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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 $$f_X:X_1\rightarrow X_2$$.

##### MSC:
 55M10 Dimension theory in algebraic topology 54F45 Dimension theory in general topology
##### Keywords:
$$Z$$-set; Nöbeling spaces
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