局部隔离与闭性-适性之间的对偶性
Localic separation and the duality between closedness and fittedness
AI总结：
本文研究局部隔离公理体系，引入一个新的一阶分离性质以揭示强Hausdorff性质与$\mathcal{F}$-分离性之间的对偶关系，并完整厘清了各类$T_1$型公理间的蕴含关系，证明强Hausdorff性质不蕴含$\mathcal{F}$-分离性。
摘要：
存在若干局部隔离公理，它们大致类似于经典拓扑中的$T_1$公理。例如，除了众所周知的次适性与适性之外，还有Rosicky-Smarda的$T_1$-locales、完全无序locales，以及更范畴化地，最近引入的$\mathcal{F}$-分离locales（即具有适性对角线的locales）——这一性质严格弱于适性。最近已证明，强Hausdorff性质与$\mathcal{F}$-分离性在某种意义上是彼此对偶的。在本文中，我们提供了这种对偶性的更多实例——例如，我们引入了一个新的一阶分离性质，它与$\mathcal{F}$-分离性的关系，正如Johnstone-Sun-shu-Hao-Paseka-Smarda保守Hausdorff公理与强Hausdorff性质的关系一样，并且该性质可能具有独立的研究价值。利用这一性质，我们通过建立这些性质与文献中出现的其他$T_1$型公理之间所有可能的蕴含关系，理顺了该理论的遗留问题。特别地，我们证明了强Hausdorff性质并不蕴含$\mathcal{F}$-分离性，这一问题此前一直悬而未决，并显示出与其在拓扑空间范畴中的对应物之间的显著差异。
英文摘要：
There are a number of localic separation axioms which are roughly analogous to the $T_1$-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosicky-Smarda's $T_1$-locales, totally unordered locales and, more categorically, the recently introduced $\mathcal{F}$-separated locales (i.e., those with a fitted diagonal) - a property strictly weaker than fitness. It has recently been shown that the strong Hausdorff property and $\mathcal{F}$-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality - e.g., we introduce a new first-order separation property which is to $\mathcal{F}$-separatedness as the Johnstone-Sun-shu-Hao-Paseka-Smarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other $T_1$-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply $\mathcal{F}$-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces.
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