Classification of Erdős type subspaces of nonseparable -spaces

Consider an arbitrary infinite cardinal number μ and the possibly nonseparable real Banach space ℓ μ p . For a fixed collection of subsets E α ⊂ R for α ∈ μ , one can study the space E μ = { x ∈ ℓ μ p : x α ∈ E α , for each α ∈ μ } as a subspace of ℓ μ p . The main result in this article states that there exist two cardinal invariants λ , κ of E μ so that whenever infinitely many of the E α are of the first category in themselves, E μ ≈ E × λ ω × κ if and only if all E α are zero-dimensional F σ δ -subsets of R and E μ 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.