To the theory of \(\mathbf{H} \)-sober spaces (Q6575420)

Sobriety is probably the most important and useful property of \(T_0\)-spaces. It has been used in the characterizations of spectral spaces and \(T_0\)-spaces that are determined by their open set lattices. With the development of domain theory and non-Hausdorff topology, two other properties also emerged as very useful and important properties for \(T_0\)-spaces: the property of being a \(d\)-space and that of well-filteredness. In [\textit{X. Xu}, Topology Appl. 289, Article ID 107548, 38 p. (2021; Zbl 1494.54015)], the reviewer provided a unified approach to \(d\)-spaces, sober spaces and well-filtered spaces, and developed a general framework for dealing with all these spaces.\N\NIn this paper, the authors contribute some results to the theory of H-sober spaces. They introduce the concepts of H-bases and H\(^*\)-bases of \(T_0\)-spaces for a general irreducible (subset) system H. Some basic properties of H-sober spaces are discussed. Using H-bases and H\(^*\)-bases, they give a characterization of H-sober spaces and a characterization of H-sobrifications of \(T_0\)-spaces.