Axioms of R-Countability Via R-Dense Sets

Abstract

In this article, we introduce a new type of r-countability axiom by using the concept of regular-dense (r-dense) sets; the so called D-r-countability axiom, which states that space is D-r-(separable, first countable, Lindel f, δ-compact, second countable) if a topological space contains an r-(separable, first countable, Lindel f, δ-compact, second countable) subspace that is r-dense.  We study the relationship between these spaces, the r-countability axiom and the d-countability axiom. Furthermore, we characterize the topological hereditary properties of these spaces using theory and some examples.  Finally, we study the behavior of the D-r-countability axiom and D-countability axiom in regular spaces.