Semialgebraic metric spcaes and resolution of singularities of definable sets

TL;DR

This paper explores the embedding of definable metric spaces into Euclidean spaces, characterizes when their closures are compact with extended metrics, and resolves singularities of definable sets using blow-up techniques.

Contribution

It provides a criterion for embedding definable metric spaces into Euclidean spaces with compact closures and extends the resolution of singularities to definable sets.

Findings

Characterization of when a definable metric space can be embedded with a compact closure

Existence of isometric but not semialgebraically isometric spaces

Resolution of singularities for definable sets using blow-up methods

Abstract

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.