On connected components of some globally semi-analytic sets

TL;DR

This paper identifies a class of global real analytic functions ensuring that semi-analytic sets defined by them have finitely many connected components, each also semi-analytic, advancing understanding of their topological structure.

Contribution

It introduces a specific class of functions, $ ext{ extbackslash mathcal{A}}$, that guarantees finiteness and semi-analyticity of connected components of certain global semi-analytic sets.

Findings

Semi-analytic sets in class $ ext{ extbackslash mathcal{A}}$ have finitely many connected components.

Each connected component of such sets is also semi-analytic and defined by $ ext{ extbackslash mathcal{A}}$.

The class $ ext{ extbackslash mathcal{A}}$ provides a structural framework for analyzing global semi-analytic sets.

Abstract

We isolate a class, say $\mathcal{A}$, of global real analytic functions such that, each global semi-analytic set defined by $\mathcal{A}$ has only finitely many connected components and each component is also a global semi-analytic set defined by $\mathcal{A}$.