Characterizing Complete Erdős Space

The space now known as complete Erdős space Ec was introduced by Paul Erdős in 1940 as the closed subspace of the Hilbert space l2 consisting of all vectors such that every coordinate is in the convergent sequence {0} ∪ {1/n : n ∈ N}. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of Ec. As an application we determine the class of factors of Ec. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces lp according to the ‘Erdős method’ are homeomorphic to Ec. A novel application states that if I is a Polishable Fσ-ideal on ω, then I with the Polish topology is homeomorphic to either Z, the Cantor set 2ω , Z × 2ω , or Ec. This last result answers a question that was asked by Stevo Todorčević.