Transfinite inductions producing coanalytic sets

@article{Vidnynszky2014TransfiniteIP,
  title={Transfinite inductions producing coanalytic sets},
  author={Zolt{\'a}n Vidny{\'a}nszky},
  journal={Fundamenta Mathematicae},
  year={2014},
  volume={224},
  pages={155-174}
}
A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set. 
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