On some open problems concerning strong d-spaces and super H-sober spaces

2021/09/23 by Jin, Mengjie, Miao, Hualin, Li, Qingguo
#FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2109.11299

Abstract

In this paper, we highlight some open problems stated by Xu and Zhao. In particular, we focus on strong d-spaces and answer two open problems concerning strong d-spaces. One is about the product space of an arbitrary family of strong d-spaces. The other concerns the reflectivity of the category S-Top_d of strong d-spaces. Meantime, for an irreducible subset system (R-subset system for short) H and a T₀ space X, we prove that H naturally satisfies property M. Based on this, we deduce that X is super H-sober iff X is H-sober and H satisfies property Q. Additionally, we give positive answers to some questions posed by Xu in \citeXiaoquan21. Furthermore, we obtain that the finite product of hyper-sober spaces is hyper-sober.

Related

• On H-sober spaces and H-sobrifications of T₀ spaces
• On strong R-spaces
• Non-reflective categories of some kinds of weakly sober spaces
• On Function Spaces Related to H-sober Spaces
• Some open problems on well-filtered spaces and sober spaces