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Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type dualitytheorems for the following four categories: all of them have as objects thelocally compact Hausdorff spaces, and their morphisms are, respectively, thecontinuous skeletal maps, the quasi-open perfect maps, the open maps, the openperfect maps. In particular, a Stone-type duality theorem for the category ofall compact Hausdorff spaces and all open maps between them is proved. We alsoobtain equivalence theorems for these four categories. The versions of thesetheorems for the full subcategories of these categories having as objects alllocally compact connected Hausdorff spaces are formulated as well.

Some Generalizations of Fedorchuk Duality Theorem -- I