Presentation ranks on Polish spaces

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}(\omeg