Nonexistence of k-bounded sobrification
Abstract
In this paper, we will focus on k-bounded sober spaces and show the existence of a T0 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 Top0 of all T0 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.
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