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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?

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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.

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  • $\begingroup$ I'm trying to show that the inverse is continuous. So I'm taking a limit in the image of $\alpha$. What else can it be? Yes, I assume that the limit is of the same form. I'm interested only in such sequences, I don't care about others. $\endgroup$ – freakish Mar 10 '18 at 17:38

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