H-sober spaces and Hofmann-Mislove type theorems
Abstract
A T0-space is called ω-sober if every irreducible closed subset having a countable dense subset is the closure of a unique point. This is precisely H-sobriety for the countable irreducible subset system ℛω. We give two natural Hofmann--Mislove representations in the general H-sobriety framework: filters induced by H-sets of X characterize H-sobriety, whereas filters induced by H-sets of the Smyth power space characterize super H-sobriety. Their relation is expressed by property Q. A minimal closed set argument with a countable selection proves that ℛω has property Q. Consequently, X is ω-sober if and only if its Smyth power space is ω-sober, equivalently, every countable irreducible compact-family filter has a compact saturated representation. We also obtain equivalent recovery, compact-intersection and representation conditions for an enlarged class containing the proper Scott-open filters with countable open bases and the filters induced by countable irreducible subsets. A further equivalent formulation uses traces on countable subspaces. The filtered-family specializations recover well-filteredness and ω-well-filteredness. Explicit examples show that ω-sobriety strictly implies ω-well-filteredness, which strictly implies representation for countable open bases alone.
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BibTeX
@misc{xu2026hsober,
title = {{H-sober spaces and Hofmann-Mislove type theorems}},
author = {Xiaoquan Xu and Xinjie Huang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05214},
eprint = {2611.05214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05214},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - H-sober spaces and Hofmann-Mislove type theorems
AU - Xiaoquan Xu
AU - Xinjie Huang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05214
UR - https://arxiv.org/abs/2611.05214
AB - A T_0-space is called ω-sober if every irreducible closed subset having a countable dense subset is the closure of a unique point. This is precisely H-sobriety for the countable irreducible subset system ℛ^ω. We give two natural Hofmann--Mislove representations in the general H-sobriety framework: filters induced by H-sets of X characterize H-sobriety, whereas filters induced by H-sets of the Smyth power space characterize super H-sobriety. Their relation is expressed by property Q. A minimal closed set argument with a countable selection proves that ℛ^ω has property Q. Consequently, X is ω-sober if and only if its Smyth power space is ω-sober, equivalently, every countable irreducible compact-family filter has a compact saturated representation. We also obtain equivalent recovery, compact-intersection and representation conditions for an enlarged class containing the proper Scott-open filters with countable open bases and the filters induced by countable irreducible subsets. A further equivalent formulation uses traces on countable subspaces. The filtered-family specializations recover well-filteredness and ω-well-filteredness. Explicit examples show that ω-sobriety strictly implies ω-well-filteredness, which strictly implies representation for countable open bases alone.
ER -