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 \(\Pi_1^1\)-complete.
"The Class of Purely Unrectifiable Sets in \(\ell_2\) is \(\Pi_1^1\)-complete." Real Anal. Exchange 39 (2) 323 - 334, 2013/2014.