Complexity of distances: Theory of generalized analytic equivalence relations
The result's identifiers
Result code in 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>
Alternative codes found
RIV/00216208:11320/23:10475539
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Complexity of distances: Theory of generalized analytic equivalence relations
Original language description
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.
Czech name
—
Czech description
—
Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Mathematical Logic
ISSN
0219-0613
e-ISSN
1793-6691
Volume of the periodical
23
Issue of the periodical within the volume
1
Country of publishing house
SG - SINGAPORE
Number of pages
45
Pages from-to
2250014
UT code for WoS article
000860032900002
EID of the result in the Scopus database
2-s2.0-85130427537