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

Harvard

Quorning, V 2022, 'Presentation ranks on Polish spaces', Fundamenta Mathematicae, bind 257, nr. 2, s. 115-140. https://doi.org/10.4064/fm633-9-2021

APA

Quorning, V. (2022). Presentation ranks on Polish spaces. Fundamenta Mathematicae, 257(2), 115-140. https://doi.org/10.4064/fm633-9-2021

Vancouver

Quorning V. Presentation ranks on Polish spaces. Fundamenta Mathematicae. 2022;257(2):115-140. https://doi.org/10.4064/fm633-9-2021

Author

Quorning, Vibeke. / Presentation ranks on Polish spaces. I: Fundamenta Mathematicae. 2022 ; Bind 257, Nr. 2. s. 115-140.

Bibtex

@article{8f4752384d964653a499a94eb4c5461a,
title = "Presentation 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 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.",
keywords = "Cantor–Bendixson derivative, Cantor–Bendixson rank, co-analytic rank, co-analytic set, Effros Borel space",
author = "Vibeke Quorning",
note = "Publisher Copyright: {\textcopyright} Instytut Matematyczny PAN, 2022.",
year = "2022",
doi = "10.4064/fm633-9-2021",
language = "English",
volume = "257",
pages = "115--140",
journal = "Fundamenta Mathematicae",
issn = "0016-2736",
publisher = "Polska Akademia Nauk Instytut Matematyczny",
number = "2",

}

RIS

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