Descriptive properties of families of autohomeomorphisms of the unit interval (Q924219)

Let \({\mathbb H}\subseteq C[0,1]\) denote the Polish space of all increasing autohomeomorphisms of \([0,1]\). An \(f\in {\mathbb H}\) is said to be strictly singular if \(f\) has no positive finite derivative at any point. Confirming a conjecture of \textit{S. Graf, R. D. Mauldin} and \textit{S. C. Williams} [``Random homeomorphisms'', Adv. Math. 60, 239--359 (1986; Zbl 0596.60005)], it is shown that SS\({\mathbb H}\), the set of all strictly singular autohomeomorphisms, is \(\Pi_1^1\)-complete. In addition, the following subsets of \({\mathbb H}\) are also shown to be \(\Pi_1^1\)-complete: SS\({\mathbb H}^+\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has no finite positive right-hand derivative at any point, SS\({\mathbb H}^-\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has no finite positive left-hand derivative at any point, \(\Delta_{<\infty}\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has finite Dini derivative at every point, \(\Delta_{>0}\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has positive Dini derivative at every point, and \(\Delta_{>0}\cap \Delta_{<\infty}\). These sets are also used to provide Borel-inseparable pairs of coanalytic sets.