Some Generalizations of Fedorchuk Duality Theorem -- I

arXiv:0709.4495v1 [math.GN] 27 Sep 2007 Some Generalizations of Fedorchuk Duality Theorem – I Georgi Dimov∗ Dept. of Math. and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria Abstract Generalizing Duality Theorem of V. V. Fedorchuk [11], we prove Stone-type duality theorems for the following four categories: all of them have as objects the locally com- pact Hausdorffspaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In parti- cular, a Stone-type duality theorem for the category of all compact Hausdorffspaces and all open maps between them is proved. We also obtain equivalence theorems for these four categories. The versions of these theorems for the full subcategories of these categories having as objects all locally compact connected Hausdorffspaces are formulated as well. MSC: primary 54D45, 18A40; secondary 06E15, 54C10, 54E05. Keywords: Normal contact algebra; Local contact algebra; Compact spaces; Locally compact spaces; Skeletal maps; (Quasi-)Open perfect maps; Open maps; Perfect maps; Duality; Equiva- lence. Introduction According to the famous Stone Duality Theorem ([22]), the category of all zero- dimensional compact Hausdorffspaces and all continuous maps between them is dually equivalent to the category Bool of all Boolean algebras and all Boolean homomorphisms between them. In 1962, H. de Vries [4] introduced the notion of compingent Boolean algebra and proved that the category of all compact Hausdorff spaces and all continuous maps between them is dually equivalent to the category ∗This paper was supported by the project MI-1510/2007 “Applied Logics and Topological Struc- tures” of the Bulgarian Ministry of Education and Science. 1E-mail address: gdimov@fmi.uni-sofia.bg 1 of all complete compingent Boolean algebras and appropriate morphisms between them. Using de Vries’ Theorem, V. V. Fedorchuk [11] showed that the category SkeC of all compact Hausdorffspaces and all quasi-open maps between them is dually equivalent to the category DSkeC of all complete normal contact algebras and all complete Boolean homomorphisms between them satisfying one simple con- dition (see Theorem 2.13 below). The normal contact algebras (briefly, NCAs) are Boolean algebras with an additional relation, called contact relation. The axioms which this contact relation satisfies are very similar to the axioms of Efremoviˇc prox- imities ([10]). The notion of normal contact algebra was introduced by Fedorchuk [11] under the name Boolean δ-algebra as an equivalent expression of the notion of compingent Boolean algebra of de Vries. We call such algebras “normal contact alge- bras” because they form a subclass of the class of contact algebras introduced in [7]. In 1997, Roeper [20] defined the notion of region-based topology as one of the possible formalizations of the ideas of De Laguna [3] and Whitehead [24] for a region-based theory of space. Following [23, 7], the region-based topologies of Roeper appear here as local contact algebras (briefly, LCAs), because the axioms which they satisfy almost coincide with the axioms of local proximities of Leader [14]. In his paper [20], Roeper proved the following theorem: there is a bijective correspondence between all (up to homeomorphism) locally compact Hausdorffspaces and all (up to isomor- phism) complete LCAs. It generalizes the theorem of de Vries [4] that there exists a bijective correspondence between all (up to homeomorphism) compact Hausdorff spaces and all (up to isomorphism) complete NCAs. Using the results of Fedorchuk [11] and Roeper [20], we show here that the bijective correspondence established by Roeper can be extended to a duality between the category SkeLC of all locally compact Hausdorffspaces and all skeletal (in the sense of Mioduszewski and Rudolf [16]) continuous maps between them and the category DSkeLC of all complete LCAs and all complete Boolean homomorphisms between them satisfying two sim- ple axioms; this is done in Theorem 2.11 which generalizes the Fedorchuk Duality Theorem cited above. Further, we regard the non-full subcategory OpLC (resp., OpC) of the category SkeLC (resp., SkeC): its objects are all locally compact (resp., all compact) Hausdorffspaces and its morphisms are all open maps. We find the corresponding subcategory DOpLC (resp., DOpC) of the category DSkeLC (resp., DSkeC) which is dually equivalent to the category OpLC (resp., OpC) (see Theorem 2.17 and Theorem 2.19); as far as we know, even the compact case (i.e. the result about the category OpC) is new. The subcategories DSkePerLC and DOpPerLC of the category DSkeLC which are dually equivalent, respectively, to the categories SkePerLC and OpPerLC of all locally compact Hausdorffspaces and all quasi-open perfect maps (respectively, all open perfect maps) between them are found as well (see Theorem 2.15 and Theorem 2.21). The versions of all men- tioned above theorems for the full su

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