A weak Asplund space whose dual is not weak fragmentable

Authors:
Petar S. Kenderov, Warren B. Moors and Scott Sciffer

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3741-3747

MSC (2000):
Primary 54C60, 46B20, 54C10

Published electronically:
May 21, 2001

MathSciNet review:
1860511

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Under the assumption that there exists in the unit interval an uncountable set with the property that every continuous mapping from a Baire metric space into is constant on some non-empty open subset of , we construct a Banach space such that belongs to Stegall's class but is not fragmentable.

**1.**George Bachman and Lawrence Narici,*Functional analysis*, Academic Press, New York-London, 1966. MR**0217549****2.**Marián J. Fabian,*Gâteaux differentiability of convex functions and topology*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. Weak Asplund spaces; A Wiley-Interscience Publication. MR**1461271****3.**Ryszard Frankiewicz and Kenneth Kunen,*Solution of Kuratowski’s problem on function having the Baire property. I*, Fund. Math.**128**(1987), no. 3, 171–180. MR**922569****4.**Ondřej Kalenda,*Stegall compact spaces which are not fragmentable*, Topology Appl.**96**(1999), no. 2, 121–132. MR**1702306**, 10.1016/S0166-8641(98)00045-5**5.**P. S. Kenderov and J. Orihuela,*A generic factorization theorem*, Mathematika**42**(1995), no. 1, 56–66. MR**1346672**, 10.1112/S0025579300011359**6.**W. B. Moors and S. D. Sciffer, Sigma-fragmentable spaces that are not countable unions of fragmentable subspaces,*Topology Appl.*(to appear.)**7.**I. Namioka and R. Pol,*Mappings of Baire spaces into function spaces and Kadec renorming*, Israel J. Math.**78**(1992), no. 1, 1–20. MR**1194955**, 10.1007/BF02801567**8.**Charles Stegall,*A class of topological spaces and differentiation of functions on Banach spaces*, Proceedings of the conferences on vector measures and integral representations of operators, and on functional analysis/Banach space geometry (Essen, 1982) Vorlesungen Fachbereich Math. Univ. Essen, vol. 10, Univ. Essen, Essen, 1983, pp. 63–77. MR**730947**

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Additional Information

**Petar S. Kenderov**

Affiliation:
Institute of Mathematics, Bulgarian Academy of Science, Acad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria

Email:
pkend@bgcict.acad.bg; pkend@math.bas.bg

**Warren B. Moors**

Affiliation:
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

Email:
moors@math.auckland.ac.nz

**Scott Sciffer**

Affiliation:
Department of Mathematics, University of Newcastle, Newcastle NSW-2308, Australia

DOI:
https://doi.org/10.1090/S0002-9939-01-06002-6

Keywords:
Stegall's class,
fragmentability,
weak Asplund space,
double arrow space,
Baire space,
minimal usco.

Received by editor(s):
February 17, 2000

Received by editor(s) in revised form:
April 22, 2000

Published electronically:
May 21, 2001

Additional Notes:
The first author was partially supported by Grant MM-701/97 of the National Fund for Scientific Research of the Bulgarian Ministry of Education, Science and Technology

The second author was supported by a Marsden fund grant, VUW 703, administered by the Royal Society of New Zealand

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2001
American Mathematical Society