Some Generalizations of Fedorchuk Duality Theorem -- I

TL;DR

This paper extends Fedorchuk's duality theorem to four categories of locally compact Hausdorff spaces with various classes of morphisms, establishing new Stone-type dualities and equivalences for these categories.

Contribution

It generalizes Fedorchuk's duality theorem to multiple categories with different morphisms, including a duality for all compact Hausdorff spaces with open maps.

Findings

Proves duality theorems for four categories of locally compact Hausdorff spaces.

Establishes equivalence theorems for these categories.

Provides versions for subcategories of connected spaces.

Abstract

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.