Cech closure spaces.

Abstract

This paper presents an introduction to Cech closure spaces. The set of all Cech closure operators on a set is closed under the operations of union and composition. An association between Cech closure operators on a finite set and zero-one relation matrices is used to present matrix operations corresponding to union and composition of Cech closure operators. Finitely generated Cech closure operators are defined, and it is shown that the set of all finitely generated Cech closure operators on a set, partially ordered in a natural way, yields a uniquely complemented, distributive, and complete lattice and is therefore a Boolean algebra. A Cech closure operator generates a semi-topology and an underlying topology; relationships between these are studied. Several separation properties are generalized to Cech closure spaces and studied in this broader context.