PreprintCC BY 4.0arXiv
The ball fixed point property for continuous functions on locally compact spaces
Abstract
Let K be a compact Hausdorff space and p∈ K. We prove that the Banach space C(K,p)={f∈ C(K,ℝ):f(p)=0} fails the ball fixed point property (BFPP) whenever p is non-isolated, answering a question of Avilés, Japón, Lennard, Martínez-Cervantes and Stawski. Consequently, C0(L) fails the BFPP for every noncompact locally compact Hausdorff space L. Together with Barroso's characterization for C(K), this yields complete topological classifications of the BFPP for C0(L) and C(K,p). We also resolve a second question of Avilés et al., showing that the BFPP of C(K) implies that of C(K,p) for every compact Hausdorff space K and every P-point p∈ K if and only if there are no measurable cardinals.
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BibTeX
@misc{dipaulo2026ball,
title = {{The ball fixed point property for continuous functions on locally compact spaces}},
author = {Lily DiPaulo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01486},
eprint = {2611.01486},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01486},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The ball fixed point property for continuous functions on locally compact spaces
AU - Lily DiPaulo
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01486
UR - https://arxiv.org/abs/2611.01486
AB - Let K be a compact Hausdorff space and p∈ K. We prove that the Banach space C(K,p)={f∈ C(K,ℝ):f(p)=0} fails the ball fixed point property (BFPP) whenever p is non-isolated, answering a question of Avilés, Japón, Lennard, Martínez-Cervantes and Stawski. Consequently, C_0(L) fails the BFPP for every noncompact locally compact Hausdorff space L. Together with Barroso's characterization for C(K), this yields complete topological classifications of the BFPP for C_0(L) and C(K,p). We also resolve a second question of Avilés et al., showing that the BFPP of C(K) implies that of C(K,p) for every compact Hausdorff space K and every P-point p∈ K if and only if there are no measurable cardinals.
ER -