Cantor-Bendixson type ranks

Speaker: Vibeke Quorning (KU)

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. We will 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 will also present characterizations of the compact and $\sigma$-compact Polish spaces in terms of the behaviour of this family and discuss some related questions.