In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following characterization of the consequence (...) operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X. (shrink)

This paper is closely related to investigations of abstract properties of basic logical notions expressible in terms of closure spaces as they were begun by A. Tarski (see [6]). We shall prove many properties of -conjunctive closure spaces (X is -conjunctive provided that for every two elements of X their conjunction in X exists). For example we prove the following theorems:1. For every closed and proper subset of an -conjunctive closure space its interior is empty (i.e. it is a boundary (...) set). 2. If X is an -conjunctive closure space which satisfies the -compactness theorem and [X] is a meet-distributive semilattice (see [3]), then the lattice of all closed subsets in X is a Heyting lattice. 3. A closure space is linear iff it is an -conjunctive and topological space. 4. Every continuous function preserves all conjunctions. (shrink)

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.

We give an idea of uniform approach to the problem of characterization of absolute extensors for categories of topological spaces [21], closure spaces [15], Boolean algebras [22], and distributive lattices [4]. In this characterization we use the notion of retract of the closure space of filters in the lattice of all subsets.

The main theorem says that a consequence operator is an effective part of the consequence operator for the classical prepositional calculus iff it is a consequence operator for a logic satisfying the compactness theorem, and in which every finitely axiomatizable theory is decidable.

The main result of this paper is the following theorem: a closure space X has an , , Q-regular base of the power iff X is Q-embeddable in It is a generalization of the following theorems:(i) Stone representation theorem for distributive lattices ( = 0, = , Q = ), (ii) universality of the Alexandroff's cube for T 0-topological spaces ( = , = , Q = 0), (iii) universality of the closure space of filters in the lattice of all (...) subsets for , -closure spaces (Q = 0). By this theorem we obtain some characterizations of the closure space given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F iff X is a consistent closure space satisfying the compactness theorem and X contains a 0, -base consisting of -prime sets. (shrink)

This paper is a continuation of investigations on Galois connections from [1], [3], [10]. It is a continuation of [2]. We have shown many results that link properties of a given closure space with that of the dual space. For example: for every -disjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the -prime sets in X (Theorem 2). Moreover the characterizations of the satisfiability relation for classical (...) logic are shown. Roughly speaking our main result here is the following: a satisfiability relation in a logic L with, a countable language is a fragment of the classical one iff the compactness theorem for L holds (Theorems 3–8). (shrink)

For a complete Heyting lattice , we define a category Etale (). We show that the category Etale () is equivalent to the category of the sheaves over , Sh(), hence also with -valued sets, see [2], [1]. The category Etale() is a generalization of the category Etale (X), see [1], where X is a topological space.