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A real analytic set is one which can be locally written as the zero set of a finite number of real analytic functions. My question is whether the logarithmic spiral, given in polar coordinates by the equation $$r=e^{\theta},$$ a real analytic set?

I am guessing that it is not since, its Cartesian counterpart involves either logarithms or arctan functions and it might not be defined at $0$. However, I am not certain.

Similarly, I wondered if the Hawaiian Earring is a real analytic set? In that case, I am not aware of any analytical representation.

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  • $\begingroup$ Can you give a reference to "Hawaiian Earring" which is far from widely known. $\endgroup$
    – Jean Marie
    Sep 6, 2016 at 22:36
  • $\begingroup$ @JeanMarie en.wikipedia.org/wiki/Hawaiian_earring Look it up in Wikipedia. $\endgroup$
    – Crostul
    Sep 6, 2016 at 23:00

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The logarithmic spiral is an analytic set covered by $$\{\,(x,y)\in\Bbb R^2\mid x\ne 0, \ln(x^2+y^2)\notin 2\pi\Bbb Z+\pi, \tan (\tfrac12\ln(x^2+y^2))=\tfrac yx\,\}$$ and $$\{\,(x,y)\in\Bbb R^2\mid y\ne 0, \ln(x^2+y^2)\notin 2\pi\Bbb Z, \cot (\tfrac12\ln(x^2+y^2))=\tfrac xy\,\}$$

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