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We will say that a tree T is almost a ray if T is the union of a ray and a finite tree. Let l be a non-degenerate labeling of the vertex set V of almost a ray T and let dₗ be the corresponding ultrametric on V. It is shown that the ultrametric space (V, dₗ) is totally bounded iff this space contains an infinite totally bounded subspace. We also prove that the last property characterizes the almost rays.
Dovgoshey et al. (Sat,) studied this question.
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