A De Vries-type Duality Theorem for Locally Compact Spaces -- III

TL;DR

This paper develops new duality theorems for subcategories of locally compact zero-dimensional spaces, extending classical results and characterizing morphisms via dual objects, with novel insights even in the compact case.

Contribution

It introduces new Stone-type duality theorems for specific subcategories of locally compact zero-dimensional spaces, generalizing existing dualities and characterizing morphisms through dual objects.

Findings

New duality theorems for subcategories of ZLC

Characterization of morphisms via dual objects

Descriptions of dual objects for various subsets of spaces

Abstract

In this paper we prove some new Stone-type duality theorems for some subcategories of the category $\ZLC$ of locally compact zero-dimensional Hausdorff spaces and continuous maps. These theorems are new even in the compact case. They concern the cofull subcategories $\SkeZLC$, $\QPZLC$, $\OZLC$ and $\OPZLC$ of the category $\ZLC$ determined, respectively, by the skeletal maps, by the quasi-open perfect maps, by the open maps and by the open perfect maps. In this way, the zero-dimensional analogues of Fedorchuk Duality Theorem and its generalization are obtained. Further, we characterize the injective and surjective morphisms of the category $\HLC$ of locally compact Hausdorff spaces and continuous maps, as well as of the category $\ZLC$, and of some their subcategories, by means of some properties of their dual morphisms. This generalizes some well-known results of M. Stone and de Vries. An analogous problem is investigated for the homeomorphic embeddings, dense embeddings, LCA-embeddings etc., and a generalization of a theorem of Fedorchuk is obtained. Finally, in analogue to some well-known results of M. Stone, the dual objects of the open, regular open, clopen, closed, regular closed etc. subsets of a space $X\in\card{\HLC}$ or $X\in\card{\ZLC}$ are described by means of the dual objects of $X$; some of these results (e.g., for regular closed sets) are new even in the compact case.