A De Vries-type Duality Theorem for Locally Compact Spaces – III

This paper is a continuation of the papers [5, 6] and also of [3, 4]. In it 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 [8] and its generalization (presented in [4]) 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 their subcategories discussed in [4, 5, 6], by means of some properties of their dual morphisms. This generalizes some well-known results of M. Stone [18] and de Vries [2]. An analogous problem is investigated for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc., and a generalization of a theorem of Fedorchuk [8, Theorem 6] is obtained. Finally, in analogue to some well-known results of M. Stone [18], the dual objects of the open, regular open, clopen, closed, regular closed etc. subsets of a space X ∈ |HLC| or X ∈ |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. MSC: primary 54D45, 18A40; secondary 54C10, 54E05.