Representing lattices by topologies has been studied to a great extent. In this paper, we prove that a complete lattice L is isomorphic to the lattice of open subsets of a P-space iff L is order generated by its countably prime elements. We establish the dual equivalence between the category of complete lattices order generated by their countably prime elements with morphisms preserving arbitrary sups and countable infs, and the category of countably sober P-spaces with morphisms of continuous maps. Finally, we show that the category of countably sober P-spaces is reflective in the category of P-spaces.

中文翻译：

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哪些分布格是 P 空间开集的格？

用拓扑表示晶格已经被大量研究。在本文中，我们证明了一个完全格 L 与一个 P 空间的开子集格同构，如果 L 是由其可数素数元素生成的阶。我们建立了由它们的可数素数元素生成的完全格阶的范畴与保留任意 sup 和可数 infs 的态射，以及具有连续映射的态射的可数清醒 P 空间的范畴之间的对偶等价。最后，我们证明可数清醒 P 空间的类别反映在 P 空间的类别中。