Topology induced bycone metric

Question

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them

Answer 1

On page 25 of this paper by Petko D. Proinov on the arXiv, it is proved that if $E$ is a solid vector space and $(X,d)$ is a cone metric space over $E$, then the open balls of $(X,d)$ form the basis of a topology of a metrisable topological space.

Answer 2

Petko D. Proinov's result is about tvs-cone metric spaces (which is a generalization of cone metric spaces) so that result holds for cone metric spaces as well. It means that there is a homeomorphism between a cone metric space and a metric space, hence cone metric spaces and metric spaces are the same from the topological view.