The generalized Borel conjecture and strongly proper orders
Author:
Paul Corazza
Journal:
Trans. Amer. Math. Soc. 316 (1989), 115140
MSC:
Primary 03E35; Secondary 04A15, 26A21, 28A05
MathSciNet review:
982239
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Abstract: The Borel Conjecture is the statement that , where is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that . We show that this statement is also independent. The construction involves forcing with an stage iteration of strongly proper orders; this latter class of orders is shown to include several wellknown orders, such as Sacks and Silver forcing, and to be properly contained in the class of proper, bounding orders. The central lemma is the observation that A. W. Miller's proof that the statement "Every set of reals of power c can be mapped (uniformly) continuously onto " holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that is not restricted by the presence of large universal measure zero sets (as it is by the presence of large sets). We also investigate the ideal and prove various consistency results concerning the relationships between , and AFC (where ). These latter results partially answer two questions of J. Brown.
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 [Br1]
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 [Br2]
 J. B. Brown and K. Prikry, Variation on Lusin's Theorem, Trans. Amer. Math. Soc. 302 (1987), 7785. MR 887497 (88e:26003)
 [BrC]
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 E. van Douwen, The integers and topology, Handbook of Set Theoretic Topology (K. Kunen and J. Vaughan, eds.), NorthHolland, 1984. MR 776619 (85k:54001)
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 D. H. Fremlin and A. W. Miller, On some properties of Hurewicz, Menger, and Rothberger Fund. Math. 129 (1988), 1733. MR 954892 (89g:54061)
 [G1]
 E. Grzegorek, Solution of a problem of Banach on fields without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 710. MR 616191 (82h:04005)
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 , Always of the first category sets, Proceedings of the 12th Winter School on Abstract Analysis, Srni (Bohemian Weald), January 1528, 1984, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo (2) 6 (1984), 139147. MR 782712
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 , Always of the first category sets. II, unpublished manuscript, 1985.
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 F. Galvin and A. W. Miller, sets and other singular sets, Topology Appl. 17 (1984), 145155. MR 738943 (85f:54011)
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 J. R. Isbell, Spaces without large projective subspaces, Math. Scand. 17 (1965), 80105. MR 0196695 (33:4882)
 [J1]
 T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
 [J2]
 , Multiple forcing, Cambridge Univ. Press, 1986. MR 895139 (89h:03001)
 [K]
 K. Kunen, Random and Cohen reals, Handbook of SetTheoretic Topology (K. Kunen and J. Vaughn, eds.), NorthHolland, 1984. MR 776619 (85k:54001)
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 K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
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 R. Laver, On the consistency of Borel's conjecture, Acta Math. 137, 151169. MR 0422027 (54:10019)
 [M1]
 A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93114. MR 613787 (84e:03058a)
 [M2]
 , Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575584. MR 716618 (84k:03125)
 [M3]
 , Special subsets of the real line, Handbook of SetTheoretic Topology (K. Kunen and J. Vaughn, eds.), NorthHolland, 1984. MR 776624 (86i:54037)
 [Ma]
 E. Szpilrajn (Marczewski), On absolutely measurable sets and functions, C. R. Soc. Sci. Varsovie (3) 30 (1937), 3968. (Polish)
 [P]
 J. Pawlikowski, Why Solovay real produces Cohen real, J. Symbolic Logic 51 (1986), 957968. MR 865922 (88e:03076)
 [R]
 F. Rothberger, Eine Verscharfung dei Eigenschaft , Fund. Math. 30 (1938), 5055.
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 G. E. Sacks, Forcing with perfect closed sets, Axiomatic Set Theory (D. Scott, ed.), Proc. Sympos. Pure Math., vol. 13, part 2, Amer. Math. Soc., Providence, R.I., 1971, pp. 331355. MR 0276079 (43:1827)
 [Sh]
 S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, SpringerVerlag, 1982. MR 675955 (84h:03002)
 [Si]
 W. Sierpiński, Sur la nonvariance topologique de la propriété , Fund. Math. 33 (1945), 264268. MR 0017332 (8:140c)
 [W]
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719890982239X
PII:
S 00029947(1989)0982239X
Keywords:
Borel's conjecture,
strong measure zero,
universal measure zero,
iterated forcing Sacks order,
uniformly continuous maps,
maps onto ,
proper orders,
bounding orders
Article copyright:
© Copyright 1989
American Mathematical Society
