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.

On Soft Ordered Maps

“…Later on, numerous studies were done on soft topological notions and their features (see, for example, [6,10,12,15,23,34,35,36]). Recently, El-Shafei et al [19] established two new soft relations, namely partial belong and total non belong. Depend on them, they introduced partial soft separation axioms, namely p-soft T i -spaces for i = 0, 1, 2, 3, 4.…”

On supra soft topological ordered spaces

“…Definition 2.12. (El-Shafei et al, 2018;Shabir & Naz, 2011) Let (G, T) be a soft set over A and let a 2 A:…”

“…Later on, many authors carried out detailed studies via soft topologies and defined most of the fundamental concepts in soft topologies analogously to them in ordinary topologies. In particular, they highlighted on soft separation axioms which are among the most interesting and common topics via soft topologies (see, e.g., Al-shami & El-Shafei, 2019a;El-Shafei, Abo-Elhamayel, & Al-shami, 2018;Singh & Noorie, 2017;Tantawy, El-Sheikh, & Hamde, 2016). Al-shami and Ko cinac (2019) proved the equivalence between enriched and extended soft topologies and derived the interchangeable attribute for soft interior and closure operators.…”

On Two Classes of Soft Sets in Soft Topological Spaces

“…The notion of soft topological spaces is introduced in [6]. Then researchers modified several concepts of classical topological spaces to include soft topological spaces, some recently published soft topological papers are appeared in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For the purpose of improving some known topological theorems, Hdeib [27] introduced the notion of ω-closed sets as a weaker notion of closed sets as follows: Let (X, ) be a topological space and A a subset of X.…”