Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 60, Number 3 (2008), 793-818.
On nonseparable Erdös spaces
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 .
J. Math. Soc. Japan, Volume 60, Number 3 (2008), 793-818.
First available in Project Euclid: 4 August 2008
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DIJKSTRA, Jan J.; VAN MILL, Jan; VALKENBURG, Kirsten I. S. On nonseparable Erdös spaces. J. Math. Soc. Japan 60 (2008), no. 3, 793--818. doi:10.2969/jmsj/06030793. https://projecteuclid.org/euclid.jmsj/1217884493