Topological elementary equivalence of closed semi-algebraic sets in the real plane (Q2710595)

The authors investigate topological properties of subsets \(S\) of the real plane, expressed by first-order sentences in the language of the reals that are augmented with a binary relation symbol \(S\). Two sets are called topologically elementary equivalent if they have the same such first-order topological properties. The subsets of the real plane that are first-order definable in the structure of the reals as a binary relation over the reals are the semi-algebraic sets. The authors' main concern is with topologically closed semi-algebraic sets and the conditions under which such sets are topologically equivalent.