2017/01/14 by Masahiro Shiota, Shiota, Masahiro
Mathematics · #03C64 #14B12 #54E45 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1701.03858
openalex publication_date 2017/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that its closure is compact and the metric on the image induced by d is extended to a definable metric on the closure if and only if the limit of d(r(t);r(t)) is 0 as t converges to 0 for any definable continuous curve r from (0, 1] to X (Theorem 1). We also find two compact semialgebraic metric spaces over the real field which are isometric but not semialgebraically isometric (Theorem 2). A version of blow up is the key to the proof of Theorem 1. Using it in the same way, we prove a resolution of singularities of definable sets (Theorem 3). We prove the theorems by a constructive procedure.