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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
##### Keywords:
$$C^ 1$$-homogeneous; compact subset; $$C^ 1$$-submanifold
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##### References:
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