Classification of Erdős type subspaces of nonseparable ℓp-spaces

Consider an arbitrary infinite cardinal number μ and the possibly nonseparable real Banach space . For a fixed collection of subsets Eα⊂R for α∈μ, one can study the space as a subspace of . The main result in this article states that there exist two cardinal invariants λ,κ of Eμ so that whenever infinitely many of the Eα are of the first category in themselves, Eμ≈E×λω×κ if and only if all Eα are zero-dimensional Fσδ-subsets of R and Eμ is at least one-dimensional. Here, E denotes the famed Erdős space introduced by Paul Erdős as ‘rational points in Hilbert space’, the subspace of Hilbert space consisting of vectors of which all coordinates are rational.