Universal meager $F_\sigma$-sets in locally compact manifolds

TL;DR

The paper constructs a universal meager $F_\sigma$-set in manifolds modeled on cubes, which can contain any meager subset via a homeomorphism, and shows all such sets are ambiently homeomorphic.

Contribution

It introduces the concept of universal meager $F_\sigma$-sets in manifolds and proves their uniqueness up to ambient homeomorphism.

Findings

Existence of universal meager $F_\sigma$-sets in manifolds.

Any two such sets are ambiently homeomorphic.

Abstract

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