Weak-invariant properties of the norm topology

Namioka, I.; Pol, R.

doi:10.1090/S0002-9939-1993-1132419-4

Weak-invariant properties of the norm topology
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by I. Namioka and R. Pol PDF

Proc. Amer. Math. Soc. 118 (1993), 507-511 Request permission

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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