Lecture 1: O-minimal structures

In Real Algebraic and Analytic Geometry the following traditional classes of sets and their geometries are considered: (1) The class of semialgebraic sets (Whitney-! Lojasiewicz, in the 50’s) (see [BCR]). (2) The class of semianalytic sets (! Lojasiewicz, in the 60’s) (see [L]). (3) The class of subanalytic sets (Gabrielov-Hironaka-Hardt and Krakovian school, in the 70’s) (see [BM], [Hi], [LZ]). These classes of sets have many nice properties. Semialgebraic and subanalytic sets form the so-called Tarski-type systems, that is the corresponding class is closed under boolean operators and under proper projections. In particular, these classes have the finiteness property: each set in these classes has locally only finite number of connected components and each of the components also belongs to the corresponding class.