On the Metrizability of TVS-Cone Metric Spaces

Shou Lin, Kedian Li, Ying Ge

Metric spaces are cone metric spaces, and cone metric spaces are TVS-cone metric spaces. We prove that TVS-cone metric spaces are paracompact. A metrization theorem of TVS-cone metric spaces is obtained by a purely topological tools. We obtaine that a homeomorphism $f$ of a compact space is expansive if and only if $f$ is TVS-cone expansive. In the end, for a TVS-cone metric topology, a concrete metric generating the topology is constructed.