One of the main difficulties in establishing that a given system is o-minimal is that one has to check that a certain finiteness condition (namely the finiteness of the number of connected components) holds for sets described by arbitrary formulas of a first-order logical language. In this lecture I shall sketch a method that reduces this task to a mathematically tractable one. As a consequence one obtains what I believe to be a natural generalization of the notion of subanalytic set to the smooth context.