and -Spaces in Smooth Topological Spaces

In (25), we introduce the notion of r-fuzzy neighborhood filters in smooth topological spaces in sense of Gahler (10), and used it to define and study separation axioms ࢀ࢏ ,࢏ = ૙ ,૚ ,૛ . Here we continue our study of the axioms of separation in smooth topological spaces. Therefore, we introduce the notion of r-fuzzy neighborhood filter at a set. Then by using this notion we define and study separation axioms ࢀ࢏ ,࢏ = ૜ ,૝ . These axioms are related only to usual points and ordinary subsets and reduce to axioms defined in (5), if ࣎ :�ࡵ ࢄ → {૙ ,૚ }. So the current separation axioms are generalization of the old one. In addition, we show ࢀ࢏ -space not necessarily be a ࢀ࢏ ૚-space for i=3,4. We give a condition for which, ࢀ࢏ -space is a ࢀ࢏ ૚-space for i=3,4. Finally, these axioms are good extension from the point of view of Aygun et al. (2).