PreprintCC BY-NC-SA 4.0arXiv
A function space characterization of RB-domains
Abstract
We prove that a domain D is an RB-domain if and only if the function space [D→ E] is continuous for every domain E, thereby resolving a conjecture of Luan and Li. More precisely, for each domain D we construct an explicit algebraic test domain AD such that continuity of the single function space [D→ AD] already forces D to be an RB-domain. For countably based domains D, the test domain can be chosen as a fixed domain that is independent of D. The analogous continuity characterization of FS-domains is false: the closed-disk domain is an FS-domain, but its function space into a suitable pointed algebraic domain is not continuous.
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BibTeX
@misc{chen2026function,
title = {{A function space characterization of RB-domains}},
author = {Yuxu Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20187},
eprint = {2609.20187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20187},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A function space characterization of RB-domains
AU - Yuxu Chen
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20187
UR - https://arxiv.org/abs/2609.20187
AB - We prove that a domain D is an RB-domain if and only if the function space [D→ E] is continuous for every domain E, thereby resolving a conjecture of Luan and Li. More precisely, for each domain D we construct an explicit algebraic test domain A_D such that continuity of the single function space [D→ A_D] already forces D to be an RB-domain. For countably based domains D, the test domain can be chosen as a fixed domain that is independent of D. The analogous continuity characterization of FS-domains is false: the closed-disk domain is an FS-domain, but its function space into a suitable pointed algebraic domain is not continuous.
ER -