Cantor-Bendixson type 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 a $\sigma$-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 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 study the behaviour of this family and compare the ranks to the original Cantor-Bendixson rank. The main results are characterizations of the compact and $\sigma$-compact Polish spaces in terms of this behaviour.