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

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 T 0 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 [12]. Furthermore, we obtain that the finite product of hyper-sober spaces is hyper-sober.