An incomplete real tree with complete segments (Q6581286)

Let \(\Lambda\) be a dense subgroup of \((\mathbb{R}, +)\) and let \(\mathcal{T}\) be a \(\Lambda\)-tree. The real tree associated to \(\mathcal{T}\) is defined as \(\mathcal{T}^{\mathrm{sc}}=\bigcup_{s \in S} \overline{s}\), where \(S\) is the set of segments in \(\mathcal{T}\) and \(\overline{s}\) is the metric completion of \(s\) in \(\mathcal{T}\).\N\NIn this paper the authors give an example of a \(\mathbb{Q}\)-tree \(\mathcal{T}\), as defined in [\textit{G. W. Brumfiel}, Contemp. Math. 74, 51--75 (1988; Zbl 0662.32022)] for which \(\mathcal{T}^{\mathrm{sc}}\) is not a complete metric space. In particular they prove that if \(\mathbb{F}\) is the field of real Puiseux series with \(\mathbb{Q}\)-valuation, and \(\mathcal{T}_{\mathbb{F}}\) is its associated \(\mathbb{Q}\)-tree, then \(\mathcal{T}^{\mathrm{sc}}_{\mathbb{F}}\) is not metrically complete.