New separation axioms on closure spaces generated by relations

Abstract

It is evident from recent developments that topological structures which are more general than usual topology has many applications in allied branches of Science and Engineering. Cech closure space dened by Cech [3] being a generalized topo- logical space has been utilized in digital topology [4, 10, 11], a theory developed in late 1960s for the study of geometric and topological properties of digital im- ages [5, 6, 8, 9]. Cech closure spaces are obtained from the Kuratowski ones by omitting the conditions of idempotency. In 2006, Allam, Bakeir and Abo-Tabl [1] introduced a new approach to dene closure spaces through relations and stud- ied lower separation axioms, continuous functions and subspaces on closure spaces generated by relations. In 2008, the same authors [2] introduced some methods to generate topologies via binary relations. G. Liu [7] in 2010, utilized the notion of aftersets and foresets [1] which is renamed as R-left and R-right relative sets in [7]. In the present paper, we introduced and studied higher separation axioms on closure spaces generated by relations. Let X be any set then a relation on X is a subset of X x X, i.e., R⊆ X x X. The formula (x; y) ∈ R is abbreviated as xRy means that x is in relation R with y.