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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: James Hagler
Journal: Trans. Amer. Math. Soc. 214 (1975), 415-428
MSC: Primary 46E15; Secondary 54A25
DOI: https://doi.org/10.1090/S0002-9947-1975-0388062-1
MathSciNet review: 0388062
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Abstract: A dyadic space S is defined to be a continuous image of ${\{ 0,1\} ^\mathfrak {m}}$ for some infinite cardinal number $\mathfrak {m}$. We deduce Banach space properties of $C(S)$ and topological properties of S. For example, under certain cardinality restrictions on $\mathfrak {m}$, we show: Every dyadic space of topological weight $\mathfrak {m}$ contains a closed subset homeomorphic to ${\{ 0,1\} ^\mathfrak {m}}$. Every Banach space X isomorphic to an $\mathfrak {m}$ dimensional subspace of $C(S)$ (for S dyadic) contains a subspace isomorphic to ${l^1}(\Gamma )$ where $\Gamma$ has cardinality $\mathfrak {m}$.


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Article copyright: © Copyright 1975 American Mathematical Society