Universality of the closure space of filters in the algebra of all subsets

Abstract

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 ℒ is a countable set and C: P(ℒ) → P(ℒ) is a closure operator on X, then C satisfies the compactness theorem iff the closure space 〈ℒ,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 2ω 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″. We assume the reader is familiar with notions in [5].