Abstract A nonempty compact saturated subset F of a topological space is called k-irreducible if it cannot be written as a union of two compact saturated proper subsets. A topological space is said to be co-sober if each of its k-irreducible compact saturated sets is the saturation of a point. Wen and Xu (2018) proved that Isbell's non-sober complete lattice equipped with the lower topology is sober but not co-sober. So far, it is still unknown whether every sober Scott space is co-sober. In this paper, we construct a dcpo whose Scott space is sober but not co-sober, which strengthens Wen and Xu's result.

中文翻译：

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清醒的斯科特空间并不总是清醒的

摘要 拓扑空间的一个非空紧饱和子集F如果不能写成两个紧饱和真子集的并集，则称其为k-不可约的。如果拓扑空间的每个 k 不可约紧凑饱和集都是一个点的饱和，则称该拓扑空间是共清醒的。Wen and Xu (2018) 证明 Isbell 的非清醒完全格配备了较低的拓扑结构是清醒的，但不是共清醒的。到目前为止，是否每个清醒的 Scott 空间都是共同清醒的仍是未知数。在本文中，我们构造了一个 dcpo，其 Scott 空间是清醒但不是共同清醒的，这加强了 Wen 和 Xu 的结果。