  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.
