Characterizing non-separable sigma-locally compact infinite-dimensional manifolds and its applications

Abstract

For an infinite cardinal τ, let ℓ₂^f(τ) be the linear span of the canonical orthonormal basis of the Hilbert space ℓ₂(τ) of weight = τ. In this paper, we give characterizations of topological manifolds modeled on ℓ₂^f(τ) and ℓ₂^f(τ) × Q, where Q = [−1,1]^ℕ is the Hilbert cube. We denote the full simplicial complex of cardinality = τ and the hedgehog of weight = τ by Δ(τ) and J(τ), respectively. Using our characterization of ℓ₂^f(τ), we prove that both the metric polyhedron of Δ(τ) and the space
   J(τ)^ℕ_f = {x ∈ J(τ)^ℕ | x(n) = 0 except for finitely many n ∈ ℕ}
are homeomorphic to ℓ₂^f(τ).