A weakly Stegall space that is not a Stegall space

Authors:
Warren B. Moors and Sivajah Somasundaram

Journal:
Proc. Amer. Math. Soc. **131** (2003), 647-654

MSC (2000):
Primary 54C60, 26E25, 54C10

DOI:
https://doi.org/10.1090/S0002-9939-02-06717-5

Published electronically:
June 27, 2002

MathSciNet review:
1933358

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Abstract: A topological space is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space and minimal usco , is single-valued at some point of . In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space whose dual, equipped with the weak topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space such that belongs to the class of weakly Stegall spaces but does not.

**1.**Marián J. Fabian,*Gâteaux differentiability of convex functions and topology. Weak Asplund spaces,*Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley-Interscience, New York, 1997. MR**98h:46009****2.**Ondrej F. K. Kalenda, Weak Stegall spaces, unpublished manuscript, Spring 1997 (3 pages).**3.**Ondrej F. K. Kalenda, Stegall compact spaces which are not fragmentable,*Topology Appl.***96**(1999), 121-132. MR**2000i:54027****4.**Ondrej F. K. Kalenda, A weak Asplund space whose dual is not in Stegall's class,*Proc. Amer. Math. Soc.***130**(2002), 2139-2143.**5.**Petar S. Kenderov, Ivaylo S. Kortezov and Warren B. Moors, Continuity points of quasi-continuous mappings,*Topology Appl.***109**(2001), 321-346. MR**2001m:54008****6.**Petar S. Kenderov and Warren B. Moors, Game characterization of fragmentability of topological spaces,*Mathematics and Education in Mathematics 8-18, Proceedings of the 25th Spring Conference of the Union of Bulgarian Mathematicians, April 1996, Kazanlak, Bulgaria*(1996).**7.**Petar S. Kenderov and Warren B. Moors, Fragmentability and sigma-fragmentability of Banach spaces,*J. London Math Soc.***60**(1999), 203-223. MR**2001f:46025****8.**Petar S. Kenderov, Warren B. Moors and Scott Sciffer, A weak Asplund space whose dual is not weak fragmentable,*Proc. Amer. Math. Soc.***129**(2001), 3741-3747.**9.**John C. Oxtoby,*Measure and category. A survey of the analogies between topological and measure spaces,*Graduate Texts in Mathematics, Vol. 2. Springer-Verlag, New York, 1971. MR**52:14213****10.**Sivajah Somasundaram,*Almost weak Asplund spaces*, PhD Thesis, The University of Waikato, Hamilton, New Zealand (in preparation).

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Additional Information

**Warren B. Moors**

Affiliation:
Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton 2001, New Zealand

Email:
moors@math.waikato.ac.nz

**Sivajah Somasundaram**

Affiliation:
Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton 2001, New Zealand

Email:
ss15@math.waikato.ac.nz

DOI:
https://doi.org/10.1090/S0002-9939-02-06717-5

Keywords:
Weak Asplund,
almost weak Asplund,
Stegall space,
weakly Stegall space

Received by editor(s):
September 12, 2001

Published electronically:
June 27, 2002

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2002
American Mathematical Society