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Abstract

For any Polish space $X$ it is well-known that the Cantor–Bendixson rank provides a co-analytic rank on $F_{\aleph _0}(X)$ if and only if $X$ is $\sigma $-compact. In the case of $\omega ^\omega $ one may recover a co-analytic rank on $F_{\aleph _0}(\omega ^\omega )$ 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 $F_{\aleph _0}(X)$ 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 $\sigma $-compact Polish spaces in terms of this behaviour.