Universal meager $F_σ$-sets in locally compact manifolds

Abstract

In each manifold M modeled on a finite or infinite dimensional cube [0, 1] n , n ≤ ω, we construct a meager Fσ-subset X ⊂ M which is universal meager in the sense that for each meager subset A ⊂ M there is a homeomorphism h : M → M such that h(A) ⊂ X. We also prove that any two universal meager Fσ-sets in M are ambiently homeomorphic.