Countable spread of exp Y and λY

Abstract

We show that it is consistent with ZFC that there exists a compact 0-dimensional Hausdorff space X for which exp X has countable spread, but X is not metrizable. This establishes the independence of Malyhin's problem. The space X also has no uncountable weakly separated subspaces, its superextension is first countable, and its square is a strong S-space. For 0-dimensional Y we prove that λY has countable spread iff Y is compact and metrizable. We show that it is consistent with ZFC that if Y is 0-dimensional and λY is first countable, then Y is compact and metrizable.