An incomplete real tree with complete segments

Abstract

Let $\mathbb{F}$ be the field of real Puiseux series and ${\mathcal{T}}_{\mathbb{F}}$ the $\mathbb{Q}$-tree defined by Brumfiel. We show that completing all the segments of ${\mathcal{T}}_{\mathbb{F}}$ does not result in a complete metric space.

9 pages, 4 figures, comments are welcome!