On subclasses of weak Asplund spaces
Authors:
Ondrej F. K. Kalenda and Kenneth Kunen
Journal:
Proc. Amer. Math. Soc. 133 (2005), 425429
MSC (2000):
Primary 46B26, 03E35
Published electronically:
September 2, 2004
MathSciNet review:
2093063
Fulltext PDF Free Access
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Abstract: Assuming the consistency of the existence of a measurable cardinal, it is consistent to have two Banach spaces, , where is a weak Asplund space such that (in the weak* topology) in not in Stegall's class, whereas is in Stegall's class but is not weak* fragmentable.
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 M. Fabian, Gâteaux differentiability of convex functions and topology: weak Asplund spaces, WileyInterscience, New York, 1997. MR 1461271 (98h:46009)
 [2]
 R. Frankiewicz and K. Kunen, Solution of Kuratowski's problem on function having the Baire property I., Fund. Math. 128 (1987), 171180.MR 0922569 (89a:03090)
 [3]
 T. Jech, M. Magidor, W. Mitchell and K. Prikry, Precipitous ideals, J. Symb. Log. 45 (1980), 18. MR 0560220 (81h:03097)
 [4]
 Y. Kakuda, On a condition for Cohen extensions which preserve precipitous ideals, J. Symbolic Logic 46 (2) (1981), 296300.MR 0613283 (82i:03062)
 [5]
 O. Kalenda, Hereditarily Baire spaces and point of continuity property, Diploma Thesis, Charles University, Prague, 1995 (Czech).
 [6]
 O. Kalenda, Stegall compact spaces which are not fragmentable, Topol. Appl. 96 (2) (1999), 121132. MR 1702306 (2000i:54027)
 [7]
 O. Kalenda, A weak Asplund space whose dual is not in Stegall's class, Proc. Amer. Math. Soc. 130 (7) (2002), 21392143. MR 1896051 (2003a:46024)
 [8]
 A. Kanamori, The higher infinite. Large cardinals in set theory from their beginnings, SpringerVerlag, Berlin, 1994. MR 1321144 (96k:03125)
 [9]
 P. Kenderov, W. Moors and S. Sciffer, A weak Asplund space whose dual is not weak* fragmentable, Proc. Amer. Math. Soc. 129 (12) (2001), 37413747. MR 1860511 (2002h:54014)
 [10]
 K. Kunen, Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179227. MR 0277346 (43:3080)
 [11]
 K. Kunen, Set theory. An introduction to independence proofs, Studies in logic and the foundations of mathematics, vol. 102, NorthHolland, 1980. MR 0597342 (82f:03001)
 [12]
 C. Kuratowski, La propriété de Baire dans les espaces métriques, Fund. Math. 16 (1930), 390394.
 [13]
 D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), 143178. MR 0270904 (42:5787)
 [14]
 J. Silver, The consistency of the GCH with the existence of a measurable cardinal, Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. XIII, Part I,, Univ. California, Los Angeles, Calif., 1967, pp. 391395.MR 0278937 (43:4663)
 [15]
 R. M. Solovay and S. Tennenbaum, Iterated Cohen extensions and Souslin's problem, Ann. of Math. 94 (1971), 201245. MR 0294139 (45:3212)
 [16]
 C. Stegall, A class of topological spaces and differentiability, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen 10 (1983), 6377. MR 0730947 (85j:46026)
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Additional Information
Ondrej F. K. Kalenda
Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email:
kalenda@karlin.mff.cuni.cz
Kenneth Kunen
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
kunen@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S0002993904077445
PII:
S 00029939(04)077445
Keywords:
Weak Asplund space,
fragmentable space,
Stegall's class of spaces,
measurable cardinal
Received by editor(s):
October 4, 2001
Published electronically:
September 2, 2004
Additional Notes:
The first author was supported by Research grants GAUK 277/2001, GAČR 201/00/1466 and MSM 113200007
The second author was supported by NSF Grant DMS0097881
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
