Analytic non-Borel sets and vertices of differentiable curves in the plane. (English) Zbl 1019.51013

Sofronidis, Nikolaos Efstathiou

Real Anal. Exch. 26(2000-2001), No. 2, 735-748 (2001).

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 \(1\leq n\leq\aleph_0\).

Reviewer: Hans Havlicek (Wien)

MSC:

51L15	\(n\)-vertex theorems via direct methods
53A04	Curves in Euclidean and related spaces
54H05	Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
03E15	Descriptive set theory
58D99	Spaces and manifolds of mappings (including nonlinear versions of 46Exx)

Keywords:

analytic sets; plane curves; vertex theorems