In 2005, Dijkstra studied subspaces of the Banach spaces that are constructed as `products' of countably many zero-dimensional subsets of , as a generalization of Erdös space and complete Erdös space. He presented a criterion for deciding whether a space of the type 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 corresponding to products of zero-dimensional subsets of in nonseparable Banach spaces. We are able to generalize both Dijkstra's criterion and his classification of closed variants of . We can further generalize the latter to complete spaces and we find that a one-dimensional complete space 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 .
Jan J. DIJKSTRA. Jan VAN MILL. Kirsten I. S. VALKENBURG. "On nonseparable Erdös spaces." J. Math. Soc. Japan 60 (3) 793 - 818, July, 2008. https://doi.org/10.2969/jmsj/06030793