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The duality theorems of M. Stone, H. de Vries and V. V. Fedorchuk show that the Boolean algebras with proximity-type relations on them can be regarded as an algebraic code of compact zero-dimensional (resp., compact) Hausdorff spaces. In the dissertation the duality theorems mentioned above are extended to the class of locally compact Hausdorff spaces; also, many duality theorems which are new even for the compact case are proved. It is demonstrated that even some relations on Boolean algebras which have a remote resemblance to proximity relations are connected with some classes of topological spaces. It is shown that the new duality theorems obtained in the dissertation have many applications in the field of General Topology; here below some of them are listed: 1) the notion of an lc-proximity is introduced and, using it, a description of the ordered set of all (up to equivalence) locally compact Hausdorff extensions of Tychonoff spaces is obtained; it is a purely proximity-type generalization of the Smirnov Compactification Theorem; 2) the maps between Tychonoff (resp., zero-dimensional) spaces which have some continuous extensions of a special type on arbitrary but fixed locally compact Hausdorff extensions of these spaces are characterized; the following kinds of extensions are regarded: open, quasi-open, skeletal, perfect, injective, surjective; with this, in particular, some results of Ponomarev, Taimanov, Poljakov, Banaschewski, Leader, Dwinger and Bezhanishvili are generalized; 3) a mathematical realization of the philosophical ideas of A. N. Whitehead and T.de Laguna about Euclidean spaces is obtained.
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