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.
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 $$\{\,(x,y)\in\Bbb R^2\mid y\ne 0, \ln(x^2+y^2)\notin 2\pi\Bbb Z, \cot (\tfrac12\ln(x^2+y^2))=\tfrac xy\,\}$$