Some generalizations of Fedorchuk duality theorem—I

Abstract

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are 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 compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. IntroductionIn the first quarter of 20th century, de Laguna [4] and Whitehead [30] initiate a new theory of space which in nowadays is well developed and is known as region-based theory of space. It is an alternative to the classical point-based theory of space and concerns not only R 3 , as it was in the very beginning, but also some large classes of topological spaces. The main ideas of the region-based theory of space can be formulated as follows:• the notion of point is too abstract to be taken as a primitive notion of the theory of space; instead of points, some more realistic spatial entities have to be put as primitives on the basis of the theory of space-in de Laguna [4] they are called solids and Whitehead [30] calls them regions;• some basic relations (like part-of, overlap, contact (or connection, in Whitehead's terminology)) between the regions has to be considered;• the points must not be disregarded; they have to be defined by means of the regions and some of the basic relations between them;• an equivalence (in some sense) between the region-based approach and the point-based approach has to be obtained. Topology and its Applications 156 (2009) 728-746 729 The notion of regular closed (or regular open) subset of a topological space is considered as a standard point-based model of the notion of region; in this model, two regular closed sets (i.e., regions) are in contact iff they have a non-empty intersection.De Laguna and Whitehead do not present their ideas in the form of a rigorous mathematical theory. This is done later by some other authors. A good survey of the region-based approach to the theory of space is given by Gerla in [17]. Recent surveys in this field, pointing out its relations with Theoretical Computer Science and AI, can be found in [2].The ideas of de Laguna and Whitehead lead naturally to the following general programme:• define in topological terms those subsets of a topological space that correspond most closely to the idea of regions;• choose some (algebraic) structure which is inherent to the family of all regions of a topological space, fix some kind of morphisms between the obtained (algebraic) objects and build in this way a category A;• find a subcategory T of the catego...

Open and other kinds of map extensions over zero-dimensional local compactifications

“…So, we can suppose that f is skeletal. Then, as it follows from the proof of [4,Theorem 2.11], ϕ f is a complete Boolean homomorphism. Thus, by [4, (33)], the map ϕ f has a left adjoint ϕ f : RC(X 1 ) −→ RC(X 2 ), F → cl X 2 (f (F )).…”

“…we describe axiomatically the restrictions of the Leader's local proximities on the Boolean algebra RC(X) of all regular closed subsets of a Tychonoff space X) and prove this new assertion independently of the Leader's theorem using only our generalization (see [5]) of de Vries Duality Theorem [3]. This permits us to use our recent general results obtained in [4,6]. Finally, on the base of our variant of Leader's Theorem, we characterize in the language of local contact algebras only (i.e.…”

“…We will recall some results from [5,4] which are basic for our investigations in the present paper. Let DHLC be the category whose objects are all complete LCAs and whose morphisms are all functions ϕ : (A, ρ, IB) −→ (B, η, IB ′ ) between the objects of DHLC satisfying conditions…”

A generalization of the Stone Duality Theorem

“…This paper is a continuation of the papers [13,15,16] and, to some extent, of the papers [7,8,9,10,11,12,14,29]. In it we prove a new duality theorem for the category of precontact algebras which implies the Stone Duality Theorem, its connected version obtained in [16], the recent duality theorems from [3,23], and some new duality theorems for the category of contact algebras and for the category of complete contact algebras.…”

A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I

“…The categories HLC and DLC are dually equivalent. In more details, let Λ t : HLC −→ DLC and Λ a : DLC −→ HLC be the contravariant functors extending, respectively, the Roeper correspondences Ψ t : |HLC| −→ |DLC| and Ψ a : |DLC| −→ |HLC| (see [4] for Ψ t and Ψ a ) to the morphisms of the categories HLC and DLC in the following way:…”