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

We investigate topological properties of subsets S of the real plane, expre ssed bq first-order logic sentences in the language of the reals augmented with a binary relation symbol for S.. Two sets are called topologically ele mentary equivalent if they have the same such first-order topological prope rties. The contribution of this paper is a natural and effective characteri zation of topological elementary equivalence of closed semi-algebraic sets.