Compact subsets of the first Baire class

Todorčević, Stevo

doi:10.1090/S0894-0347-99-00312-4

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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
Posted: 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.

[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), no. 2, 395–446. MR 1226181 (94i:46016), http://dx.doi.org/10.1216/rmjm/1181072569
[Ba] R. Baire, Sur les fonctions des variables rèelles, Ann. Mat. Pura Appl. 3 (1899), 16-30.
[Bl] Andreas Blass, A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82 (1981), no. 2, 271–277. MR 609665 (83k:03063), http://dx.doi.org/10.1090/S0002-9939-1981-0609665-0
[Bo] J. Bourgain, Some remarks on compact sets of first Baire class, Bull. Soc. Math. Belg. 30 (1978), no. 1, 3–10. MR 549647 (80j:54008)
[BFT] J. Bourgain, D. H. Fremlin, and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), no. 4, 845–886. MR 509077 (80b:54017), http://dx.doi.org/10.2307/2373913
[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] Gabriel Debs, Effective properties in compact sets of Borel functions, Mathematika 34 (1987), no. 1, 64–68. MR 908840 (89b:03082), http://dx.doi.org/10.1112/S0025579300013280
[F1] D. H. Fremlin, Consequences of Martin’s axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cambridge, 1984. MR 780933 (86i:03001)
[F] D. H. Fremlin, On compact spaces carrying Radon measures of uncountable Maharam type, Fund. Math. 154 (1997), no. 3, 295–304. MR 1475869 (99d:28019)
[Go] Gilles Godefroy, Compacts de Rosenthal, Pacific J. Math. 91 (1980), no. 2, 293–306 (French, with English summary). MR 615679 (82f:54030)
[GL] Gilles Godefroy and Alain Louveau, Axioms of determinacy and biorthogonal systems, Israel J. Math. 67 (1989), no. 1, 109–116. MR 1021365 (91k:03126), http://dx.doi.org/10.1007/BF02764903
[GT] Gilles Godefroy and Michel Talagrand, Espaces de Banach représentables, Israel J. Math. 41 (1982), no. 4, 321–330 (French, with English summary). MR 657864 (84g:46019), http://dx.doi.org/10.1007/BF02760538
[Gr] Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636 (92c:54001)
[H] F. Hausdorff, Die Graduierung nach dem Endrerlauf, Abl. Königl. Sächs Gesell. Wiss Math.-Phys. Kl. 31 (1909), 296-334.
[J] Robert C. James, A separable somewhat reflexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. MR 0417763 (54 #5811), http://dx.doi.org/10.1090/S0002-9904-1974-13580-9
[Ke] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
[Kr] Adam Krawczyk, On the Rosenthal compacta and analytic sets, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1095–1100. MR 1086583 (92j:54050), http://dx.doi.org/10.1090/S0002-9939-1992-1086583-5
[LS] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain ℓ₁ and whose duals are non-separable, Studia Math. 54 (1975), no. 1, 81–105. MR 0390720 (52 #11543)
[LT] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056 (58 #17766)
[LSV] A. Louveau, S. Shelah, and B. Veličković, Borel partitions of infinite subtrees of a perfect tree, Ann. Pure Appl. Logic 63 (1993), no. 3, 271–281. MR 1237234 (94g:04003), http://dx.doi.org/10.1016/0168-0072(93)90151-3
[Lo] Stanisław Łojasiewicz, An introduction to the theory of real functions, 3rd ed., A Wiley-Interscience Publication, John Wiley & Sons Ltd., Chichester, 1988. With contributions by M. Kosiek, W. Mlak and Z. Opial; Translated from the Polish by G. H. Lawden; Translation edited by A. V. Ferreira. MR 952856 (89e:26001)
[M1] Witold Marciszewski, On a classification of pointwise compact sets of the first Baire class functions, Fund. Math. 133 (1989), no. 3, 195–209. MR 1065902 (91k:54026)
[M2] Witold Marciszewski, On properties of Rosenthal compacta, Proc. Amer. Math. Soc. 115 (1992), no. 3, 797–805. MR 1096213 (92i:54023), http://dx.doi.org/10.1090/S0002-9939-1992-1096213-4
[MN] Miroslav Hušek and Jan van Mill (eds.), Recent progress in general topology, North-Holland Publishing Co., Amsterdam, 1992. Papers from the Symposium on Topology (Toposym) held in Prague, August 19–23, 1991. MR 1229121 (95g:54004)
[Mi] Arnold W. Miller, Infinite combinatorics and definability, Ann. Pure Appl. Logic 41 (1989), no. 2, 179–203. MR 983001 (90b:03070), http://dx.doi.org/10.1016/0168-0072(89)90013-4
[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 0370466 (51 #6693)
[OR] E. Odell and H. P. Rosenthal, A double-dual characterization of separable Banach spaces containing 𝑙¹, Israel J. Math. 20 (1975), no. 3-4, 375–384. MR 0377482 (51 #13654)
[Pa] Janusz Pawlikowski, Parametrized Ellentuck theorem, Topology Appl. 37 (1990), no. 1, 65–73. MR 1075374 (91j:04002), http://dx.doi.org/10.1016/0166-8641(90)90015-T
[Po1] Roman Pol, Note on the spaces 𝑃(𝑆) of regular probability measures whose topology is determined by countable subsets, Pacific J. Math. 100 (1982), no. 1, 185–201. MR 661448 (83g:54024)
[Po2] R. Pol, On pointwise and weak topology in function spaces, Warszaw University, 1984.
[Po3] Roman Pol, Note on pointwise convergence of sequences of analytic sets, Mathematika 36 (1989), no. 2, 290–300 (1990). MR 1045789 (91j:54070), http://dx.doi.org/10.1112/S0025579300013140
[R1] Haskell P. Rosenthal, A characterization of Banach spaces containing 𝑙¹, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 0358307 (50 #10773)
[R2] Haskell P. Rosenthal, Point-wise compact subsets of the first Baire class, Amer. J. Math. 99 (1977), no. 2, 362–378. MR 0438113 (55 #11032)
[R3] Haskell P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), no. 5, 803–831. MR 499730 (80d:46023), http://dx.doi.org/10.1090/S0002-9904-1978-14521-2
[Ro] Fritz Rothberger, Sur les familles indénombrables de suites de nombres naturels et les problèmes concernant la propriété 𝐶, Proc. Cambridge Philos. Soc. 37 (1941), 109–126 (French). MR 0004281 (2,352a)
[S] Dana Scott, A proof of the independence of the continuum hypothesis, Math. Systems Theory 1 (1967), 89–111. MR 0218233 (36 #1321)
[St] Charles Stegall, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223. MR 0374381 (51 #10581), http://dx.doi.org/10.1090/S0002-9947-1975-0374381-1
[Ste] Jacques Stern, A Ramsey theorem for trees, with an application to Banach spaces, Israel J. Math. 29 (1978), no. 2-3, 179–188. MR 0476554 (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 0227743 (37 #3327)
[Ta] Michel Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 756174 (86j:46042)
[To1] Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949 (90d:04001)
[To2] Stevo Todorčević, Free sequences, Topology Appl. 35 (1990), no. 2-3, 235–238. MR 1058803 (91f:54003), http://dx.doi.org/10.1016/0166-8641(90)90108-E
[To3] Stevo Todorčević, Analytic gaps, Fund. Math. 150 (1996), no. 1, 55–66. MR 1387957 (98j:03070)
[W] H. E. White Jr., Variants of Blumberg’s theorem, Illinois J. Math. 26 (1982), no. 3, 359–373. MR 658447 (83g:54034)

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