London Mathematical Society Lecture Note Series. 248. Cambridge: Cambridge University Press. x, 180 p. £24.95; $ 39.95 (1998).

This book gives a beautiful introduction to aspects of o-minimality, in the spirit of Grothendieck’s ‘tame topology’. Although, as the author comments, the subject was developed in close contact with model theory, no model-theoretic background is needed, and many of the methods come from real algebraic geometry. Much of the material is not previously published. The book begins with a definition of o-minimality, a proof that o-minimal ordered groups and ordered fields are divisible abelian and real closed respectively, and a proof that the real field

$(\mathbb{R},<,+,\xb7)$ is o-minimal (via the Tarski -Seidenberg Theorem which is proved via a cell decomposition). The o-minimal Monotonicity and Cell Decomposition Theorems (developed in papers of Pillay and Steinhorn, and one also with Knight) are then proved in Ch. 3. Dimension and Euler characteristic and their basic properties are introduced in the next chapter. In Ch. 5 the author shows that in an o-minimal structure any definable family of definable sets is a Vapnik-Cervonenkis class (a notion from probability theory, relevant also to neural networks). This is equivalent to the fact that o-minimal structures do not have the independence property. In the next two chapters some basic point set topology is developed, followed (for o-minimal expansions of fields) by some theory of differentiation: a Mean Value Theorem, an Implicit Function Theorem, and a Cell Decomposition with

${C}^{1}$-cells and maps. A Triangulation Theorem is proved in Ch. 8, via a Good Directions Lemma. This leads to a proof that in an o-minimal expansion of an ordered field, two definable sets have the same dimension and Euler characteristic if and only if there is a definable bijection between them. Under the same assumptions, a Trivialisation Theorem is proved in Section 9. It follows that given a definable family of definable sets, the sets fall into finitely many embedded definable homeomorphism types. This and Wilkie’s proof of the o-minimality of the reals with exponentiation are applied to prove a conjecture of Benedetti and Risler: roughly speaking, if we consider semialgebraic subsets of

${\mathbb{R}}^{n}$ defined by a bounded number of polynomial equalities and inequalities, and the polynomials are built from monomials by a bounded number of additions, then the semialgebraic sets fall into finitely many embedded homeomorphism types. Finally, in Ch. 10 the author moves from definable sets to definable spaces, given by an atlas of charts, and constructs definable quotients. The book is an elegant and lucid account, well-suited to a beginning graduate student, with a number of exercises. No attempt is made to cover recent material on o-minimality, for example on o-minimal expansions of the reals, or on the Trichotomy Theorem of Peterzil and Starchenko and its applications to definable groups.