PreprintCC BY 4.0arXiv
Function spaces with compact-open topology need not be sober
Abstract
We exhibit sober spaces X and Y such that the function space [X → Y]K of continuous maps, equipped with the compact-open topology, is not sober; the codomain can be chosen to be the Sierpiński space S. Our construction builds on Isbell's complete lattice M, whose Scott space ΣM is non-sober. Let H = Q(M) be the set of nonempty compact saturated subsets of ΣM, ordered by reverse inclusion. Since H is well known to be a spatial frame, the space X = pt(H) is sober with 𝒪(X) ≅ H. The key technical result is that sobriety of the space H would entail sobriety of the original Scott space ΣM. This contradicts Isbell's example, and the desired counterexample follows.
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BibTeX
@misc{liu2026function,
title = {{Function spaces with compact-open topology need not be sober}},
author = {Bei Liu and Qingguo Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36551},
eprint = {2610.36551},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36551},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Function spaces with compact-open topology need not be sober
AU - Bei Liu
AU - Qingguo Li
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36551
UR - https://arxiv.org/abs/2610.36551
AB - We exhibit sober spaces X and Y such that the function space [X → Y]_K of continuous maps, equipped with the compact-open topology, is not sober; the codomain can be chosen to be the Sierpiński space S. Our construction builds on Isbell's complete lattice M, whose Scott space ΣM is non-sober. Let H = Q(M) be the set of nonempty compact saturated subsets of ΣM, ordered by reverse inclusion. Since H is well known to be a spatial frame, the space X = pt(H) is sober with 𝒪(X) ≅ H. The key technical result is that sobriety of the space H would entail sobriety of the original Scott space ΣM. This contradicts Isbell's example, and the desired counterexample follows.
ER -