PreprintCC BY-NC-SA 4.0arXiv
Hamel-based TVG Stronger than the Usual Topology on R
Abstract
For every nonzero rational-valued additive functional on the real line, we construct a Hausdorff nondiscrete topological vector group topology strictly finer than the usual topology. We prove that two such topologies coincide exactly when their coefficient functionals differ by a nonzero real rescaling; every other pair is incomparable. The family contains as many distinct topologies as there are subsets of the real line. Their supremum is still nondiscrete and strictly finer than every individual member. We also prove failure of first countability and existence of maximal nondiscrete extensions.
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BibTeX
@misc{yaroshevskaya2026hamelbased,
title = {{Hamel-based TVG Stronger than the Usual Topology on R}},
author = {Irina Yaroshevskaya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24132},
eprint = {2610.24132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24132},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Hamel-based TVG Stronger than the Usual Topology on R
AU - Irina Yaroshevskaya
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24132
UR - https://arxiv.org/abs/2610.24132
AB - For every nonzero rational-valued additive functional on the real line, we construct a Hausdorff nondiscrete topological vector group topology strictly finer than the usual topology. We prove that two such topologies coincide exactly when their coefficient functionals differ by a nonzero real rescaling; every other pair is incomparable. The family contains as many distinct topologies as there are subsets of the real line. Their supremum is still nondiscrete and strictly finer than every individual member. We also prove failure of first countability and existence of maximal nondiscrete extensions.
ER -