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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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DOI: https://doi.org/10.1090/S0002-9947-1975-0388062-1
Article copyright: © Copyright 1975 American Mathematical Society

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