Preliminaries

Abstract

In this section, we give definitions of a Whitney stratification, a tube system, a vector field on a stratification, isomorphisms between Whitney stratifications, etc., and show their properties needed later, particularly I.1.13, with complete proofs. Some of our definitions are a little different from the usual ones, e.g. [G-al]. We modify the definitions to suit them to our purpose. We treat special topics unknown even to singularity specialists. We use them in Chapter III. For this, we need the method of integration of vector fields, which may contradict our philosophy. This is because the theorems in Chapter III are stated in a more general situation than X. The X-versions of the results of this section and Chapter III, except 1.1.6 and 1.1.7, can be proved without the method of integration. Note that the X-versions work in the C^r category, r a positive integer (see Chapter II).