Semi-linear stars are contractible

TL;DR

This paper proves that in ordered vector spaces, every definable set can be decomposed into simply-connected parts, and bounded stars of cells are contractible, confirming a previous conjecture.

Contribution

It establishes that all definable sets in ordered vector spaces are unions of simply-connected parts and proves bounded stars are contractible, resolving a conjecture.

Findings

Every definable set is a finite union of relatively open, simply-connected definable subsets.

Stars of cells in a special linear decomposition are definably simply-connected.

Bounded stars are definably contractible.

Abstract

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The proof goes through the stronger statement that the star of a cell in a special linear decomposition of $X$ is definably simply-connected. In fact, if the star is bounded, then it is definably contractible.