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A \(Z\)-set unknotting theorem for Nöbeling spaces. (English) Zbl 1163.55001
All spaces in this paper are separable metric spaces. A complete \(n\)-dimensional space \(X\) is said to be an \(n\)-dimensional Nöbeling space if the following conditions are satisfied:
(i) \(X\) is an absolute extensor in dimension \(n\);
(ii) For every map \(f:Y\rightarrow X\) from a complete metric space \(Y\) of dimension \(\leq n\) and for every open cover \(\mathcal {U}\) of \(X\) there is a closed embedding \(g:Y\rightarrow X\) such that for each \(y\in Y\) there is \(U \in \mathcal {U}\) that contains both \(f(y)\) and \(g(y)\).
The main result in this paper is the following:
Let \(X_1\) and \(X_2\) be \(n\)-dimensional Nöbeling spaces and let \(A_1\) and \(A_2\) be homeomorphic \(Z\)-sets in \(X_1\) and \(X_2\) such that \(X_1\setminus A_1\) and \(X_2\setminus A_2\) are homeomorphic to Nöbeling spaces modeled on triangulated manifolds. Then every homeomorphism \(f_A:A_1\rightarrow A_2\) extends to a homeomorphism \(f_X:X_1\rightarrow X_2\).

55M10 Dimension theory in algebraic topology
54F45 Dimension theory in general topology
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