Inverses of Borel functions

A classical result from Lusin and Souslin tells us that should a Borel function f:X→Y be a bijection, then its inverse f−1:Y→X must also be a Borel function. Let bB={f:[0,1]−Q→[0,1]−Q:fis Borel measurable}. Here, we consider the sets bB′={f∈bB:f is one-to-one}, and bH={f∈bB′:f:[0,1]−Q→[0,1]−Qis a bijection}, each endowed with the metric d(f,g)=supx∈[0,1]−Q⁡|f(x)−g(x)|, and the latter also with the metric d⁎(f,g)=d(f,g)+d(f−1,g−1). Let α