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Generalizing de Vries' duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. IntroductionIn the first quarter of 20th century, de Laguna [4] and Whitehead [30] initiate a new theory of space which in nowadays is well developed and is known as region-based theory of space. It is an alternative to the classical point-based theory of space and concerns not only R 3 , as it was in the very beginning, but also some large classes of topological spaces. The main ideas of the region-based theory of space can be formulated as follows:• the notion of point is too abstract to be taken as a primitive notion of the theory of space; instead of points, some more realistic spatial entities have to be put as primitives on the basis of the theory of space-in de Laguna [4] they are called solids and Whitehead [30] calls them regions;• some basic relations (like part-of, overlap, contact (or connection, in Whitehead's terminology)) between the regions has to be considered;• the points must not be disregarded; they have to be defined by means of the regions and some of the basic relations between them;• an equivalence (in some sense) between the region-based approach and the point-based approach has to be obtained. Topology and its Applications 156 (2009) 728-746 729 The notion of regular closed (or regular open) subset of a topological space is considered as a standard point-based model of the notion of region; in this model, two regular closed sets (i.e., regions) are in contact iff they have a non-empty intersection.De Laguna and Whitehead do not present their ideas in the form of a rigorous mathematical theory. This is done later by some other authors. A good survey of the region-based approach to the theory of space is given by Gerla in [17]. Recent surveys in this field, pointing out its relations with Theoretical Computer Science and AI, can be found in [2].The ideas of de Laguna and Whitehead lead naturally to the following general programme:• define in topological terms those subsets of a topological space that correspond most closely to the idea of regions;• choose some (algebraic) structure which is inherent to the family of all regions of a topological space, fix some kind of morphisms between the obtained (algebraic) objects and build in this way a category A;• find a subcategory T of the catego...

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This paper is a continuation of [7], where a duality theorem for the category HLC of locally compact Hausdorff spaces and continuous maps is proved. In the present paper, we characterize the injective and surjective morphisms of the category HLC, as well as of its cofull subcategories determined by the open, by the skeletal or by the perfect maps, by means of some corresponding properties of their dual morphisms. This is in analogy with some well-known results of M. H. Stone [17] and generalizes some similar results of de Vries [4]. Such characterizations are also obtained for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc.; in particular a theorem of Fedorchuk [10, Theorem 6] is generalized. Again in analogy with some well-known results of M. H. Stone [17], the dual objects of the open, regular open, clopen, compact open, regular closed etc.subsets of a locally compact Hausdorff space X are directly described by means of the dual object of X; some of these results are new even in the compact case. An explicit description of the products of local contact algebras (= LC-algebras) in the category DHLC dual to the category HLC is given and a completion theory for LC-algebras is developed.