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This is the problem from Curves and Surfaces(Montiel, Ros). The hint suggests the surface $S$ parametrized as

$$X(u,v) = (e^{-u}cos(u), e^{-u}sin(u), v)$$

to be the answer. Though I understand that the surface is not a closed subset of $\mathbb{R}^3$, I don't understand why $S$ does not have a larger connected surface that contains $S$. I guess it should have something to do with the limit points of $S$, but I am not perfectly sure. Could anybody help me with this problem?

MMH..
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  • Welcome to Math.SE! <> To start, are you happy with the picture of the logarithmic spiral $r = e^{-\theta}$ in polar coordinates, and do you see how this curve is not part of a larger connected curve? – Andrew D. Hwang Apr 02 '22 at 14:14
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    I actually came up with a solution, using the fact that the surface that is containing $S$ should have a contain a limit point $p$ of $S$, which leads to a contradiction because there is no inverse projection as a homeomorphism that covers p. – MMH.. Apr 02 '22 at 17:31
  • Btw, I appreciate your help! – MMH.. Apr 02 '22 at 17:32

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