On the Hyperbolizing metric spaces

Abstract

In this paper, we prove that the metric space $(Z\setminus M,u_Z)$ defined by Z.Ibragimov is asymptotically $P{T}_{-1}$ if the metric space $(Z,d)$ is $P{T}_0$, 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 $P{T}_{-1}$ without the assumption of $P{T}_0$ on the metric space $(Z,d)$.

arXiv commentThis paper has been withdrawn by the author due to a crucial sign error in equation 1