The Class of Purely Unrectifiable Sets in the Hilbert Space is coanalytic-complete

Abstract

The space $F({\ell }_2)$ of all closed subsets of ${\ell }_2$ is a Polish space. We show that the subset $P\subset F({\ell }_2)$ consisting of the purely 1-unrectifiable sets is $\Pii$-complete.

10 pages