TITLE: G is homeomorphic to $G/H\times H$
QUESTION [6 upvotes]: Let G be a topological group and $H\le G$.
Let $\pi: G\to G/H$ be the canonical projection and a continuous $\sigma: G/H\to G$ such that $\pi \circ \sigma = Id$.
Prove that G is  homeomorphic to $G/H\times H$.
I am confused by this question because $\pi$ is a bijective morphism so we should rather have $G\cong G/H$.
Thank you for your help.
Edit
Thanks to your comments/answers I cleared my confusion. Now if I go back to the original question. Wy is $G$ homeomorphic to  $H\times G/H$.
Factorization theorem gives an isomorphism $G\cong H\times G/H$ because $H=\ker \pi$ and $G/H=\text{Image}(\pi)$.
I don't see how to build this homeomorphism by hand. 
Thanks again.

REPLY [5 votes]: Hint: Consider $f:G/H\times H\rightarrow G$ defined by $f(x,y)=\sigma(x)y$ and $h:G\rightarrow G/H\times H$ defined by $h(z)=(\pi(z), (\sigma(\pi(z))^{-1}z)$ and show that $f$ is the inverse of $h$.