Logarithmic spiral is an embedding

Question

Consider the logarithmic spiral $\alpha(t) = e^{-t} (cos(t),sin(t))$.

When $\alpha:\mathbb{R} \to \alpha(\mathbb{R})$, I have shown that this is a bijective continuous mapping.

I would like to prove that it is an homeomorphism.

What results should I be using for this?

Thoughts

Perhaps this has to do with polar coordinates?

Answer 1

Take a sequnce $x_n=e^{-a_n}(\cos a_n, \sin a_n)$ convergent to some $x=e^{-a}(\cos a, \sin a)$. It follows that $\lVert x_n\rVert$ is convergent to $\lVert x\rVert$. But $\lVert x_n\rVert =e^{-a_n}$ and $\lVert x\rVert=e^{-a}$. So we get that $e^{-a_n}$ converges to $e^{-a}$ and so $a_n$ converges to $a$. This proves that the inverse is continuous.