A conjunction in closure spaces

Abstract

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 ω-conjunctive closure spaces (X is ω-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 ω-conjunctive closure space its interior is empty (i.e. it is a boundary set). 2. If X is an ω-conjunctive closure space which satisfies the ω-compactness theorem and \(\hat P\)[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 ω-conjunctive and topological space. 4. Every continuous function preserves all conjunctions.