Partial soft separation axioms and soft compact spaces

Abstract

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft T i -spaces (i = 0, 1, 2, 3, 4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft T i -spaces are stronger than soft T i -spaces, for i = 0, 1, 4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft T i -spaces, for i = 0, 1, 2. Furthermore, we prove the equivalent among finite p-soft T i -spaces, for i = 1, 2, 3 and derive that a finite product of p-soft T i -spaces is p-soft T i , for i = 0, 1, 2, 3, 4. In the last section, we show the relationships which associate some p-soft T i -spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T 2 -space is soft compact and prove that every soft compact p-soft T 2 -space is soft T 3 -space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.