Recent zbMATH articles in MSC 57M99https://zbmath.org/atom/cc/57M992023-11-13T18:48:18.785376ZWerkzeugTopological elementary equivalence of regular semi-algebraic sets in three-dimensional spacehttps://zbmath.org/1521.031142023-11-13T18:48:18.785376Z"Geerts, Floris"https://zbmath.org/authors/?q=ai:geerts.floris"Kuijpers, Bart"https://zbmath.org/authors/?q=ai:kuijpers.bart-h-mSummary: We consider semi-algebraic sets and properties of these sets that are expressible by sentences in first-order logic over the reals. We are interested in first-order properties that are invariant under topological transformations of the ambient space. Two semi-algebraic sets are called topologically elementarily equivalent if they cannot be distinguished by such topological first-order sentences. So far, only semi-algebraic sets in one and two-dimensional space have been considered in this context. Our contribution is a natural characterisation of topological elementary equivalence of regular closed semi-algebraic sets in three-dimensional space, extending a known characterisation for the two-dimensional case. Our characterisation is based on the local topological behaviour of semi-algebraic sets and the key observation that topologically elementarily equivalent sets can be transformed into each other by means of geometric transformations, each of them mapping a set to a first-order indistinguishable one.