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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

Citations:

Zbl 1152.54347
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References:

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