Bornologies and Kuratowski-Painlevé Convergence to the Empty Set Revisited
Abstract
Let (X,τ) be a noncompact metrizable space and let ⟨ Tn⟩ be a sequence of nonempty closed sets converging to the empty set in the sense of Kuratowski--Painlevé. The subsets of X whose closures meet at most finitely many terms form a bornology ℬτ(⟨ Tn⟩). We give an internal criterion for inclusion between two such bornologies and recover each of them from a canonical free filter of closed sets. In fact, a nontrivial bornology is sequence-induced exactly when it has the enlargement property and its closed polar has countable character. Our principal result says that the bornology of relatively compact sets is sequence-induced precisely when X is locally compact and separable; in the locally compact case the character of its closed polar is the density of X. In the penultimate section, we make a clever use of the sequence Tn={n} in the real line to give a negative answer to a conjecture posed in 2001 about a coarsest topology making the associated bornology a nontrivial metric bornology. Finally, we end the paper with a more general result: a coarsest such supertopology exists if and only if the original topology already makes the induced bornology a nontrivial metric bornology, in which case the coarsest supertopology is the original topology itself.
Related
- Painlevé II connection problem revisited
- On Banach and Kuratowski Theorem and generalized strong sequences
- A 2+1-dimensional extended Dym equation: moving boundary problems solvable via Painlevé II symmetry reduction
- An Extended 2+1-Dimensional Gardner Equation: On Moving Boundary Problems Solvable via Ermakov-Painlevé II Symmetry Reduction
- Multivariable Painleve'-II equation: connection formulas for arbitrary system size
BibTeX
@misc{ryke2026bornologies,
title = {{Bornologies and Kuratowski-Painlevé Convergence to the Empty Set Revisited}},
author = {Basile Beyer de Ryke and Gerald Beer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35907},
eprint = {2610.35907},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35907},
}
RIS
TY - UNPB
TI - Bornologies and Kuratowski-Painlevé Convergence to the Empty Set Revisited
AU - Basile Beyer de Ryke
AU - Gerald Beer
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.35907
UR - https://arxiv.org/abs/2610.35907
AB - Let (X,τ) be a noncompact metrizable space and let ⟨ T_n⟩ be a sequence of nonempty closed sets converging to the empty set in the sense of Kuratowski--Painlevé. The subsets of X whose closures meet at most finitely many terms form a bornology ℬ_τ(⟨ T_n⟩). We give an internal criterion for inclusion between two such bornologies and recover each of them from a canonical free filter of closed sets. In fact, a nontrivial bornology is sequence-induced exactly when it has the enlargement property and its closed polar has countable character. Our principal result says that the bornology of relatively compact sets is sequence-induced precisely when X is locally compact and separable; in the locally compact case the character of its closed polar is the density of X. In the penultimate section, we make a clever use of the sequence T_n={n} in the real line to give a negative answer to a conjecture posed in 2001 about a coarsest topology making the associated bornology a nontrivial metric bornology. Finally, we end the paper with a more general result: a coarsest such supertopology exists if and only if the original topology already makes the induced bornology a nontrivial metric bornology, in which case the coarsest supertopology is the original topology itself.
ER -