Weak-invariant properties of the norm topology

Authors:
I. Namioka and R. Pol

Journal:
Proc. Amer. Math. Soc. **118** (1993), 507-511

MSC:
Primary 46B20; Secondary 54H05

MathSciNet review:
1132419

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A property (P) relative to the norm topology of a Banach space is a weak-invariant if, whenever and are weakly homeomorphic subsets of (possibly different) Banach spaces and (, norm) has property (P), then (, norm) has property (P). We show that the property of being -discrete and the property of being an absolute Souslin- space of weight , both relative to the norm topology, are weak-invariants. These conclusions are obtained from a result concerning maps of metrizable spaces into function spaces.

**[1]**William G. Fleissner,*An axiom for nonseparable Borel theory*, Trans. Amer. Math. Soc.**251**(1979), 309–328. MR**531982**, 10.1090/S0002-9947-1979-0531982-9**[2]**R. W. Hansell,*On characterizing non-separable analytic and extended Borel sets as types of continuous images*, Proc. London Math. Soc. (3)**28**(1974), 683–699. MR**0362269****[3]**J. E. Jayne and C. A. Rogers, -*analytic sets*, Analytic Sets (Conf., University of Coll., Univ. of London, 1978), Academic Press, London and New York, 1980, pp. 1-181.**[4]**J. L. Kelley et al.,*Linear topological spaces*, Graduate Texts in Math., vol. 27, Springer-Verlag, New York, Heidelberg, and Berlin, 1975.**[5]**R. Pol,*Note on decompositions of metrizable spaces. I*, Fund. Math.**95**(1977), no. 2, 95–103. MR**0433371****[6]**A. H. Stone,*On 𝜎-discreteness and Borel isomorphism*, Amer. J. Math.**85**(1963), 655–666. MR**0156789****[7]**M. H. Stone,*The generalized Weierstrass approximation theorem*, Math. Mag.**21**(1948), 167–184, 237–254. MR**0027121****[8]**Waclaw Sierpinski,*General topology*, Mathematical Expositions, No. 7, University of Toronto Press, Toronto, 1952. Translated by C. Cecilia Krieger. MR**0050870**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46B20,
54H05

Retrieve articles in all journals with MSC: 46B20, 54H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1132419-4

Keywords:
Weak topology of Banach spaces,
-discreteness,
Souslin sets

Article copyright:
© Copyright 1993
American Mathematical Society