On the structure of 𝑆 and 𝐶(𝑆) for 𝑆 dyadic

Abstract

ABSTRACT. A dyadic space S is defined to be a continuous image of {o, l}m for some infinite cardinal number m. We deduce Banach space properties of C(S) and topological properties of S. For example, under certain cardinality restrictions on m, we show: Every dyadic space of topological weight m contains a closed subset homeomorphic to {o, l}m. Every Banach space X isomorphic to an m dimensional subspace of C(S) (for S dyadic) contains a subspace isomorphic to /'(r) where r has cardinality m.Introduction. In this paper we examine the relationships between dyadic spaces S and the Banach spaces C(S) of continuous real valued functions on S. The main result is

On Corson compacta and embeddings of 𝐶(𝐾) spaces

“…The last corollary is essentially known. It can be derived from a result due to Hagler [10], which, in particular, says that if K is not metrizable and C(K) embeds into C(2 ω 1 ), then C(K) contains an isomorphic copy of the Banach space l 1 (ω 1 ).…”

Some remarks on countably determined measures and uniform distribution of sequences

“…We remark that every dyadic space belonging to the class of Cg is metrizable. Indeed, suppose that X is not metrizable then by a result of Hagler [11] the classical Banach space ]1(co +) is embedded in C(X), and hence X carries a regular Borel probability measure # of uncountable type, that is dimLl(#)=co + (cf. [21], Sect.…”

On nonseparable Banach spaces

“…An example. The results of [3], like those of [6,9], and [10] depended on the existence of independent famihes of sets in order to get embedding of / X(T) spaces. The following example shows that such techniques would not be adequate to yield the result of 5.1.…”

“…(1) a family (F£: £ < co + } and We set P = U£<w+ P( and for each £ < co+ % = U"2£ %f, and for (F,f), (G, g) in P we set (G, g) < (F,/) iff F c G and/ c g. Obviously the set (P, <) is a partially ordered set and each %£ is a dense subset of P. 6.10. Lemma.…”