Not every countable complete distributive lattice is sober

Abstract The study of the sobriety Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that space every continuous directed complete poset (usually called domain) is sober. Johnstone constructed first whose non-sober. Soon after, Isbell gave lattice with non-sober space. Based on Isbell’s example, Xu, Xi, Zhao showed there even Heyting algebra Achim Jung then asked whether countable sober main aim this paper to answer Jung’s problem by constructing This modified obtain distributive In addition, we prove topology product $\Sigma P\times \Sigma Q$ coincides $P\times if set Id ( P ) Q all incremental ideals posets are both countable. this, it deduced space, P$ coherent well filtered. particular, L