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

TL;DR

This paper proves that the set of purely 1-unrectifiable closed subsets in the Hilbert space  is a highly complex coanalytic-complete set within the space of all closed subsets.

Contribution

It establishes the coanalytic-completeness of the class of purely 1-unrectifiable sets in the Hilbert space, revealing its descriptive set-theoretic complexity.

Findings

The set of purely 1-unrectifiable sets is -complete in the projective hierarchy.

The space of all closed subsets of  is a Polish space.

The class of purely 1-unrectifiable sets has maximal complexity among coanalytic sets.

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.