2026/07/28 by Chong Shen, Xiaoyong Xi, Dongsheng Zhao
#math.GN
A poset model of a topological space X is a poset P such that X is homeomorphic to the maximal point space of P (the set Max(P) of all maximal points of P equipped with the relative Scott topology of P). Xi and Zhao proved that if a space has a dcpo model satisfying Lawson condition, it must be coherent and well-filtered. It is still open wether every coherent and well filtered space T1 has a dcpo model satisfying Lawson condition. In this paper, we answer this problem. In another paper, Xi and Zhao proved that every Hausdorff k-space has a bounded complete dcpo model. It is, however, still unknown whether it is true that if a Hausdorff space is a k-space if it has a bounded complete dcpo model. We will construct a Hausdorff space which is not a k-space but has a bounded complete dcpo model.