TITLE: Does homeomorphic and isomorphic always imply homeomorphically isomorphic?
QUESTION [51 upvotes]: 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 $(G,\cdot,T)$ and $(H,\star,S)$ are isomorphic as topological groups?

If no, what if they are both Hausdorff?  What if they are both Hausdorff and two-sided complete?

REPLY [16 votes]: Any two $2$-dimensional rational vector spaces in $\mathbb R$ are isomorphic as groups and also order-isomorphic (I believe), hence homeomorphic when given the relative topology from $\mathbb R$. But any isomorphism as topological groups would have to be given by multiplication by a real number. So for example $\mathbb Q+\mathbb Q\pi$ and $\mathbb Q+\mathbb Q\sqrt 5$ are a counterexample.
EDIT: Or the same thing with $\mathbb Z$ instead of $\mathbb Q$.