## Journal of the Mathematical Society of Japan

### 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 $\bm{R}$, 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 $\mathscr{E}_\mu$ corresponding to products of $\mu$ zero-dimensional subsets of $\bm{R}$ 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 $\mathscr{E}_\mu$ 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 $\mathscr{E}_\mu$.

#### Article information

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

Dates
First available in Project Euclid: 4 August 2008

https://projecteuclid.org/euclid.jmsj/1217884493

Digital Object Identifier
doi:10.2969/jmsj/06030793

Mathematical Reviews number (MathSciNet)
MR2440414

Zentralblatt MATH identifier
1154.54021

#### Citation

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

#### References

• M. Abry, J. J. Dijkstra and J. van Mill, On one-point connectifications, Topology Appl., 154 (2007), 725–733.
• J. J. Dijkstra, A criterion for Erdős spaces, Proc. Edinb. Math. Soc., 48 (2005), 595–601.
• J. J. Dijkstra, Characterizing stable complete Erdős space, preprint.
• J. J. Dijkstra and J. van Mill, Homeomorphism groups of manifolds and Erdős space, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 29–38.
• J. J. Dijkstra and J. van Mill, Characterizing complete Erdős space, Canad. J. Math., in press.
• J. J. Dijkstra and J. van Mill, Erdős space and homeomorphism groups of manifolds, Mem. Amer. Math. Soc., in press.
• R. Engelking, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics, 10, Heldermann Verlag, Lemgo, 1995.
• P. Erdős, The dimension of the rational points in Hilbert space, Ann. of Math., 41 (1940), 734–736.
• W. Hurewicz, Sur la dimension des produits Cartésiens, Ann. of Math., 36 (1935), 194–197.
• K. Kawamura, L. G. Oversteegen and E. D. Tymchatyn, On homogeneous totally disconnected 1-dimensional spaces, Fund. Math., 150 (1996), 97–112.
• A. Lelek, On plane dendroids and their end points in the classical sense, Fund. Math., 49 (1960/1961), 301–319.
• P. Roy, Failure of equivalence of dimension concepts for metric spaces, Bull. Amer. Math. Soc., 68 (1962), 609–613.