Analytic non-Borel sets and vertices of differentiable curves in the plane (Q1852413)

Let \({\mathcal P}^{r}\) be the space formed by all differentiable paths of class \(C^r\) from \([0,1]\) to \({\mathbb R}^{2}\) such that the arc length is the parameter of the curve which is traced by the path. For any cardinal number \(n\), where \(1\leq n\leq\aleph_{0}\), and any number \(r\), where \(2\leq r\leq\infty\), the author considers the set of paths in \({\mathcal P}^r\) tracing curves which have at least \(n\) vertices. For \(r=2\) and arbitrary \(n\) this set is analytic non-Borel in \({\mathcal P}^2\). On the other hand, for \(2<r\leq\infty\), this set is \(F_\sigma\) (a countable union of closed sets) in \({\mathcal P}^r\), if \(n\) is finite, and \(F_{\sigma\delta}\) (a countable intersection of \(F_\sigma\)-sets) in \({\mathcal P}^r\), if \(n=\aleph_0\). The result is based upon the the set of continuous paths tracing curves having at least \(n\) tangents in a fixed direction. This set is analytic and non-Borel for \(1\leq n\leq\aleph_0\).