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Homeomorphic neighborhoods in \(\mu^{n+1}\)-manifolds. (English) Zbl 0915.55005

This paper is devoted to the study of shape properties of \(Z\)-sets in manifolds modeled on the \((n+1)\)-dimensional universal Menger compactum (denoted by \(\mu^{n+1})\). The author uses his theory of proper \(n\)-shape, which is a generalization to locally compact spaces of a theory previously introduced by Chigogidze for compact spaces. The main result in the paper is the following \(\mu^{n+1}\)-manifold version of a result of R. B. Sher:
Let \(X\) and \(Y\) be \(Z\)-sets in \(\mu^{n+1}\)-manifolds \(M\) and \(N\) respectively, such that \(n\text{-Sh}_p(X)\leq n\text{-Sh}_p(Y)\). Then, for each neighborhood \(U\) of \(X\) in \(M\) and each neighborhood \(V\) of \(Y\) in \(N\), there exists an open neighborhood \(V'\) of \(Y\) such that for every \(\mu^{n+1}\)-manifold closed neighborhood \(S\) of \(Y\) in \(V'\cap V\), there exists a \(\mu^{n+1}\)-manifold closed neighborhood \(R\) of \(X\) in \(U\) which is homeomorphic to \(S\).
Another interesting result is also presented in the paper concerning the existence of special kinds of neighborhoods of \(Z\)-sets shape dominated by polyhedra.

MSC:

55P55 Shape theory
54C56 Shape theory in general topology
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