Ramsey theorem for trees with an application to Banach spaces, Israel J. Math. 29 (1978), 179-188. MR 57:16114
  • [Sz] W. Szlenk, The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia. Math. 30 (1968), 53-61. MR 37:3327
  • [Ta] M. Talagrand, Pettis integral and measure theory, Memoirs of the AMS, Vol 51, No. 307, 1984. MR 86j:46042
  • [To1] S. Todorcevic, Partition problems in topology, Amer. Math. Soc., Providence, 1989. MR 90d:04001
  • [To2] S. Todorcevic, Free sequences, Topology and its Appl. 35 (1990), 235-238. MR 91f:54003
  • [To3] S. Todorcevic, Analytic gaps, Fund. Math. 150 (1996), 55-66. MR 98j:03070
  • [W] H.E. White, Jr., Variants of Blumberg's Theorem, Illinois J. Math. 26 (1982), 359-373. MR 83g:54034

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Additional Information

Stevo Todorcevic
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3

Received by editor(s): January 20, 1997
Received by editor(s) in revised form: April 20, 1999
Published electronically: July 6, 1999
Article copyright: © Copyright 1999 American Mathematical Society