Analytic Non-Borel Sets and Vertices of Differentiable Curves in the Plane

Abstract

The purpose of this paper is to show that given any non-zero cardinal number $n \leq {\aleph }_{0}$, the set of differentiable paths of class $C^{2}$ and of unit length in the plane having their arc length as the parameter in $[0,1]$ and tracing curves which have at least $n$ vertices is analytic non-Borel, while for any $r \in ({\N } \cup \{ \infty \} ) \setminus \{ 0,1,2 \} $, the set of differentiable paths of class $C^{r}$ and of unit length in the plane having their arc length as the parameter in $[0,1]$ and tracing curves which have at least $n$ vertices is $F_{\sigma }$ if $n<{\aleph }_{0}$ and $F_{\sigma \delta }$ if $n={\aleph }_{0}$.