Nonexistence of k-bounded sobrification
General Topology
2020-11-24 v1
Abstract
In this paper, we will focus on k-bounded sober spaces and show the existence of a T_0 space X not admitting any k-bounded sobrification. This strengthens a result of Zhao, Lu and Wang, who proved that the canonical k-bounded sobrification does not exist. Our work provides a complete solution to a question of Zhao and Ho, and shows that unlike Sob and BSob,the category KBSob of all k-bounded sober spaces is not a reflective subcategory of the category Top_0 of all T_0 spaces. Furthermore, we introduce the notion of qk-bounded sober spaces and prove that the category KBSob is a full reflective subcategory of the category QKBSob of all qk-bounded sober spaces and continuous mappings preserving existing irreducible suprema.
Cite
@article{arxiv.2011.11606,
title = {Nonexistence of k-bounded sobrification},
author = {Jing Lu and Kaiyun Wang and Guohua Wu and Bin Zhao},
journal= {arXiv preprint arXiv:2011.11606},
year = {2020}
}