On metric properties of stratified sets
Topological properties of Thom-Whitney stratified spaces have been much studied by several mathematicians. The first author [Lect. Notes Math. 1462, 42--62 (1991; Zbl 0733.58003)] showed that after replacing Whitney's condition (b) by a weaker condition, called (C)-regularity, one still obtains most of the topological properties of Whitney stratified spaces, e.g., existence of a Thom-Mather structure as an abstract stratified set defined by a system of control data, invariance by transverse intersection and continuous controlled extension of stratified vector fields. In the paper under review, the authors study the class of locally radial stratified spaces which are (C)-regular for standard control functions. A locally compact regular stratified space is locally homeomorphic to a cone on a regular stratified space. The volume is locally finite and in fact equivalent to that of a ball of the same dimension. Stokes' theorem for certain regular stratified spaces is proved. Finally, the weakly Whitney stratified spaces are described. This class is intermediate between the classical Whitney stratified spaces and locally radial (C)-regular spaces.
scientific article; zbMATH DE number 1945297