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

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 $\mathfrak {m}$.