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Published online by Cambridge University Press: 20 November 2018
The space now known as complete Erdős space ${{\mathfrak{E}}_{\text{c}}}$ was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space
${{\ell }^{2}}$ consisting of all vectors such that every coordinate is in the convergent sequence
$\left\{ 0 \right\}\cup \left\{ 1/n:n\in \mathbb{N}\\right\}$ . In a solution to a problem posed by Lex
$G$ . Oversteegen we present simple and useful topological characterizations of
${{\mathfrak{E}}_{\text{c}}}$ . As an application we determine the class of factors of
${{\mathfrak{E}}_{\text{c}}}$ . In another application we determine precisely which of the spaces that can be constructed in the Banach spaces
${{\ell }^{p}}$ according to the ‘Erdős method’ are homeomorphic to
${{\mathfrak{E}}_{\text{c}}}$ . A novel application states that if
$I$ is a Polishable
${{F}_{\sigma }}$ -ideal on
$\omega $ , then
$I$ with the Polish topology is homeomorphic to either
$\mathbb{Z}$ , the Cantor set
${{2}^{\omega }},\,\mathbb{Z}\,\times \,{{2}^{\omega }}$ , or
${{\mathfrak{E}}_{\text{c}}}$ . This last result answers a question that was asked by Stevo Todorčević.