Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

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