  Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type duality
theorems for the following four categories: all of them have as objects 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
all compact Hausdorff spaces 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 Hausdorff spaces are formulated as well.
