An incomplete real tree with complete segments

TL;DR

This paper investigates the properties of a specific real tree constructed from Puiseux series, demonstrating that its segment completion does not yield a complete metric space, thus revealing limitations in the structure's completeness.

Contribution

It provides a counterexample showing that completing segments in a certain real tree does not guarantee metric completeness, highlighting a nuanced aspect of real tree structures.

Findings

Completing segments of the tree does not produce a complete space.

The constructed tree from Puiseux series is incomplete after segment completion.

The result challenges assumptions about the completeness of real trees derived from algebraic structures.

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.