Submission date: 28 December 2020

Abstract:

Let R be any real closed field expanded by some o-minimal structure. Let f :A → R^d be a definable and continuous mapping defined on a definable, closed, bounded subset A of R^n. Let E be a finite family of definable subsets of R^n contained in A. Let p be any positive integer. We prove that then there exists a finite simplicial complex T in R^n and a definable homeomorphism h : |T | → A, where |T| := UT , such that for each simplex S from T , the restriction of h to its relative interior ri(S) is a C^p-embedding of ri(S) into R^n and moreover both h and fh are of class C^p in the sense that they have definable C^p-extensions defined on an open definable neighborhood of |T | in R^n. Then we call a pair (T , h) a strict C^p-triangulation of A. In addition, this triangulation can be made compatible with E in the sense that for each B belonging to E, h^{-1} (B) is a union of some ri(S), where S are from T. We also give an application to approximation theory.

Mathematics Subject Classification: Primary 32B25. Secondary 32S45, 03C64, 14P10, 32B20, 57R05.

Keywords and phrases: o-minimal structure, semialgebraic set, C^p-triangulation, strict C^p-triangulation, capsule, detector.

Full text, revised version, 15 March 2021: pdf. (Versions of 28 December 2020 and 28 January 2021).