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Weak-invariant properties of the norm topology

I. Namioka, R. Pol
1993 Proceedings of the American Mathematical Society  
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 cr-discrete and the property of being an absolute Souslin-^ space of weight < Hx , both relative to the norm topology, are weak-invariants. These conclusions are obtained from a result concerning maps of metrizable spaces
more » ... trizable spaces into function spaces. It is a well-known fact that a weakly separable subset of a Banach space is separable relative to the norm topology. One can formalize this observation as follows: a property (P) relative to the norm topology is called a weak-invariant if, whenever A and B are subsets of (possibly different) Banach spaces such that (A, weak) and (B, weak) are homeomorphic and (A, norm) has property (P), then (B, norm) also has property (P). Thus the separability relative to the norm topology and, indeed, the weight relative to the norm topology are weakinvariants. One of the consequences of the theorem in this note is that the a -discreteness relative to the norm topology is a weak-invariant. By combining this result and those in [5, 2], we also show that the property of being an absolute Souslin-Jj?" space (defined later) of weight < N* in the norm topology is also a weakinvariant. A subset A of a metrizable space M is said to be a-discrete if A = [Jn^Li An with each A" relatively discrete. If p is a metric on M compatible with the topology, then the set A is cr-discrete if and only if A = U~ , B" , where, for each n, there exists an en > 0 such that p(x, y) > e" whenever x ,y e B" and x ^ y (cf. [6]). We use repeatedly the fact that a subset A of a metrizable space M is cr-discrete if it is locally cr-discrete, i.e., for each x e M there is a neighborhood U of jc such that U n A is cr-discrete [6, Lemma 2]. In this note, all topological spaces are assumed to be Hausdorff. Let K be a compact space. Then C(K) denotes the Banach space of all realvalued continuous functions on K with the supremum norm. The topology of pointwise convergence is denoted by rp . The following is the main result of the note.
doi:10.1090/s0002-9939-1993-1132419-4 fatcat:74s23z2gszb3tpd6mighdkge7m