## On nonseparable Erdős spaces.(English)Zbl 1154.54021

Let $$\mu$$ be an infinite cardinal and $$p\geq1$$. Consider the space $\ell_{\mu}^{\,p}=\{x=(x_{\alpha})_{\alpha\in\mu}\in{\mathbb R}^{\mu}: \sum_{\alpha\in\mu}| x_{\alpha}| ^p<\infty\}$ with the topology generated by the norm $$| | x_{\alpha}| | _p= (\sum_{\alpha\in\mu}| x_{\alpha}| ^p)^{1/p}$$, which is a generalization of the Banach space $$\ell^{\,p}(=\ell_{\omega}^{\,p})$$. In this paper, the authors study subspaces $${\mathcal E}_{\mu}=(\prod_{\alpha\in\mu}E_{\alpha})\cap \ell_{\mu}^{\,p}$$ of $$\ell_{\mu}^{\,p}$$, where each $$E_{\alpha}$$ is a subset of $$\mathbb R$$. General separable metrizable spaces $${\mathcal E}_{\omega}$$ were studied by the first author in [Proc. Edinb. Math. Soc., II. Ser. 48, No. 3, 595–601 (2005; Zbl 1152.54347)]. The authors generalize the results obtained there to an arbitrary cardinal number $$\mu$$. Indeed, they prove the following: Assume that $${\mathcal E}_{\mu}$$ is not empty and that ind$$\,E_{\alpha}=0$$ for every $$\alpha\in\mu$$. For each $$m\in{\mathbb N}$$ let $$\eta(m)=(\eta(m)_{\alpha})_{\alpha\in\mu}\in {\mathbb R}^{\mu}$$, where each $$\eta(m)_{\alpha}=\sup\{| a| :a\in E_{\alpha} \cap [-1/m,\,1/m]\}$$ and $$\sup\emptyset=0$$. Then the following statements are equivalent: (1) $$| | \eta(m)| | _p=\infty$$ for each $$m\in{\mathbb N}$$; (2) there exists an $$x\in\prod_{\alpha\in\mu}E_{\alpha}$$ with $$| | x| | _p=\infty$$ and $$\lim_{\alpha\in\mu}x_{\alpha}=0$$; (3) every nonempty clopen subset of $${\mathcal E}_{\mu}$$ is unbounded; and (4) ind$$\,{\mathcal E}_{\mu}>0$$. They also obtain a classification theorem for complete spaces $${\mathcal E}_{\mu}$$: If $${\mathcal E}_{\mu}$$ is topologically complete, ind$$\,{\mathcal E}_{\mu}>0$$ and each $$E_{\alpha}$$ is zero-dimensional, then there exist discrete spaces $$X$$ and $$Y$$ such that $${\mathcal E}_{\mu}$$ is homeomorphic to $${\mathfrak E}_c\times X^{\omega}\times Y$$, where $${\mathfrak E}_c$$ means complete Erdős space. Furthermore, as an application of the results, they show that ind$$\,{\mathcal E}_{\mu}=\text{Ind}\,{\mathcal E}_{\mu}=\dim{\mathcal E}_{\mu}$$.

### MSC:

 54F45 Dimension theory in general topology 54F65 Topological characterizations of particular spaces

Zbl 1152.54347
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### References:

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