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Ensembles semi-algébriques symétriques par arcs. (Semi-algebraic sets symmetric by arcs). (French) Zbl 0686.14027
The author defines and studies a natural and interesting class of semi- algebraic sets, that is, the class of semi-algebraic sets symmetric by arcs. A semi-algebraic set E in $${\mathbb{R}}^ n$$ is called symmetric by arcs if, for any analytic arc $$\gamma:\quad (-1,1)\to {\mathbb{R}}^ n,$$ $$\gamma$$ ((-1,0))$$\subset E$$ implies $$\gamma$$ ((-1,1))$$\subset E$$. Then, define AR-topology of $${\mathbb{R}}^ n$$ such that $$E\subset {\mathbb{R}}^ n$$ is closed if and only if E is a semi-algebraic set symmetric by arcs.
In this paper, for example, it is given a characterization of connected components of desingularization of an algebraic set, in the ward of AR- irreducible decomposition. - Further, the author defines the notion of arc-analytic mapping and he studies, for a semi-algebraic set X symmetric by arcs, the structure of the ring $$A_ a(X)$$ of functions $$X\to {\mathbb{R}}$$ semi-algebraic and arc-analytic.
Reviewer: G.Ishikawa

##### MSC:
 14Pxx Real algebraic and real-analytic geometry 58A07 Real-analytic and Nash manifolds 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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##### References:
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