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

TL;DR

This paper addresses open problems related to strong d-spaces and super H-sober spaces, providing new results on product spaces, reflectivity, and conditions for super H-sobriety, advancing the understanding of these topological concepts.

Contribution

It solves open problems about the product of strong d-spaces and the reflectivity of the category of strong d-spaces, and characterizes super H-sobriety in terms of H-sobriety and property Q.

Findings

Product of an arbitrary family of strong d-spaces remains strong d-spaces.

Finite products of hyper-sober spaces are hyper-sober.

Conditions under which a space is super H-sober are established.

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 $\mathbf{S}$-$\mathbf{Top}_{d}$ of strong $d$-spaces. Meantime, for an irreducible subset system ($R$-subset system for short) $\mathrm{H}$ and a $T_{0}$ space $X$, we prove that $\mathrm{H}$ naturally satisfies property $M$. Based on this, we deduce that $X$ is super $\mathrm{H}$-sober iff $X$ is $\mathrm{H}$-sober and $\mathrm{H}$ satisfies property $Q$. Additionally, we give positive answers to some questions posed by Xu in \cite{Xiaoquan21}. Furthermore, we obtain that the finite product of hyper-sober spaces is hyper-sober.