Abstract: The notion of irreducibility in the conventional Zariski topology is too coarse for real algebraic sets. For example, the hyperbola xy=1 is an irreducible algebraic set which is not even connected in the Euclidean topology. Cartan umbrella is another irreducible algebraic set, which is connected, but decomposes into the union of a 'sheet' and a 'stick.' Inspired by Nash and his notion of 'sheets,' one might require then to distinguish, as called by Kurdyka, the 'rigid pieces' of a real algebraic set. We will review different approaches to defining irreducibility and irreducible components in real algebraic geometry, in which more 'good' functions than just polynomials should be considered. We may work with the ring of Nash functions, continuous rational functions, or, most notably, arc-analytic functions. Nash functions are able to detect two components for the hyperbola xy=1. Continuous rational functions are able to detect a sheet and a stick component for Cartan umbrella. But we find examples in which only the phenomenon called 'arc-symmetricity' can among the others realize a decomposition.

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