关于dcpo模型两个问题的答案
The answers to two problems on dcpo models
AI总结：
本文回答了两个关于dcpo模型的问题，一是凝聚且良滤的\(T_1\)空间是否有满足劳森条件的dcpo模型，二是有有界完备dcpo模型的豪斯多夫空间是否为\(k -\)空间，通过构造反例给出了否定答案。
AI中文摘要：
拓扑空间\(X\)的偏序集模型是一个偏序集\(P\)，使得\(X\)与\(P\)的最大点空间（\(P\)的所有最大点的集合\(\text{Max}(P)\)配备\(P\)的相对斯科特拓扑）同胚。奚和赵证明了若一个空间有满足劳森条件的dcpo模型，则它一定是凝聚且良滤的。每个凝聚且良滤的\(T_1\)空间是否有满足劳森条件的dcpo模型仍未解决，本文回答了这个问题。另外，奚和赵证明了每个豪斯多夫\(k -\)空间有有界完备dcpo模型，但有有界完备dcpo模型的豪斯多夫空间是否是\(k -\)空间未知，我们构造了一个非\(k -\)空间但有有界完备dcpo模型的豪斯多夫空间。
英文摘要：
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 $T_1$ 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.
