PreprintarXiv licencearXiv
The rank problem for metric spaces
Abstract
We construct a complete, second countable metric space with uncountable Scott rank. This answers a well-known question posed by Fokina, Friedman, K{ö}rwien and Nies, and shows that the known upper bound ω1 on the rank, proved by Doucha, is sharp. Further, a Baire category based variant of the rank of metric spaces is developed, and this is used to give new proofs of the known rank bounds based only on standard methods of descriptive set theory.
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BibTeX
@misc{tornquist2026rank,
title = {{The rank problem for metric spaces}},
author = {Asger Tornquist},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22351},
eprint = {2610.22351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22351},
}
RIS
TY - UNPB
TI - The rank problem for metric spaces
AU - Asger Tornquist
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22351
UR - https://arxiv.org/abs/2610.22351
AB - We construct a complete, second countable metric space with uncountable Scott rank. This answers a well-known question posed by Fokina, Friedman, K{ö}rwien and Nies, and shows that the known upper bound ω_1 on the rank, proved by Doucha, is sharp. Further, a Baire category based variant of the rank of metric spaces is developed, and this is used to give new proofs of the known rank bounds based only on standard methods of descriptive set theory.
ER -