On nonseparable Erdös spaces

Abstract

In 2005, Dijkstra studied subspaces $E$ of the Banach spaces ${\ell }^p$ that are constructed as `products' of countably many zero-dimensional subsets of $R$, as a generalization of Erdös space and complete Erdös space. He presented a criterion for deciding whether a space of the type $E$ has the same peculiar features as Erdös space, which is one-dimensional yet totally disconnected and has a one-dimensional square. In this paper, we extend the construction to a nonseparable setting and consider spaces $E_{\mu }$ corresponding to products of $\mu$ zero-dimensional subsets of $R$ in nonseparable Banach spaces. We are able to generalize both Dijkstra's criterion and his classification of closed variants of $E$. We can further generalize the latter to complete spaces and we find that a one-dimensional complete space $E_{\mu }$ is homeomorphic to a product of complete Erdös space with a countable product of discrete spaces. Among the applications, we find coincidence of the small and large inductive dimension for $E_{\mu }$.