Presentation ranks on Polish spaces
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Presentation ranks on Polish spaces. / Quorning, Vibeke.
I: Fundamenta Mathematicae, Bind 257, Nr. 2, 2022, s. 115-140.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Presentation ranks on Polish spaces
AU - Quorning, Vibeke
N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2022.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
KW - Cantor–Bendixson derivative
KW - Cantor–Bendixson rank
KW - co-analytic rank
KW - co-analytic set
KW - Effros Borel space
U2 - 10.4064/fm633-9-2021
DO - 10.4064/fm633-9-2021
M3 - Journal article
AN - SCOPUS:85149874171
VL - 257
SP - 115
EP - 140
JO - Fundamenta Mathematicae
JF - Fundamenta Mathematicae
SN - 0016-2736
IS - 2
ER -
ID: 343213219