I. Preliminaries.- 1.1. Whitney stratifications.- 1.2. Subanalytic sets and semialgebraic sets.- 1.3. PL topology and C? triangulations.- II. X-Sets.- 11.1. X-sets.- 11.2. Triangulations of X-sets.- 11.3. Triangulations of X functions.- 11.4. Triangulations of semialgebraic and X0 sets and functions.- 11.5. Cr X-manifolds.- 11.6. X-triviality of X-maps.- 11.7. X-singularity theory.- III. Hauptvermutung For Polyhedra.- III.1. Certain conditions...