The answers to some questions on H-sober spaces

Abstract

In this paper, we consider and solve several open problems posed by Xu in [11, 14]. Those open questions concern different categorical constructions of H-sober spaces and hyper-sober spaces.
First, for an irreducible subset system H and a T0 space X, we prove that H automatically satisfies property M, which was unknown before, hence we deduce that X is super H-sober iff X is H-sober and H satisfies property Q in the sense of [14]. Beyond the aforementioned work, many questions asked by Xu in [14] are also solved in the paper. Second, we derive the concrete forms of coequalizers in H-Sob. Finally, we obtain that the finite product of hyper-sober spaces is hyper-sober, which gives a positive answer to a question posed in [11].