Totally bounded ultrametric spaces generated by labeled rays - Dovgoshey, Oleksiy; Vito, Valentino (Editorial de la Universidad Politécnica de Valencia, 2025)
We will say that an infinite 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 if this space contains an infinite totally bounded subspace.  We also prove that the last property characterizes the almost rays.