Connected components in d-minimal structures

TL;DR

The paper addresses expanding a d-minimal structure by adjoining definable connected components while preserving tameness. It introduces the R^natural construction and proves it yields a d-minimal expansion (also applicable to almost o-minimal expansions) with a clear description of definable sets as finite unions of unions of connected components from R-definable sets. The core technical tool is a multi-cell decomposition for d-minimal structures, which is then leveraged to prove a generalized theorem characterizing R^natural-definable sets. The results clarify how connected components contribute to definability without destroying the underlying tame topology, offering a robust framework for analyzing expansions by connected components.

Abstract

For a given d-minimal expansion $\mathfrak R$ of the ordered real field, we consider the expansion $\mathfrak R^\natural$ of $\mathfrak R$ generated by the sets of the form $\bigcup_{S \in \mathcal C}S$, where $\mathcal C$ is a subfamily of the collection of connected components of an $\mathfrak R$-definable set. We prove that $\mathfrak R^{\natural}$ is d-minimal. A similar assertion holds for almost o-minimal expansions of ordered groups.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: Key Lemma
• proof
• proof : Proof of Claim 1
• proof : Proof of Claim 2