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.