Universal nowhere dense and meager sets in Menger manifolds

TL;DR

This paper constructs universal nowhere dense and meager sets in Menger manifolds, showing their properties and proving that all such sets are ambiently homeomorphic, advancing the understanding of universal sets in topological manifolds.

Contribution

It introduces the first constructions of universal nowhere dense and meager sets in Menger manifolds, and proves their ambient homeomorphism equivalence.

Findings

Existence of a universal nowhere dense set homeomorphic to the manifold.

Existence of a universal meager $F_\sigma$-set in the manifold.

All universal meager $F_\sigma$-sets are ambiently homeomorphic.

Abstract

In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_\sigma$-set $\Sigma_0\subset M$ which is universal meager in the sense that for each meager subset $B\subset M$ there is a homeomorphism $h$ of $M$ such that $h(B)\subset \Sigma_0$. Also we prove that any two universal meager $F_\sigma$-sets in $M$ are ambiently homeomorphic.