# Is $G/H\times H\simeq G$ topologically?

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)

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).
• If $H$ is nontrivial and discrete, you're going to have problems with the number of connected components. – Randall Apr 23 '18 at 13:35