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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
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Abstract: A property (P) relative to the norm topology of a Banach space is a weak-invariant if, whenever $ A$ and $ B$ are weakly homeomorphic subsets of (possibly different) Banach spaces and ($ A$, norm) has property (P), then ($ B$, norm) has property (P). We show that the property of being $ \sigma $-discrete and the property of being an absolute Souslin- $ \mathcal{F}$ space of weight $ \leqslant {\aleph _1}$, both relative to the norm topology, are weak-invariants. These conclusions are obtained from a result concerning maps of metrizable spaces into function spaces.

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Keywords: Weak topology of Banach spaces, $ \sigma $-discreteness, Souslin sets
Article copyright: © Copyright 1993 American Mathematical Society

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