Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that
$(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$.
Does it follow that $(G,\cdot,T)$ is homeomorphically isomorphic to and $(H,\star,S)$?
(H,\star,S)$ are isomorphic as topological groups?
If no, what if they are both Hausdorff? What if they are both Hausdorff and two-sided complete?
