I am confused by the terminology. $\displaystyle S^1$ is a circle, i.e., a one-dimensional curve, without the interior points. Now, how can

be easily bent upward into a hemi-sphere, which is a two-dimensional surface? In my opinion, relating a two-dimensional set, like $\displaystyle \mathbb{R}^2$, with one-dimensional set, like $\displaystyle \mathbb{R}$, is the main difficulty of this problem.

Also, I am confused by the notation $\displaystyle S^1=(-1,1)\times(-1,1)$. Doesn't $\displaystyle \times$ mean Cartesian product? Again, relating two-dimensional set $\displaystyle (-1,1)\times(-1,1)$ to one-dimensional $\displaystyle S^1$ is nontrivial.

I assume that by $\displaystyle S^1$ you do mean one-dimensional curve because you say that a projection from half of $\displaystyle S^1$ into $\displaystyle (-1,1)$ is a bijection.