TITLE: Is $G/H\times H\simeq G$ topologically?
QUESTION [6 upvotes]: It is not true for $G/H\times H\simeq G$ to hold for a subgroup $H\leq G$ when we talk about group isomorphisms. 
However, what about topological groups and a homeomorphism in the above? (without being a group morphism)

REPLY [13 votes]: No.  Take $G=\mathbb{R}$ and $H = \mathbb{Z}$.  If you consider the quotient $G/H$ as the group version correctly topologized, then $\mathbb{R}/\mathbb{Z} \times \mathbb{Z}$ is not homeo to $\mathbb{R}$.  For one, the former isn't even connected (it's a stack of circles).
I'm also sure there are simpler examples than this one.