3 problems

Let $(X,d)$ and $(Y,\rho)$ be ultrametric spaces, with completions $(\tilde{X},\tilde{d})$ and $(\tilde{Y},\tilde{\rho})$. Let $\bar{\mathbf{B}}_X^0$ and $\bar{\mathbf{B}}_Y^0$ den…

Let $(X,d)$ be an ultrametric space, and let $(\tilde{X},\tilde{d})$ be its completion. Write $\bar{\mathbf{B}}_X$ and $\bar{\mathbf{B}}_{\tilde{X}}$ for the balleans of closed bal…

Let $(X,d)$ be an infinite ultrametric space. A labeled ray is a ray equipped with a labeling that generates the indicated ultrametric space. A tree is starlike if it has exactly o…