A CRITERION FOR ERDŐS SPACES

[摘要] In 1940 Paul ErdÅ‘s introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $mathcal{E}$ of the Banach spaces $ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $mathcal{E}$ is closed in $ell^p$, then it is homeomorphic to complete ErdÅ‘s space if and only if $dimmathcal{E}>0$ and every $E_n$ is zero dimensional.

[学科分类] 数学（综合）

[关键词] Primary 54F45;54F65;ErdÅ‘s space;complete ErdÅ‘s space;Lelek fan;cohesive;topological dimension;almost zero dimensional