Let $G$ be a topological group and its two-side uniformity $\mathcal{U}$ (that is the uniformity generated by right uniformity and left uniformity of $G$) coincides with the uniformity of a metric $d$ on $G$.
Evidently, the topological group is metrizable since the group topology is identical with the topology induced by $d$. Then the topological group $G$ equipped with two-side uniformity is isomorphic to (relatively to topological group structures) a dense subgroup of its completion $\hat{G}$ equipped with two side uniformity. I was wondering whether $\hat{G}$ is metrizable?
My attempt: Since metric $d$ is uniformly continuous on $G\times G$, we can uniquely extend this mapping to a uniformly continuous mapping $k:\hat{G}\times\hat{G}\rightarrow\mathbb{R}$. I proved $k$ is a pseudo-metric of $\hat{G}$ and the topology induced by $k$ is coarser than the topology of $\hat{G}$.
Any help is appreciated. Thanks.
(cited from Alexander V. Arhangel'skii, Mikhail G. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008).