Triangulations of X-Maps

Abstract

Let X ⊂ Rⁿ and Y ⊂ R^m be locally closed X-sets and let f : X → Y be an X-map. A C⁰ X-triangulation of f is a quadruplet of X-polyhedra X₀ ⊂ Rⁿ’ and Y₀ ⊂ R^m’ and X-homeomorphisms π : X₀ → X and τ : Y₀ → Y such that τ^-1 o f o π is PL. We call a C⁰ X-triangulation (X₀, X₀, π, τ) a C⁰ R-X-triangulation if Y is a polyhedron, Y₀ = Y and τ = id. In connection with the preceeding chapters, it may be natural to treat an (R-)X-triangulation of f which we define assuming, in addition, that π and τ are of class C^r, r > 0, on each simplex of some simplicial decompositions of X₀ and Y₀ respectively. However, I cannot prove the results of this chapter in the terms of an (R-)X-triangulation, and I think that a C⁰ (R-) X-triangulation is more natural than an (R-) Xtriangulation in itself. We define naturally a C⁰ X-stratification of an X-set in a Euclidean space. Here note that the stratification is finite locally at each point of the Euclidean space and each stratum is not only an X-set and a C⁰ manifold but also a C⁰ X-submanifold of the Euclidean space (i.e., locally X-homeomorphic to a Euclidean space).