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A note on power spaces and function spaces of strongly well-filtered spaces
Abstract
Strong well-filteredness is a refinement of well-filteredness arising naturally in non-Hausdorff topology and domain theory. In this paper, we construct a countable locally compact strong R-space X such that neither its Smyth power space nor its Scott power space is a strong d-space. This example gives negative answers to the questions concerning the preservation of strong well-filteredness and the strong R-space property under power space constructions. Furthermore, for the two-point discrete space \two, the function space (\two,X) is not a strong d-space under any of the pointwise convergence, compact-open, Isbell, core-open or Scott topologies, where the Scott topology is induced by the pointwise order.
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BibTeX
@misc{yuan2026note,
title = {{A note on power spaces and function spaces of strongly well-filtered spaces}},
author = {Zhenzhu Yuan and Xiangrui Li and Bei Liu and Qinggguo Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02725},
eprint = {2610.02725},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02725},
}
RIS
TY - UNPB
TI - A note on power spaces and function spaces of strongly well-filtered spaces
AU - Zhenzhu Yuan
AU - Xiangrui Li
AU - Bei Liu
AU - Qinggguo Li
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02725
UR - https://arxiv.org/abs/2610.02725
AB - Strong well-filteredness is a refinement of well-filteredness arising naturally in non-Hausdorff topology and domain theory. In this paper, we construct a countable locally compact strong R-space X such that neither its Smyth power space nor its Scott power space is a strong d-space. This example gives negative answers to the questions concerning the preservation of strong well-filteredness and the strong R-space property under power space constructions. Furthermore, for the two-point discrete space \two, the function space (\two,X) is not a strong d-space under any of the pointwise convergence, compact-open, Isbell, core-open or Scott topologies, where the Scott topology is induced by the pointwise order.
ER -