The instability of nonseparable complete Erdős spaces and representations in $\mathbb R$-trees

Jan J. Dijkstra¹ ; Kirsten I. S. Valkenburg¹

¹ Faculteit der Exacte Wetenschappen//Afdeling Wiskunde Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands

Fundamenta Mathematicae, Tome 207 (2010) no. 3, pp. 197-210

Résumé

One way to generalize complete Erdős space ${\mathfrak E}_{\rm c}$ is to consider uncountable products of zero-dimensional $G_\delta$-subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise about analogies between the behaviour of complete Erdős space and its generalizations. The discovery that ${\mathfrak E}_{\rm c}$ is unstable, by which we mean that the space is not homeomorphic to its infinite power, by Dijkstra, van Mill, and Steprāns, led to the solution of a series of problems in the literature. In the present paper we prove by a different method that our nonseparable complete Erdős spaces are also unstable. Another application of ${\mathfrak E}_{\rm c}$ is that it is homeomorphic to the endpoint set of the universal separable $\mathbb R$-tree. Our standard models can also be represented as endpoint sets of more general $\mathbb R$-trees, but some universality properties are lost

Détail

DOI : 10.4064/fm207-3-1

Keywords: generalize complete erd space mathfrak consider uncountable products zero dimensional delta subsets real line intersected appropriate banach space resulting nonseparable complete erd spaces fully classified only cardinal invariants done earlier paper authors together van mill think correct generalize concept complete erd space nonseparable setting natural questions arise about analogies between behaviour complete erd space its generalizations discovery mathfrak unstable which mean space homeomorphic its infinite power dijkstra van mill stepr led solution series problems literature present paper prove different method nonseparable complete erd spaces unstable another application mathfrak homeomorphic endpoint set universal separable mathbb r tree standard models represented endpoint sets general mathbb r trees universality properties lost