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PreprintCC BY 4.0arXiv

A Necessary Condition for F-Spaces to be Separable

  • Hector Salas
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Abstract

A topological vector space E has the countable intersection property for subspaces if every family of closed subspaces intersecting to $0$ contains a countable subfamily that also intersects to $0$. We prove that every separable F-space satisfies this property, whereas a TVS with an uncountable Markushevich basis does not. Finally, we construct a separable TVS that fails to satisfy this property.

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Cite

BibTeXRIS
BibTeX
@misc{salas2026necessary,
  title = {{A Necessary Condition for F-Spaces to be Separable}},
  author = {Hector Salas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24065},
  eprint = {2609.24065},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24065},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Necessary Condition for F-Spaces to be Separable
AU  - Hector Salas
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24065
UR  - https://arxiv.org/abs/2609.24065
AB  - A topological vector space E has the countable intersection property for subspaces if every family of closed subspaces intersecting to $0$ contains a countable subfamily that also intersects to $0$. We prove that every separable F-space satisfies this property, whereas a TVS with an uncountable Markushevich basis does not. Finally, we construct a separable TVS that fails to satisfy this property.
ER  -