R-Countability Axioms

In this article, we use the concept of regular open sets to define a generalization of the countability axioms; namely regular countability axioms, and they are denoted by r-countability axioms. This class of axioms includes r-separable spaces, r-first countable spaces, r-Lindelöf spaces, r--compact spaces and r-second countable spaces. We investigate their fundamental properties, and study the implication of the new axioms among themselves and with the known axioms. Moreover, we study the hereditary properties for r-countability axioms, also we consider some related functions in terms of r-open sets, which preserve these spaces. Finally, we prove that in regular space r-countability axioms and countability axioms are equivalent, while in locally compact T_2 space, the spaces: Lindelöf, r- Lindelöf, -compact and r--compact are all equivalent.