For any Polish space X it is well-known that the Cantor–Bendixson rank provides a co-analytic rank on F_ℵ0(X) if and only if X is σ-compact. In the case of ω^ω one may recover a co-analytic rank on F_ℵ0(ω^ω) by considering the Cantor–Bendixson rank of the induced trees instead. In this paper, we generalize this idea to arbitrary Polish spaces and thereby construct a family of co-analytic ranks on F_ℵ0(X) for any Polish space X. We study the behaviour of this family and compare the ranks to the Cantor–Bendixson rank. The main results are characterizations of the compact and σ-compact Polish spaces in terms of this behaviour.