On soft p c -separation axioms

Abstract Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X. Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft p c − T i {p}_{c}-{T}_{i} and soft p c − T i ⁎ {p}_{c}-{T}_{i}^{\ast } , i = 0 , 1 , 2 i=0,1,2 by using the concept of soft p c {p}_{c} -open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.


Introduction
After the introduction of soft set theory for the first time by Molodtsov [1] in 1999 as a new tool in mathematics to deal with several kinds of vagueness in complicated problems in sciences, the study of soft sets and their properties was applied to many branches of mathematics such as probability theory, algebra, operation research, and mathematical analysis. In [2,3], some types of soft Baire spaces and some other mathematical structures were studied and investigated. However, there are analogous theories that can be taken into account as mathematical tools for dealing with uncertainties but each theory has its own difficulties. In the last two decades, mathematicians turned their studies towards soft topological spaces and they reported in several papers different and many interesting topological concepts. Shabir and Naz [4] in 2011 introduced the concept of soft topological spaces which are defined over an initial universe with fixed set of parameters. They indicated that a soft topological space gives a parameterized family of topological spaces and introduced the concept of soft open sets, soft closed sets, soft interior point, soft closure and soft separation axioms. Shi and Pang [5] reported some important results on soft topological spaces. It is noticed that a soft topological space gives a parametrized family of topologies on the initial universe but the converse is not true, i.e., if some topologies are given for each parameter, we cannot construct a soft topological space from the given topologies. Consequently, we can say that the soft topological spaces are more generalized than the classical topological spaces. Georgiou et al. [6] in 2013 defined and studied some soft separation axioms, soft continuity in soft topological spaces using ordinary points of a topological space X.
Zorlutuna et al. in [7] and [8] defined and introduced soft neighbourhood and soft continuity in soft spaces using soft points. Hussain and Ahmad [9] continued investigating the properties of soft open (soft closed), soft neighbourhood and soft closure. They also defined and discussed the properties of soft interior, soft exterior and soft boundary.
Husain and Ahmed [10] in 2015 introduced separation axioms by using distinct point in the universal set, while in 2018, Bayramov and Aras [11] defined some separation axioms by using distinct soft points. Also, El-shafei et al. in [12] introduced different types of soft separation axioms.
Hamko and Ahmed [13] introduced the concepts of soft p c -open (soft p c closed) sets, soft p c -neighbourhood and soft p c -closure. They also defined and discussed the properties of soft p c -interior, soft p c -exterior and soft p c -boundary. Also, they defined and studied soft continuity and almost soft continuity in soft spaces using soft points and soft p c -open sets in a soft topological space. Recently, several types of soft separation axioms were studied by Al-shami et al. [14][15][16]. Also, Al-shami and El-shafei [17,18] introduced other types of soft separation axioms and obtained many characterization theorems, while in [19] some relations on soft Hausdorff spaces are given and corrected some other relations that were written before by other authors. The aim of this paper is to introduce and discuss a study of soft separation axioms, soft − P T c i , soft − ( = ) P T i 0, 1, 2 c i ⁎ , soft p c -regular and soft p c -normal spaces, which are defined over an initial universe with a fixed set of parameters by using soft points defined in [11]. Characterizations and properties of these spaces are discussed.
Throughout the present paper, X is a nonempty initial universal set and E is a set of parameters. A pair . The collection of soft sets ( ) F E , over a universal set X with the parameter set E is denoted by ( ) X SP E . Any logical operation ( ) λ on soft sets in soft topological spaces is denoted by usual set of theoretical operations with symbol ( ( )) s λ .

Preliminaries
For the definitions and results on the soft set theory and soft topological spaces, we refer to [7][8][9][10][11] and [20,21]. However, we recall some definitions and results on soft topology, which are used in the following sections.
2. The soft interior of ( ) G E , is the soft set ,˜SO˜: , , .
, is soft pre-open.
, containing x e and ′ y e , respectively.
In [4], a soft regular space is defined by using ordinary points as follows.
,˜then X is called soft regular.
In [9], a soft regular space is defined by using soft points as follows.
,˜then X is called soft regular.
. The soft complement of each soft p c -open set is called the soft p c -closed set.
The family of all soft p c -open (resp., soft p c -closed) sets in a soft topological space ( ) be a soft topological space and let ( ) G E , be a soft set. Then, ,˜˜PC˜: , , .
2. The soft pre-interior of ( ) G E , is the soft set ,˜˜PO˜: , , .
, containing x e and ′ y e , respectively.
if and only if the sp c -closure of any two soft points is distinct.  . Therefore, Proof. We shall prove the case when ( ) is a soft open set containing ′ y e but not x e . Therefore, ( ) The next example shows that the converse of Proposition 3.4 is not true in general.
Then, it can be checked that  . Therefore, , then it is a soft T 0 (resp., soft T 1 ) space due to [10].
Proof. It is similar to the proof of Proposition 4.10. □ Remark 4.17. In Example 4.6, it is easy to see that the converse of Proposition 4.16 is not true in general.
The following statements are true: Proof.
, then by Remark 2.8, it is clear that ( , is an sp c -closed set or alternatively Proof. Let for each pair of distinct points x, ∈ y X, there exists an sp c -clopen set (F, E) containing x but not y, which implies that ( ) X F E \ , is also an sp c -open set and ∈ ( ) y X F Ẽ˜\ , , since ( ) ⊓ ( ) = F E X F E ϕ , \ ,˜, X is an − sp T