Cantor-Bendixson type ranks on Polish spaces

2018/06/08 by Quorning, Vibeke
#FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)

paper · doi:10.48550/arxiv.1806.03206

Abstract

For any Polish space X it is well-known that the Cantor-Bendixson rank provides a co-analytic rank on F_ℵ0(X) if and only if X is a σ-compact. In the case of ω^ω one may recover a co-analytic rank on F_ℵ0(ω^ω) 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_ℵ0(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 σ-compact Polish spaces in terms of this behaviour.

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