Semilinear stars are contractible
Volume 241 / 2018
Fundamenta Mathematicae 241 (2018), 291-312
MSC: Primary 03C64.
DOI: 10.4064/fm394-10-2017
Published online: 29 January 2018
Abstract
Let $ {\mathcal {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 of Edmundo et al. (2013). 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.
