Journal of the Mathematical Society of Japan

On nonseparable Erdös spaces


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In 2005, Dijkstra studied subspaces E of the Banach spaces 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 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 E . We can further generalize the latter to complete spaces and we find that a one-dimensional complete space 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 E μ .

Article information

J. Math. Soc. Japan, Volume 60, Number 3 (2008), 793-818.

First available in Project Euclid: 4 August 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54F45: Dimension theory [See also 55M10] 54F65: Topological characterizations of particular spaces

complete Erdös space Lelek fan topological dimension almost zero-dimensional nonseparable Banach space


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.

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