Universal nowhere dense and meager sets in Menger manifolds (Q386185)

The authors construct universal nowhere dense and universal meager sets in \(\mu^n\)-manifolds, i.e., in manifolds modeled on Menger cubes. Such manifolds were characterized by \textit{M. Bestvina} [Mem. Am. Math. Soc. 380, 110 p. (1988; Zbl 0645.54029)]. Theorems 1.4 and 1.5 of the current paper provide us with such characterizations respectively of the Menger cube and Menger cube manifolds.NEWLINENEWLINEIn this work, a \(\mu^n\)-manifold is a paracompact space having a cover by open subsets homeomorphic to a Menger cube. Here are the main results. Let \(M\) be a Menger manifold.NEWLINENEWLINE\textbf{(1).} There exists a closed nowhere dense subset \(M_0\) of \(M\) 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 \(h:M\to M\) with \(h(A)\subset M_0\).NEWLINENEWLINE\textbf{(2).} There is a meager \(\mathrm{F}_\sigma\)-set \(\Sigma_0\subset M\) which is universal meager in the sense that for each meager subset \(B\subset M\), there is a homeomorphism \(h:M\to M\) with \(h(B)\subset\Sigma_0\).NEWLINENEWLINE\textbf{(3).} Any two universal meager \(\mathrm{F}_\sigma\)-sets in \(M\) are ambiently homeomorphic.