Let $ \sigma : \mathbb{R} \to \mathbb{R}^2$ that maps t to $(\frac{t}{1+t^4}, \frac{t}{1+t^2})$. Show that this curve is regular and injective, but it is not homeomorphic to $Im( \sigma )$.
I've found this exercise on 'curve e superfici', a book written by Marco Abate. I've already shown that is an injection and regular, but i'm not able to prove the last statement.
Thank you
The following figure shows ${\rm Im}(\sigma)$ when the domain of $\sigma$ is restricted to $[{-15},15]$. One can see that the full $B:={\rm Im}(\sigma)$ will be a figure eight with crossing point at $(0,0)$.
It is easy to verify that $\sigma: \>{\mathbb R}\to{\mathbb R}^2$ is a regular and injective immersion. It is claimed that the image set $B\subset{\mathbb R}^2$ is not homeomorphic to ${\mathbb R}$. In order to prove this it is not sufficient to look at $\sigma^{-1}$. This inverse is (maybe $\ldots$) not continuous at $(0,0)$. But there might be other maps doing the trick. Here is the way out: The set $B$ is closed and bounded in ${\mathbb R}^2$, hence compact. It therefore cannot be homeomorphic to the noncompact ${\mathbb R}$.