Presentation ranks on Polish spaces
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For any Polish space X it is well-known that the Cantor–Bendixson rank provides a co-analytic rank on Fℵ0(X) if and only if X is σ-compact. In the case of ωω one may recover a co-analytic rank on Fℵ0(ωω) by considering the Cantor–Bendixson rank of the induced trees instead. In this paper, we generalize this idea to arbitrary Polish spaces and thereby construct a family of co-analytic ranks on Fℵ0(X) for any Polish space X. We study the behaviour of this family and compare the ranks to the Cantor–Bendixson rank. The main results are characterizations of the compact and σ-compact Polish spaces in terms of this behaviour.
Original language | English |
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Journal | Fundamenta Mathematicae |
Volume | 257 |
Issue number | 2 |
Pages (from-to) | 115-140 |
ISSN | 0016-2736 |
DOIs | |
Publication status | Published - 2022 |
Bibliographical note
Publisher Copyright:
© Instytut Matematyczny PAN, 2022.
- Cantor–Bendixson derivative, Cantor–Bendixson rank, co-analytic rank, co-analytic set, Effros Borel space
Research areas
ID: 343213219