PF-well-filtered spaces
Abstract
This paper solves two open problems concerning the well-filteredness of topological spaces. The first, due to Shen, Xi and Zhao\cite{shen2023further}, asks whether a space must be open well-filtered whenever its upper space is so; the answer is negative; the Hofmann-Lawson space provides a counterexample. The second, posed by Li, Yuan and Zhao\cite{li2020unified}, inquires whether PF-well-filteredness and PF- sobriety are equivalent under the assumption of local compactness; we answer it in the affirmative. Subsequently, we give characterizations of PF-sobriety and PF-well-filteredness for T0 spaces, respectively. On the basis of the characterizations, we obtain the following result: PF-sobriety and PF-well-filteredness coincide on the Rudin spaces, which extends the previously established equivalence for locally compact spaces, as every locally compact space is a Rudin space. Furthermore, we give a counterexample to show that for nonempty T0 topological spaces X and Y, if Y is a PF-well-filtered (PF-sober) space, then the function space consisting of all continuous maps from X to Y equipped with the Isbell topology may not be a PF-well-filtered (PF-sober) space even when X is a core-compact space. We also prove that if the function space comprising all continuous maps from nonempty X to Y endowed with the Isbell topology is PF-well-filtered (PF-sober), then Y is also PF-well-filtered (PF-sober) even though PF-sobriety and PF-well-filteredness are not preserved under retracts. Finally, we show that the reflective hull of the category of PF-well-filtered (PF-sober) spaces with continuous maps in TOP0 is TOP0.
Related
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BibTeX
@misc{chen2026pfwellfiltered,
title = {{$PF$-well-filtered spaces}},
author = {Ruoxi Chen and Qingguo Li and Longchun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21578},
eprint = {2610.21578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21578},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - PF-well-filtered spaces
AU - Ruoxi Chen
AU - Qingguo Li
AU - Longchun Wang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21578
UR - https://arxiv.org/abs/2610.21578
AB - This paper solves two open problems concerning the well-filteredness of topological spaces. The first, due to Shen, Xi and Zhao\cite{shen2023further}, asks whether a space must be open well-filtered whenever its upper space is so; the answer is negative; the Hofmann-Lawson space provides a counterexample. The second, posed by Li, Yuan and Zhao\cite{li2020unified}, inquires whether PF-well-filteredness and PF- sobriety are equivalent under the assumption of local compactness; we answer it in the affirmative. Subsequently, we give characterizations of PF-sobriety and PF-well-filteredness for T_0 spaces, respectively. On the basis of the characterizations, we obtain the following result: PF-sobriety and PF-well-filteredness coincide on the Rudin spaces, which extends the previously established equivalence for locally compact spaces, as every locally compact space is a Rudin space. Furthermore, we give a counterexample to show that for nonempty T_0 topological spaces X and Y, if Y is a PF-well-filtered (PF-sober) space, then the function space consisting of all continuous maps from X to Y equipped with the Isbell topology may not be a PF-well-filtered (PF-sober) space even when X is a core-compact space. We also prove that if the function space comprising all continuous maps from nonempty X to Y endowed with the Isbell topology is PF-well-filtered (PF-sober), then Y is also PF-well-filtered (PF-sober) even though PF-sobriety and PF-well-filteredness are not preserved under retracts. Finally, we show that the reflective hull of the category of PF-well-filtered (PF-sober) spaces with continuous maps in TOP_0 is TOP_0.
ER -