the space now known as complete erdos space ec was introduced by paul erdos in 1940 as the closed subspace of the hilbert space ?2 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 ?2 according to the ‘erdos method’ are homeomorphic to ec. a novel application states that if i is a polishable f? -ideal on ?, then i with the polish topology is homeomorphic to either z, the cantor set 2?, z × 2?, or ec. this last result answers a question that was asked by stevo todorcevic.