Totally bounded ultrametric spaces generated by labeled rays

Abstract

<p>We will say that an infinite tree <i>T</i> is almost a ray if <i>T</i> is the union of a ray and a finite tree. Let <i>l</i> be a non-degenerate labeling of the vertex set <i>V</i> of almost a ray <i>T</i> and let<i> d</i>_<i>l</i>be the corresponding ultrametric on <i>V</i>. It is shown that the ultrametric space <i>(V, d_l</i>) is totally bounded if this space contains an infinite totally bounded subspace. We also prove that the last property characterizes the almost rays.<br></p>