In definition $3.11$, the author defined the following rank:
If the sets $A$ and $B$ can be separated by a transfinite difference of closed sets, then let $\alpha_1(A,B)$ denote the length of the shortest such sequence, otherwise let $\alpha_1(A,B)=\omega_1$.
Question: What is the definition of 'length of a sequence'? Does it mean the index $\theta$ where the sequence starts to stabilize, i.e. $F_\theta = F_{\theta+1}$
Remark: Let $(F_\eta)_{\eta < \lambda}$ be a decreasing sequence of sets. Assume that $F_0 = X$ (Polish space) and $F_\eta=\bigcap_{\theta<\eta}F_\theta$ for every limit $\eta$ and if $\lambda$ is a limit, then $\bigcap_{\eta<\lambda}F_\eta=\emptyset$. A set $H$ is the transfinite difference of $(F_\eta)_{\eta < \lambda}$ if $H = \bigcup_{\eta<\lambda, even \eta} {(F_\eta \backslash F_{\eta+1}})$.
The length of the sequence $(F_\eta)_{\eta < \lambda}$ is just $\lambda$.
Maybe it's nitpicky, but I'd prefer to call this a "$\lambda$-sequence" and reserve the word "sequence" for the case where $\lambda = \omega$ (the set of natural numbers).