局部紧空间的De Vries型对偶定理-- III
A De Vries-type Duality Theorem for Locally Compact Spaces -- III
摘要：
本文证明了局部紧零维Hausdorff空间和连续映射范畴$\ZLC$的某些子范畴的Stone型对偶定理。这些定理是新的，即使在紧凑的情况下。它们涉及范畴$\ZLC$的共满子范畴$\SkeZLC$、$\QPZLC$、$\OZLC$和$\OPZLC$，它们分别由骨架映射、准开完美映射、开映射和开完美映射确定。这样，就得到了Fedorchuk对偶定理及其推广的零维类似物。进一步，我们利用局部紧Hausdorff空间和连续映射范畴$\HLC$的对偶态射的性质，刻画了它们的内射和满射态射，以及$\ZLC$范畴和它们的一些子范畴的内射和满射态射.这推广了M.斯通和德弗里斯。研究了同胚嵌入、稠密嵌入、LCA-嵌入等的类似问题，并推广了Fedorchuk的一个定理。最后，与M. Stone，空间$X\in\card{\HLC}$或$X\in\card{\ZLC}$的开、正则开、闭、正则闭等子集的对偶对象是借助于$X$的对偶对象来描述的;这些结果中的一些（例如，对于正则闭集）是新的，即使在紧凑的情况下。
英文摘要：
In this paper we prove some new Stone-type duality theorems for some subcategories of the category $\ZLC$ of locally compact zero-dimensional Hausdorff spaces and continuous maps. These theorems are new even in the compact case. They concern the cofull subcategories $\SkeZLC$, $\QPZLC$, $\OZLC$ and $\OPZLC$ of the category $\ZLC$ determined, respectively, by the skeletal maps, by the quasi-open perfect maps, by the open maps and by the open perfect maps. In this way, the zero-dimensional analogues of Fedorchuk Duality Theorem and its generalization are obtained. Further, we characterize the injective and surjective morphisms of the category $\HLC$ of locally compact Hausdorff spaces and continuous maps, as well as of the category $\ZLC$, and of some their subcategories, by means of some properties of their dual morphisms. This generalizes some well-known results of M. Stone and de Vries. An analogous problem is investigated for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc., and a generalization of a theorem of Fedorchuk is obtained. Finally, in analogue to some well-known results of M. Stone, the dual objects of the open, regular open, clopen, closed, regular closed etc. subsets of a space $X\in\card{\HLC}$ or $X\in\card{\ZLC}$ are described by means of the dual objects of $X$; some of these results (e.g., for regular closed sets) are new even in the compact case.
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