O-minimal Hauptvermutung for polyhedra II

Hilbert initiated the standpoint in foundations of mathematics. From this standpoint, we allow only a finite number of repetitions of elementary operations when we construct objects and morphisms. When we start from a subset of a Euclidean space. Then we assume that any element of the line has only a finite number of connected components. We call the set tame if the assumption is satisfied, and define a tame morphism in the same way. In this paper we will show that a tame topological manifold is carried by a tame homeomorphism to the interior of a compact piecewise linear manifolds possibly with boundary and such a piecewise linear manifold possibly with boundary is unique up to piecewise linear homeomorphisms in the sense that if two manifolds are such PL manifolds possibly with boundary then they are the same as piecewise linear manifolds. We modify this to Theorem 2 so that argument of model theory works, and we prove it. We also consider the differentiable case.Comment: 45 page

C1C^1-triangulations of semialgebraic sets

We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is $C^1$ differentiable. As an application, we give a straightforward definition of the integration $\int_X \omega$ over a compact semialgebraic subset $X$ of a differential form $\omega$ on an ambient algebraic manifold, that provides a significant simplification of the theory of semialgebraic singular chains and integrations. Our results hold over every (possibly non-archimedian) real closed field.Comment: 12 page