In 1962, H. de Vries proved a duality theorem for the category {\bf HC} of compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual category obtained by him differs from the set-theoretic one. Here we obtain a new category dual to the category {\bf HLC} of locally compact Hausdorff spaces and continuous maps for which the composition of the morphisms is a natural one but the morphisms are multi-valued maps.
In 1962, de Vries [2] proved a duality theorem for the category HC of compact Hausdorff spaces and continuous maps. This theorem was the first realization in a full extent of the ideas of the so-called region-based theory of space, although, as it seems, de Vries did not know of the existence of such a theory. The region-based theory of space is a kind of point-free geometry and can be considered as an alternative to the well known Euclidean point-based theory of space. Its main idea goes back to Whitehead [23] (see also [22]) and de Laguna [1] and is based on a certain criticism of the Euclidean approach to the geometry, where the points (as well as straight lines and planes) are taken as the basic primitive notions. A. N. Whitehead and T. de Laguna noticed that points, lines and planes are quite abstract entities which have not a separate existence in reality and proposed to put the theory of space on the base of some more realistic spatial entities. In Whitehead [23], the notion of region is taken as a primitive notion: it is an abstract analog of a spatial body; also some natural relations between regions are regarded. In [22] Whitehead considers only some mereological relations like "part-of" and "overlap", while in [23] he adopts from de Laguna [1] the relation of "contact" ("connectedness" in Whitehead's original terminology) as the only primitive relation between regions. In this way the region-based theory of space appeared as an extension of mereology -a philosophical discipline of "parts and wholes".
Let us note that neither A. N. Whitehead nor T. de Laguna presented their ideas in a detailed mathematical form. Their ideas attracted some mathematicians and mathematically oriented philosophers to present various versions of region-based theory of space at different levels of abstraction. Here we can mention A. Tarski [20], who rebuilt Euclidean geometry as an extension of mereology with the primitive notion of sphere. Remarkable is also Grzegorczyk’s paper [13]. Models of Grzegorczyk’s theory are complete Boolean algebras of regular closed sets of certain topological spaces equipped with the relation of separation which in fact is the complement of Whitehead’s contact relation. On the same line of abstraction is also the point-free topology [14].
Let us mention that Whitehead’s ideas of region-based theory of space flourished and in a sense were reinvented and applied in some areas of computer science: Qualitative Spatial Reasoning (QSR), knowledge representation, geographical information systems, formal ontologies in information systems, image processing, natural language semantics etc. The reason is that the language of region-based theory of space allows us to obtain a more simple description of some qualitative spatial features and properties of space bodies. One of the most popular among the community of QSR-researchers is the system of Region Connection Calculus (RCC) introduced by Randell, Cui and Cohn [18].
A celebrated duality for the category HC is the Gelfand Duality Theorem [9,10,11,12]. The de Vries Duality Theorem, however, is the first complete realization of the ideas of de Laguna [1] and Whitehead [23]: the models of the regions in de Vries’ theory are the regular closed sets of compact Hausdorff spaces (regarded with the well known Boolean structure on them) and the contact relation ρ between these sets is defined by
The composition of the morphisms of de Vries’ category DHC dual to the category HC differs from their set-theoretic composition. In 1973, V. V. Fedorchuk [8] noted that the complete DHC-morphisms (i.e., those DHC-morphisms which are complete Boolean homomorphisms) have a very simple description and, moreover, the DHC-composition of two such morphisms coincides with their set-theoretic composition. He considered the cofull subcategory (i.e. such a subcategory which has the same objects as the whole category) DQHC of the category DHC determined by the complete DHC-morphisms. He proved that the restriction of de Vries’ duality functor to it produces a duality between the category DQHC and the category QHC of compact Hausdorff spaces and quasi-open maps (a class of maps introduced by Mardešic and Papic in [16]).
It is natural to try to extend de Vries’ Duality Theorem to the category HLC of locally compact Hausdorff spaces and continuous maps. An important step in this direction was done by Roeper [19]. Being guided by the ideas of de Laguna [1] and Whitehead [23], he defined the notion of region-based topology which is now known as local contact algebra (briefly, LCA or LC-algebra) (see [5]), because the axioms which it satisfies almost coincide with the axioms of local proximities of Leader [15]. In his paper [19], Roeper proved the following theorem: there is a bijective correspondence between all (up to homeomorphism) locally compact Hausdorff spaces and all (up to isomorphism) complete LC-algebras. In [4], using Roeper’s theorem, the Fedorchuk Duality Theorem w
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