Cantor-Bendixson type ranks on Polish spaces

TL;DR

This paper generalizes the construction of co-analytic ranks on finite subsets of Polish spaces, extending the Cantor-Bendixson rank to arbitrary Polish spaces and characterizing compactness properties.

Contribution

It introduces a family of co-analytic ranks on finite subsets of any Polish space, generalizing previous results and providing characterizations of compactness and sigma-compactness.

Findings

Constructed a family of co-analytic ranks for all Polish spaces.

Compared these ranks to the classical Cantor-Bendixson rank.

Characterized compact and sigma-compact Polish spaces via rank behaviour.

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.