Compact localization and co-sobriety of Smyth power spaces
Abstract
We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a T0-space X is stably compact---in particular, if every such subspace is Hausdorff---then PSn(X) is co-sober for every finite n≥ 1. This applies to every k1-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each k-irreducible compact saturated family is generated by its union.
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BibTeX
@misc{ji2026compact,
title = {{Compact localization and co-sobriety of Smyth power spaces}},
author = {Wei Ji and Xiaoquan Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14446},
eprint = {2610.14446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14446},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact localization and co-sobriety of Smyth power spaces
AU - Wei Ji
AU - Xiaoquan Xu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14446
UR - https://arxiv.org/abs/2610.14446
AB - We establish a compact-localization theorem for finite iterates of the Smyth power space. Successive union maps assign to every finite-level Smyth object a canonical least compact saturated carrier in the original space, and co-sobriety at each positive finite level is equivalent to co-sobriety of the corresponding iterates over all compact saturated carriers. Consequently, if every compact saturated subspace of a T_0-space X is stably compact---in particular, if every such subspace is Hausdorff---then PS^n(X) is co-sober for every finite n≥ 1. This applies to every k_1-Hausdorff space and hence to every Hausdorff or metrizable space. For the first iterate, we also give a direct full-Vietoris argument, independent of stable compactness, which shows that each k-irreducible compact saturated family is generated by its union.
ER -