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\(C^ 1\)-homogeneous compacta in \(\mathbb{R}^ n\) are \(C^ 1\)-submanifolds of \(\mathbb{R}^ n\). (English) Zbl 0863.53004
A subset \(K\subset\mathbb{R}^n\) is called \(C^1\)-homogeneous if for every pair \(x,y\in K\) there are neighbourhoods \(O_x,O_y\subset\mathbb{R}^n\) of \(x\) and \(y\), respectively, and a \(C^1\)-diffeomorphism \(h:(O_x,O_x\cap K)\to(O_y,O_y\cap K)\). In this paper, \(C^1\)-homogeneous compacta in \(\mathbb{R}^n\) are characterized: Let \(K\) be a locally compact subset of \(\mathbb{R}^n\). Then \(K\) is \(C^1\)-homogeneous if and only if \(K\) is a \(C^1\)-submanifold of \(\mathbb{R}^n\).

MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
54F65 Topological characterizations of particular spaces
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
53C40 Global submanifolds
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
28A15 Abstract differentiation theory, differentiation of set functions
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