Complexity of distances: Theory of generalized analytic equivalence relations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F23%3A00570779" target="_blank" >RIV/67985840:_____/23:00570779 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11320/23:10475539

Výsledek na webu

<a href="https://doi.org/10.1142/S0219061322500143" target="_blank" >https://doi.org/10.1142/S0219061322500143</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1142/S0219061322500143" target="_blank" >10.1142/S0219061322500143</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Complexity of distances: Theory of generalized analytic equivalence relations

Popis výsledku v původním jazyce

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov-Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that E1 is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.

Název v anglickém jazyce

Complexity of distances: Theory of generalized analytic equivalence relations

Popis výsledku anglicky

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov-Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that E1 is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Mathematical Logic

ISSN

0219-0613

e-ISSN

1793-6691

Svazek periodika

23

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

SG - Singapurská republika

Počet stran výsledku

45

Strana od-do

2250014

Kód UT WoS článku

000860032900002

EID výsledku v databázi Scopus

2-s2.0-85130427537