标题： 局部紧性并不总意味着空间性 显示英文标题

标题： Local compactness does not always imply spatiality

摘要： 在无点拓扑学中，一个众所周知的结果是每个局部紧致的框架都是空间的。 这个结果是否可以推广到MT-代数（麦金西-塔斯基代数）是一个开放问题。 我们通过构造一个局部紧致的清晰MT-代数但不是空间的反例来否定地解决这个问题。 我们还重新审视了Nöbeling在1950年代对无点拓扑学的被广泛忽视的方法。 我们证明了他的分离公理与MT-代数理论中的分离公理密切相关，值得注意的是不包括豪斯多夫性。 我们证明了Nöbeling的空间定理蕴含了众所周知的Isbell空间定理。 然后我们通过证明每个局部紧致的$T_{1/2}$-代数都是空间的，来推广Nöbeling的空间定理。 该证明利用了每个非平凡的$T_{1/2}$-代数都包含一个闭合原子的事实，我们证明这等价于选择公理。 显示英文摘要

摘要： It is a well-known result in pointfree topology that every locally compact frame is spatial. Whether this result extends to MT-algebras (McKinsey-Tarski algebras) was an open problem. We resolve it in the negative by constructing a locally compact sober MT-algebra which is not spatial. We also revisit N\"obeling's largely overlooked approach to pointfree topology from the 1950s. We show that his separation axioms are closely related to those in the theory of MT-algebras with the notable exception of Hausdorffness. We prove that N\"obeling's Spatiality Theorem implies the well-known Isbell Spatiality Theorem. We then generalize N\"obeling's Spatiality Theorem by proving that each locally compact $T_{1/2}$-algebra is spatial. The proof utilizes the fact that every nontrivial $T_{1/2}$-algebra contains a closed atom, which we show is equivalent to the axiom of choice.