On the Hyperbolizing metric spaces

In this paper, we prove that the metric space $(Z\setminus M,u_Z)$ defined by Z.Ibragimov is asymptotically $PT_{-1}$ if the metric space $(Z,d)$ is $PT_{0}$, where $M$ is a nonempty closed proper subset of $Z$. Secondly, based on the metric $u_Z$, we define a new kind of metric $k_{z}$ on the set $Z\setminus M$ and show that the new metric space $(Z\setminus M,k_{Z})$ is also asymptotically $PT_{-1}$ without the assumption of $PT_{0}$ on the metric space $(Z,d)$.