On nonseparable Erdös spaces

Abstract

In 2005, Dijkstra studied subspaces $¥mathscr{E}$ 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 $¥mathscr{E}$ 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 $¥mathscr{E}$_μ corresponding to products of μ 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 $¥mathscr{E}$. We can further generalize the latter to complete spaces and we find that a one-dimensional complete space $¥mathscr{E}$_μ 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 $¥mathscr{E}$_μ.