Georgi D. Dimov

A Proximity Approach to Some Region-Based Theories of Space

This paper is a continuation of [VAK 01]. The notion of local connection algebra, based on the primitive notions of connection and boundedness, is introduced. It is slightly different but equivalent to Roepers notion of region-based topology [ROE 97]. The similarity between the local proximity spaces of Leader [LEA 67] and local connection algebras is emphasized. Machinery, analogous to that introduced by Efremovi?c [EFR 51],[EFR 52], Smirnov [SMI 52] and Leader [LEA 67] for proximity and local proximity spaces, is developed. This permits us to give new proximity-type models of local connection algebras, to obtain a representation theorem for such algebras and to give a new shorter proof of the main theorem of Roepers paper [ROE 97]. Finally, the notion of MVD-algebra is introduced. It is similar to Mormanns notion of enriched Boolean algebra [MOR 98], based on a single mereological relation of interior parthood. It is shown that MVD-algebras are equivalent to local connection algebras. This means that the connection relation and boundedness can be incorporated into one, mereological in nature relation. In this way a formalization of the Whiteheadian theory of space based on a single mereological relation is obtained.

Topological representation of precontact algebras

The notions of 2-precontact and 2-contact spaces as well as of extensional (and other kinds) 3-precontact and 3-contact spaces are introduced. Using them, new representation theorems for precontact and contact algebras (satisfying some additional axioms) are proved. They incorporate and strengthen both the discrete and topological representation theorems from [3, 1, 2, 4, 10]. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. In particular, such a bijective correspondence for the RCC systems of [8] is obtained, strengthening in this way the previous representation theorems from [4, 1].

A Generalization of De Vries Duality Theorem

Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of locally compact Hausdorff spaces and perfect maps.

On Scott Consequence Systems

The notion of Scott consequence system (briefly, S-system) was introduced by D. Vakarelov in [32] in an analogy to a similar notion given by D. Scott in [26]. In part one of the paper we study the category SSyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category SSyst. Extending the representation theory of D. Vakarelov [32] for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.

Compactifications and A-compactifications of frames. Proximal frames

Introduction The aim of this paper is to give two new descriptions of the ordered set (^J^(F), ^ ) of all (up to equivalence) regular compactifications of a completely regular frame F and to introduce and study the notion oi A-frame as a generalization of the notion of Alexandroff space (known also as zero-set space) (Alexandroff [1], Gordon[15]). A description of the ordered set of all (up to equivalence) Acompactifications of an A-frame by means of an ordered by inclusion set of some distributive lattices (called AP-sublattices) is obtained. It implies that any A-frame has a greatest A-compactification and leads to the descriptions of (J^Jf(-F), ^ ) . A new category U©5 isomorphic to the category ©ros^rm of proximal frames is introduced. A question for compactifications of frames analogous to the R. Chandlers question [8, p. 71] for compactifications of spaces is formulated and solved. Many results of [1], [3], [15], [23], [9], [10] and [11] are generalized. We first fix some notation. If C denotes a category, we write Xe \C\ if X is an object of C, and feC(X, Y) if / is a morphism of C with domain X and codomain Y. All lattices will be with top and bottom elements, denoted respectively by 1 and 0, and T)£at will stand for the category of distributive lattices and lattice homomorphisms. By an ordered set (M, ^ ) we mean a partially ordered set (i.e. ^ is a reflexive and transitive binary relation onM) for which ^ is also antisymmetric. If X is a set then we write expX for the set of all subsets of X. We denote by I the unit closed interval [0,1] with the natural topology, by Q the set of all rational numbers, by D> the set of all dyadic numbers in the interval (0,1), by N the set of all positive natural numbers and by 2 the simplest Boolean algebra.

Compactifications, A-compactifications and proximities

A functor S from the category RegσFrm of regular σ-frames to the category DLat of distributive lattices is defined. The notion of AP-sublattice of S(α), for α ∈ ¦RegσFrm¦, is introduced and it is shown that, for every Alexandroff space (X, α), the ordered set of all (up to equivalence) A-compactifications of (X, α) is isomorphic to the set of all AP-sublattices of S(α) ordered by inclusion. This gives, in particular, a description of the ordered set of z-compactifications of a T3 1/2-space. Further, for any Tychonoff space X, the notion of a P-sublattice of S (CozX) is defined and it is proved that the ordered set of all (up to equivalence) T2-compactifications of X is isomorphic to the set of all P-sublattices of S(CozX) ordered by inclusion. We construct proximities by means of AP- and P-sublattices. Moreover, using these notions, we introduce two concrete categories PHA and PASF which are respectively isomorphic and dual to the category Prox of proximity spaces.

Frames and Grids

M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adámek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal characterization of all grids which correspond (in the above sense) to the frames having T1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.