On soft pc-separation axioms

Qumri H. Hamko 1 , Nehmat K. Ahmed 1  and Alias B. Khalaf 2
  • 1 Department of Mathematics, College of Education, Salahaddin University, Kurdistan-Region, Iraq
  • 2 Department of Mathematics, College of Science, University of Duhok, Kurdistan-Region, Iraq
Qumri H. Hamko, Nehmat K. Ahmed and Alias B. Khalaf
  • Corresponding author
  • Department of Mathematics, College of Science, University of Duhok, Kurdistan-Region, Iraq
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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 pcTi and soft pcTi, i=0,1,2 by using the concept of soft pc-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.

1 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 pc-open (soft pc closed) sets, soft pc-neighbourhood and soft pc-closure. They also defined and discussed the properties of soft pc-interior, soft pc-exterior and soft pc-boundary. Also, they defined and studied soft continuity and almost soft continuity in soft spaces using soft points and soft pc-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 PcTi, soft PcTi(i=0,1,2), soft pc-regular and soft pc-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 (F, E) is called a soft set over X, where F is a mapping F:EP(X). The collection of soft sets (F,E) over a universal set X with the parameter set E is denoted by SP(X)E. Any logical operation (λ) on soft sets in soft topological spaces is denoted by usual set of theoretical operations with symbol (s˜(λ)).

2 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.

Definition 2.1

A soft set (F,E) over X is said to be an empty soft set denoted by ϕ˜, if for all eE, F(e)=ϕ, and (F,E) over X is said to be an absolute soft set denoted by X˜, if for all eE, F(e)=X.

Definition 2.2

The complement of a soft set (F,E) is denoted by (F,E)c or X˜\(F,E) and is defined by (F,E)c=(Fc,E), where Fc:EP(X) is a mapping given by Fc(e)=X\F(e), for all eE.

It is clear that ((F,E)c)c=(F,E), ϕ˜c=X˜ and X˜c=ϕ˜.

Definition 2.3

For two soft sets (F,E) and (G,B) over a common universe X, we say that (F,E) is a soft subset of (G,B), if

  1. EB and
  2. for all eE, F(e)G(e).

We write (F,E)(G,B).

Definition 2.4

The union of two soft sets of (F,E) and (G,B) over the common universe X is the soft set (H,C)=(F,E)(G,B), where C=EB and for all eC,

H(e)={F(e):ifeEB,G(e):ifeBE,F(e)G(e):ifeEB.

Definition 2.5

The intersection (H,C) of two soft sets (F,E) and (G,B) over a common universe X, denoted (F,E)(G,B), is defined as C=ABϕ, and H(e)=F(e)G(e) for all eC.

Definition 2.6

Let xX, then (x,E) denotes the soft set over X for which x(e)={x}, for all eE. Let (F,E) be a soft set over X and xX. We say that x˜(F,E) means x belongs to the soft set (F,E) whenever xF(e) for all eE.

Definition 2.7

The soft set (F,E) is called a soft point, denoted by (xe,E) or xe, if for the element eE, F(e)={x} and F(e)=ϕ for all eE\{e}.

We say that xe˜(G,E) if xG(e).

Two soft points xe and ye are distinct if either xy or ee.

Remark 2.8

From Definitions 2.6 and 2.7, it is clear that:

  1. (x,E) is the smallest soft set containing x.
  2. If x˜(F,E), then xe˜(F,E) and xe˜(x,E) always.
  3. (F,E)={(xe,E):eE}.

Definition 2.9

[4] Let τ˜ be a collection of soft sets over a universe X with a fixed set E of parameters. Then, τ˜SP(X)E is called a soft topology if,

  1. ϕ˜ and X˜ belong to τ˜.
  2. The union of any number of soft sets in τ˜ belongs to τ˜.
  3. The intersection of any two soft sets in τ˜ belongs to τ˜.

The triplet (X,τ˜,E) is called a soft topological space over X. The members of τ˜˜ are called soft open sets in X˜ and complements of them are called soft closed sets in X˜ and they are denoted by SO(X˜) and SC(X˜), respectively. Soft interior and soft closure are denoted by s˜int and s˜cl, respectively.

Definition 2.10

[4] Let (X,τ˜,E) be a soft topological space and let (G,E) be a soft set. Then,

  1. The soft closure of (G,E) is the soft set
    s˜cl(G,E)={(K,B)˜SC(X˜):(G,E)(K,B)}.
  2. The soft interior of (G,E) is the soft set
s˜int(G,E)={(H,B)˜SO(X˜):(H,B)(G,E)}.

Definition 2.11

[8] Let (X,τ˜,E) be a soft topological space, (G,E) be a soft set over X˜ and xe˜X˜. Then, (G,E) is said to be a soft neighbourhood of xe if there exists a soft open set (H,E) such that xe˜(H,E)((G,E).

Proposition 2.12

[4] Let(Y,τ˜Y,E)be a soft subspace of a soft topological space(X,τ˜,E)and(F,E)˜SP(X)E. Then,

  1. If(F,E)is a soft open set inY˜andY˜˜τ˜, then(F,E)˜τ˜.
  2. (F,E)is a soft open set inY˜if and only if(F,E)=Y˜(G,E)for some(G,E)˜τ˜.
  3. (F,E)is a soft closed set inY˜if and only if(F,E)=Y˜(H,E)for some soft closed(H,E)inX˜.

Definition 2.13

[22] A soft subset (F,E) of a soft space X˜ is said to be soft pre-open if (F,E)s˜int[s˜cl(F,E)]. The complement of soft pre-open set is said to be soft pre-closed. The family of soft pre-open set and soft pre-closed set is denoted by s˜PO(X) and s˜PC(X), respectively.

Lemma 2.14

[22] Arbitrary union of soft pre-open sets is a soft pre-open set.

Lemma 2.15

[21] A subset(F,E)of a soft topological space(X,τ˜,E)is a soft pre-open set if and only if there exists a soft open set(G,E)such that(F,E)(G,E)s˜cl(F,E).

Lemma 2.16

[21] Let(F,E)Y˜X˜, where(X,τ˜,E)is a soft topological space andY˜is a soft pre-open subspace ofX˜. Then(F,E)˜s˜pO(X), if and only if(F,E)˜s˜pO(Y).

Theorem 2.17

[23] If(U,E)is soft open and(F,E)is soft pre-open in(X,τ˜,E), then(U,E)(F,E)is soft pre-open.

Lemma 2.18

[23] Let(F,E)Y˜X˜, where(X,τ˜,E)is a soft topological space andY˜is a soft subspace ofX˜. If(F,E)˜s˜pO(X), then(F,E)˜s˜pO(Y).

Definition 2.19

[25] A soft topological space (X,τ˜,E) is said to be:

  1. Soft T0, if for each pair of distinct soft points x, y˜X, there exist soft open sets (F,E) and (G,E) such that either x˜(F,E) and y˜(F,E) or y˜(G,E) and x˜(G,E).
  2. Soft T1, if for each pair of distinct soft points x, yX, there exist two soft open sets (F,E) and (G,E) such that x˜(F,E) but y˜(F,E) and y˜(G,E) but x˜(G,E).
  3. Soft T2, if for each pair of distinct soft points x, yX, there exist two disjoint soft open sets (F,E) and (G,E) containing x and y, respectively.

In [11], Bayramov and Aras redefined soft Ti-spaces as in the following definition.

Definition 2.20

[11] A soft topological space (X,τ˜,E) is said to be:

  1. Soft T0, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist soft open sets (F,E) and (G,E) such that either xe˜(F,E) and ye˜(F,E) or ye˜(G,E) and xe˜(G,E).
  2. Soft T1, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist two soft open sets (F,E) and (G,E) such that xe˜(F,E) but ye˜(F,E) and ye˜(G,E) but xe˜(G,E).
  3. Soft T2, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist two disjoint soft open sets (F,E) and (G,E) containing xe and ye, respectively.

Proposition 2.21

[11]

  1. Every softT2-spacesoftT1-spacesoftT0-space.
  2. A soft topological space(X,τ˜,E)is softT1if and only if each soft point is soft closed.

In [4], a soft regular space is defined by using ordinary points as follows.

Definition 2.22

[4] If for every xX and every soft closed set (F,E) not containing X, there exist two soft open sets (G,E) and (H,E) such that x˜(G,E), (F,E)(H,E) and (G,E)(H,E)=ϕ˜ then X˜ is called soft regular.

In [9], a soft regular space is defined by using soft points as follows.

Definition 2.23

[9] If for everyxe˜X˜and every soft closed set(F,E) not containing xe, there exist two soft open sets (G,E) and (H,E) such that xe˜(G,E), (F,E)(H,E) and (G,E)(H,E)=ϕ˜ then X˜ is called soft regular.

Definition 2.24

[24] A soft pre-open set (F,E) in a soft topological space (X,τ˜,E) is called soft pc-open if for each xe˜(F,E), there exists a soft closed set (K,E) such that xe˜(K,E)(F,E). The soft complement of each soft pc-open set is called the soft pc-closed set.

The family of all soft pc-open (resp., soft pc-closed) sets in a soft topological space (X,τ˜,E) is denoted by s˜pcO(X,τ˜,E) (resp., s˜pcC(X,τ˜,E)) or s˜pcO(X) (resp., s˜pcC(X)).

Definition 2.25

[21] Let (X,τ˜,E) be a soft topological space and let (G,E) be a soft set. Then,

  1. The soft pre-closure of (G,E) is the soft set
    s˜pcl(G,E)={(K,B)˜s˜PC(X˜):(G,E)(K,B)}.
  2. The soft pre-interior of (G,E) is the soft set
s˜pint(G,E)={(H,B)˜s˜PO(X˜):(H,B)(G,E)}.

Definition 2.26

[13] Let (X,τ˜,E) be a soft topological space and let (G,E) be a soft set. Then,

  1. A soft point xe˜X˜ is said to be a soft pc-limit soft point of a soft set (F,E) if for every soft pc-open set (G,E) containing xe, (G,E)[(F,E)\{xe}]ϕ˜.The set of all soft pc-limit soft points of (F,E) is called the soft pc-derived set of (F,E) and is denoted by s˜pcD(F,E).
  2. The soft pc-closure of (G,E) is the soft set
    s˜pccl(G,E)={(K,B)˜s˜PcC(X˜):(G,E)(K,B)}.
  3. The soft pc-interior of (G,E) is the soft set
s˜pcint(G,E)={(H,B)˜s˜PO(X˜):(H,B)(G,E)}.

Lemma 2.27

[24] If(F,E)Y˜X˜andY˜is soft clopen, then(F,E)˜s˜pcO(Y)if and only if(F,E)˜s˜pcO(X).

Lemma 2.28

[24] Let(F,E), Y˜X˜andY˜be soft clopen. If(F,E)˜s˜pcO(X), then(F,E)Y˜˜s˜pcO(Y).

Lemma 2.29

[13] Let(F,E)Y˜X˜. IfY˜is soft clopen, thens˜pcclY(F,E)=s˜pcclX(F,E)Y˜.

Definition 2.30

[21] A soft topological space (X,τ˜,E) is said to be:

  1. Soft P0, if for each pair of distinct soft points x, yX, there exist soft pre-open sets (F,E) and (G,E) such that either x˜(F,E) and y˜(F,E) or y˜(G,E) and x˜(G,E).
  2. Soft P1, if for each pair of distinct soft points x, yX, there exist two soft pre-open sets (F,E) and (G,E) such that x˜(F,E) but y˜(F,E) and y˜(G,E) but x˜(G,E).
  3. Soft P2, if for each pair of distinct soft points x, yX, there exist two disjoint soft pre-open sets (F,E) and (G,E) containing x and y, respectively.

3 Soft pcTi spaces for (i=0,1,2)

In this section, we define s˜pcTi spaces for (i=0,1,2) by using s˜pc-open sets and separating the soft points of the soft topological space. Several relations between these soft spaces and other types of soft separation axioms are investigated.

Definition 3.1

A soft topological space (X,τ˜,E) is said to be

  1. s˜pcT0, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist s˜pc-open sets (F,E) and (G,E) such that xe˜(F,E) and ye˜(F,E) or ye˜(G,E) and xe˜(G,E).
  2. s˜pcT1, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist two s˜pc-open sets (F,E) and (G,E) such that xe˜(F,E) but ye˜(F,E) and ye˜(G,E) but xe˜(G,E).
  3. s˜pcT2, if for each pair of distinct soft points xe, ye˜SP(X)E, there exist two disjoint s˜pc-open sets (F,E) and (G,E) containing xe and ye, respectively.

Proposition 3.2

A soft topological space(X,τ˜,E)iss˜pcT0if and only if thes˜pc-closure of any two soft points is distinct.

Proof

Let (X,τ˜,E) be an s˜pcT0 space and xe,ye˜SP(X)E with xeye Then, there exist an s˜pc-open set (F,E) containing one of the soft points, say xe, but not the other. Then, X˜\(F,E) is an s˜pc-closed set which does not contain xe but contains ye. Since, s˜pccl({ye}) is the smallest s˜pc-closed set containing ye, s˜pccl({ye})X\(F,E) and therefore xe˜s˜pccl({ye}). Consequently, s˜pccl({xe})s˜pccl({ye}).

Conversely, suppose that xe, ye˜SP(X)E such that xeye and s˜pccl({xe})s˜pccl({ye}). Let zα be a soft point in SP(X)E such that zα˜s˜pccl({xe}), but zα˜s˜pccl({ye}). We claim that xe˜s˜pccl({ye}). For, if xe˜s˜pccl({ye}), then s˜pccl({xe})s˜pccl({ye}). This contradicts the fact that zα˜s˜pccl({ye}). Consequently, xe belongs to the s˜pc-open set (G,E)=X\s˜pccl({ye}). Then, (G,E) being the complement of s˜pc-closed set. Thus (G,E) is an s˜pc-open set which contains xe but not ye. Hence, (X,τ˜,E) is an s˜pcT0.□

Proposition 3.3

If(X,τ˜,E)iss˜pcT0space, thens˜pccl(s˜pcint(xe}))s˜pccl(s˜pcint(ye}))=ϕ˜, for each pair of distinct soft pointsxe,ye˜SP(X)E.

Proof

Let (X,τ˜,E) be an s˜pcT0 space and xe,ye˜SP(X)E such that xeye. Then, there exists an s˜pc-open set (F,E) containing xe or ye, say xe but not ye, which implies that xe˜(F,E) and ye˜(F,E), then ye˜X\(F,E) and X\(F,E) is s˜pc-closed. Now s˜pcint({ye})s˜pccl(s˜pcint({ye}))X\(F,E), which implies that (F,E)s˜pccl(s˜pcint({ye}))=ϕ˜, then (F,E)X\s˜pccl(s˜pcint({ye})). Since xe˜(F,E)X\s˜pccl(s˜pcint({ye})), then s˜pccl({xe})X\s˜pccl(s˜pcint({ye})), which implies that s˜pccl(s˜pcint({xe}))s˜pccl({xe})X\s˜pccl(s˜pcint({ye})). Therefore, s˜pccl(s˜pcint({xe}))s˜pccl(s˜pcint({ye}))=ϕ˜.□

Proposition 3.4

Every softs˜pcTispace is softTi for i=0,1.

Proof

We shall prove the case when (X,τ˜,E) is s˜pcT0, the other proof is similar.

Let (X,τ˜,E) be an s˜pcT0 space and xe,ye˜SP(X)E, with xeye, so there exists an s˜pc-open set (F,E) containing one of them say xe. Since (F,E) is an s˜pc-open set, there exists a soft closed set (K,E) such that xe˜(K,E)(F,E), so X˜\(K,E) is a soft open set containing ye but not xe. Therefore, (X,τ˜,E) is a soft T0-space.□

The next example shows that the converse of Proposition 3.4 is not true in general.

Example 3.5

Let X={x1,x2}, E={e1,e2} and let τ˜={X˜,ϕ˜,(F1,E),(F2,E),(F3,E),(F4,E),(F5,E),(F6,E)}, where (F1,E)={(e1,{x1}),(e2,ϕ)}, (F2,E)={(e1,X),(e2,{x2})}, (F3,E)={(e1,ϕ),(e2,X)}, (F4,E)={(e1,{x1}),(e2,X), (F5,E)={(e1,ϕ),(e2,{x2})} and (F6,E)={(e1,{x1}),(e2,{x2})}. Then, s˜pcO(X)={X˜,ϕ˜}. It is easy to show that this space is a soft T0-space but it is not s˜pcT0.

Proposition 3.6

Every softTispace iss˜pcT1fori=1,2.

Proof

If a soft space is soft T2 or soft T1, then by Proposition 2.21(2) every soft point is soft closed and hence every soft open set is an s˜pc-open set. Therefore, if (X,τ˜,E) is soft T2 (resp., soft T1), then it is an s˜pcT2 (resp., s˜pcT1) space.□

Corollary 3.7

A soft topological space(X,τ˜,E)is softT1if and only if it iss˜pcT1.

Proof

Follows directly from Propositions 3.4 and 3.6.□

Proposition 3.8

A space(X,τ˜,E)iss˜pcT1if and only if every soft point of the space(X,τ˜,E)is ans˜pc-closed set.

Proof

Let (X,τ˜,E) be an s˜pcT1 space, so by Proposition 3.4, (X,τ˜,E) is soft T1 and by Proposition 2.21(2), every soft point is soft closed hence soft pre-closed. Since every soft point is closed, s˜OP(X)=s˜OPc(X). Hence, every soft point is an s˜pc-closed set.

Conversely, let xe be a soft point of (X,τ˜,E) which is s˜pc-closed, then X˜\{xe} is an s˜pc-open. Then, for distinct soft points xe and ye, X˜\{xe} and X˜\{ye} are s˜pc-open sets such that xe˜X˜\{ye} but ye˜X˜\{ye} and ye˜X˜\{xe} but xe˜X˜\{xe}. Thus, (X,τ˜,E) is an s˜pcT1 space.□

Proposition 3.9

A soft topological space(X,τ˜,E)is ans˜pcT1space if and only ifs˜pcD({xe})=ϕ˜, for eachxe˜X˜.

Proof

Let X˜ be an s˜pcT1 space. To prove s˜pcD({xe})=ϕ˜, for each xe˜X˜. If s˜pcD({xe})ϕ˜, for some xe˜X˜, then there is a soft point say ye˜s˜pcD({xe}) and xeye. Since X˜ is s˜pcT1, then there exists an s˜pc-open set (F,E) such that ye˜(F,E) and xe˜(F,E), then (F,E){xe}=ϕ˜ and hence ye˜s˜pcD({xe}), which is a contradiction. Thus, s˜pcD({xe})=ϕ˜, for each xe˜X˜.

Conversely, let s˜pcD({xe})=ϕ˜, for each xe˜X˜. Since s˜pccl({xe})={xe}s˜pcD({xe}) and s˜pcD({xe})=ϕ˜, s˜pccl({xe})={xe}, which implies that {xe} is an s˜pc-closed set and hence by Proposition 3.8, X˜ is an s˜pcT1 space.□

Proposition 3.10

For a soft space(X,τ˜,E), the following statements are equivalent.

  1. X˜iss˜pcT2,
  2. For eachxe˜X˜and eachxeye, there exists ans˜pc-open set(F,E)ofxesuch thatye˜s˜pcCl(F,E).
  3. For eachxe˜X˜, {s˜pccl(F,E):xe˜(F,E)˜s˜pcO(X)}={xe}.

Proof

(1) ⇒ (2). Since X˜ is an s˜pcT2 space, there exists two disjoint s˜pc-open sets (F,E) and (G,E) such that xe˜(F,E) and ye˜(G,E). This implies that (F,E)X\(G,E). Therefore, s˜pccl(F,E)X\(G,E). So, ye˜s˜pccl(F,E).

(2) (3). Let for some xeye, we have ye˜s˜pccl(F,E) for every s˜pc-open set (F,E) containing xe, which contradicts (2).

(3) (1). Let xe, ye˜X˜ with xeye. Then, there exists an s˜pc-open set (F,E) containing xe such that ye˜s˜pccl(F,E). Let (G,E)=X\s˜pccl(F,E), then ye˜(G,E) and xe˜(F,E) and so (F,E)(G,E)=ϕ˜.□

Proposition 3.11

A soft spaceX˜iss˜pcT2if for each pair of distinct soft pointsxe, ye˜X˜there exists ans˜pc-clopen set(F,E)containing one of them but not the other.

Proof

Let for each pair of distinct soft points xe, ye˜X˜, there exists an s˜pc-clopen set (F,E) containing xe but not ye, which implies that X\(F,E) is also an s˜pc-open set and ye˜X\(F,E), since (F,E)X\(F,E)=ϕ˜, X˜ is an s˜pcT2 space.□

Proposition 3.12

Every soft clopen subspace of ans˜pcTispace is ans˜pcTispace for(i=0,1,2).

Proof

We prove only the case for s˜pcT0 space and the other cases are similar. Let (Y,τ˜Y,E) be a soft clopen subspace of s˜pcT0 space (X,τ˜,E) and xe, ye be two distinct soft points in X˜. Since (X,τ˜,E) is an s˜pcT0 space and xeye, then there exists an s˜pc-open set (F,E) containing one of them say xe but not ye. So, by Lemma 2.28, (F,E)Y˜ is an s˜pc-open set in (Y,τ˜Y,E), which contains xe but not ye. Hence, (Y,τ˜Y,E) is s˜pcT0.□

Proposition 3.13

If for eachxe˜X˜, there exists a soft clopen set(F,E)containingxesuch that(F,E)is an s˜pcTisubspace ofX˜, then the soft spaceX˜is also an s˜pcTispace for(i=0,1,2).

Proof

We prove only the case for the s˜pcT0 space and the proofs of other cases are similar. Let xe and ye be two distinct soft points in X˜, then by hypothesis there exist soft regular open sets (F,E) and (G,E) such that xe˜(F,E), ye˜(G,E) and (F,E), (G,E) are s˜pcT0 subspaces of X˜. Now, if ye˜(F,E), then the proof is completed, but if ye˜(F,E) and since (F,E) is s˜pcT0 subspace of X˜, there exists an s˜pc-open set (H,E) in (F,E) such that ye˜(H,E) and xe˜(H,E) and since (F,E) is soft regular open, by Lemma 2.27, (H,E) is an s˜pc-open set in X˜ containing ye but not xe. Therefore, X˜ is an s˜pcT0 space.□

4 Soft pcTi spaces for (i=0,1,2)

In this section, we define s˜pcTi spaces for (i=0,1,2) by using s˜pc-open sets and separating the usual points of the space. Several relations between these soft spaces and other types of soft separation axioms are investigated. Relations between s˜pcTi spaces and s˜pcTi spaces for (i = 0, 1, 2) are discussed.

Definition 4.1

A soft topological space (X,τ˜,E) is said to be

  1. s˜pcT0, if for each pair of distinct points x,yX, there exist s˜pc-open sets (F,E) and (G,E) such that x˜(F,E) and y˜(F,E) or y˜(G,E) and x˜(G,E).
  2. s˜pc-T1, if for each pair of distinct points x,yX, there exist two s˜pc-open sets (F,E) and (G,E) such that x˜(F,E) but y˜(F,E) and y˜(G,E) but x˜(G,E).
  3. s˜pc-T2, if for each pair of distinct soft points x,yX, there exist two disjoint s˜pc-open sets (F,E) and (G,E) containing x and y, respectively.

Proposition 4.2

Every softs˜pcTispace is a softpi-space fori=0,1,2.

Proposition 4.3

If a soft topological space(X,τ˜,E)is as˜pcT1space, then it is softs˜pcT1.

Proof

Let (X,τ˜,E) be an s˜pcT1 space and x,yX with xy, then xeye for all eE. Since (X,τ˜,E) is an s˜pcT1, by Proposition 3.8, {xe},{ye} are s˜pc-closed sets for all eE. Hence, X˜\{ye} and X˜\{xe} are the required s˜pc-open sets containing x and y, respectively. Therefore, (X,τ˜,E) is soft s˜pcT1.□

From Definition 3.1, Definition 4.1, Proposition 4.2, Corollary 3.7 and Proposition 4.3, we obtain the following diagram of implications.

softP2softP1softP0s˜pcT2s˜pcT1s˜pcT0s˜pcT2s˜pcT1s˜pcT0s˜T2s˜T1s˜T0

Any other implication except those resulting from transitivity is not true in general as it is shown in the following examples.

Example 4.4

Let X={x,y}, E={e1,e2} and let τ˜={X˜,ϕ˜,(F1,E),(F2,E)}, where

(F1,E)={(e1,{x}),(e2,{x}}, (F2,E)={(e1,{y}),(e2,{y}}.

Then, it can be checked that s˜pcO(X)=τ˜. Hence, the space is s˜PcTi but it is not s˜pcTi for i=0,1,2.

Example 4.5

Let X={x,y}, E={e1,e2} and let

τ˜={X˜,ϕ˜,(F1,E),(F2,E),(F3,E)},
where (F1,E)={(e1,{x}),(e2,{x,y}}, (F2,E)={(e1,{y}),(e2,{x,y}}(F3,E)={(e1,ϕ),(e2,{x,y}}. Then, it can be checked that s˜pcO(X)={X˜,ϕ˜}. Also, both the soft sets (F4,E)={(e1,{x}),(e2,{x}} and (F5,E)={(e1,{y}),(e2,{y}} are soft pre-open. Hence, this space is not s˜PcTi for i=0,1,2, but it is s˜Pi for i=0,1,2.

Example 4.6

Let X={x,y}, E={e1,e2} and let

τ˜={X˜,ϕ˜,(F1,E),(F2,E),(F3,E),(F4,E),(F5,E),(F6,E)},
where (F1,E)={(e1,{x}),(e2,ϕ)}, (F2,E)={(e1,{x}),(e2,{x,y})}, (F3,E)={(e1,{y}),(e2,ϕ)}, (F4,E)={(e1,{y}),(e2,{x,y})}, (F5,E)={(e1,{x,y}),(e2,ϕ)}, (F6,E)={(e1,ϕ),(e2,{x,y})}. Then, it can be checked that s˜pcO(X)=τ˜ and s˜pO(X)=SP(X)E. Since xe2ye2 and there is no soft open set containing one of them but not the other, it is not s˜pcTi for i=0,1,2. This space is s˜pcTi for i=0,1 but it is not s˜pcT2.

Example 4.7

Let X be any infinite set, E consists of infinite parameters and let Pe be a fixed soft point in X˜ and let τ˜ be the family of all soft subsets (F,E) such that either Pe˜(F,E) or if Pe˜(F,E), then

eEX\F(e)
is finite. This space is both s˜T2 and s˜pcT2, but it is not s˜pcT2 because if xy in X and xe=Pe, if the soft open set containing x is (F,E), which implies that xe=Pe˜(F,E), hence
eEX\F(e)
is finite but there is no soft open set containing y which is disjoint from (F,E).

From Examples 4.6 and 4.7, we conclude that s˜pcT2 and s˜pcT2 are incomparable spaces.

Example 4.8

Let X={1,2,3,}, E={0,1,2} and τ˜={Gn,E):n{1,2,}}{ϕ˜,X˜}, such that Gn:EP(X), where Gn(e)={n,n+1,} for every eE. The triplet (X,τ˜,E) is a soft topological space and s˜pcO(X)={ϕ˜,X˜}.

Let x and y be two distinct points of X and eE. We suppose that y<x. Then, x{x,x+1,}=Gx(e) and y{x,x+1,}. This means that x˜(Gx(e),E) and y˜(Gx(e),E). Thus, the soft topological space (X,τ˜,E) is a soft T0-space. Also, we observe that this soft topological space is not an s˜pcT0-space.

Example 4.9

Let X be any infinite set, E={e1,e2} and τ˜ a topology consists of ϕ˜ and all soft sets (F,E), where (F,E) is defined as: X\F(e) is a finite subset of X for each eE. Then, (X,τ˜,E) is a soft topological space over X. It can be easily shown that this space is s˜pcT1 and s˜pcT1 space which is not s˜pcT2 and not s˜pcT2.

Proposition 4.10

Let(X,τ˜,E)be a soft topological space andx,yXsuch thatxy. If there exists˜pc-open sets(F,E)and(G,E)such thatx˜(F,E)andy˜X˜\(F,E)ory˜(G,E)andx˜X˜\(G,E), then(X,τ˜,E)iss˜pcT0.

Proof

Let x,yX such that xy and (F,E), (G,E) are s˜pc-open sets such that x˜(F,E) and y˜X˜\(F,E) or y˜(G,E) and x˜X˜\(G,E). If y˜X˜\(F,E), then yX\F(e) for each eE. This implies that yF(e) for each eE. Therefore, y˜(F,E). Similarly, we can show that if x˜X˜\(G,E), then x˜(G,E). Hence, (X,τ˜,E) is s˜pcT0.□

Remark 4.11

In Example 4.6, it can be easily seen that the converse of Proposition 4.10 is not true in general. Because x˜(F2,E) but y˜X˜\(F2,E) and also y˜(F4,E) but x˜X˜\(F4,E).

Proposition 4.12

A soft topological space(X,τ˜,E)iss˜pcT0if and only if for each pair of distinct pointsx,yX, s˜pccl(x,E)s˜pccl(y,E).

Proof

Let (X,τ˜,E) be an s˜pcT0 space and x,y be any two distinct points of X. There exists an s˜pc-open set (F,E) containing x but not y. Then, X˜\(F,E) is an s˜pc-closed set which does not contain x but contains y. Since s˜pccl(y,E) is the smallest s˜pc-closed set containing y, s˜pccl(y,E)X˜\(F,E) and therefore x˜s˜pccl(y,E). Consequently, s˜pccl(x,E)s˜pccl(y,E).

Conversely, suppose that x,yX, xy and s˜pccl(x,E)s˜pccl(y,E). Let z be a point of X such that z˜s˜pccl(x,E), but z˜s˜pccl(y,E). We claim that x˜s˜pccl(y,E). For, if x˜s˜pccl(y,E), then s˜pccl(x,E)s˜pccl(y,E). This contradicts the fact that z ˜s˜pccl(y,E). Consequently, x belongs to the s˜pc-open set X˜\s˜pccl(y,E) and y does not belong to it.□

Proposition 4.13

For each pair of distinct pointsx,yX. If a soft topological space(X,τ˜,E)iss˜pcT0, then s˜pccl(s˜pcint(x,E))s˜pccl(s˜pcint(y,E))=ϕ˜.

Proof

Let (X,τ˜,E) be s˜pcT0 and x,yX such that xy. Then, there exists an s˜pc-open set (F,E) containing one of the points, say x, and does not contain the other, which implies that x˜(F,E) and y˜(F,E), then y˜X˜\(F,E) and X\(F,E) is s˜pc-closed. Now we have,

s˜pcint(y,E)s˜pccl(s˜pcint(y,E))X\(F,E),
which implies that (F,E)s˜pccl(s˜pcint(y,E))=ϕ˜, then (F,E)X\s˜pccl(s˜pcint(y,E). So, x˜(F,E)X\s˜pccl(s˜pcint(y,E)), then s˜pccl(x,E)X\s˜pccl(s˜pcint(y,E)), which implies that s˜pccl(s˜pcint(x,E))X\s˜pccl(s˜pcint(y,E)). Therefore,
s˜pccl(s˜pcint(x,E))s˜pccl(s˜pcint(y,E))=ϕ˜.

Proposition 4.14

If a soft topological space(X,τ˜,E)iss˜pcT0 (resp., s˜pcT1), then it is a softT0(resp., softT1) space due to [10].

Proof

Let (X,τ˜,E) be an s˜pcT0 space and x,y be any two distinct points in X, there exists an s˜pc-open set (F,E) containing one of them say x. Since (F,E) is an s˜pc-open set, there exists a soft closed set (K,E) such that x˜(K,E)(F,E), so X\(K,E) is a soft open set containing y but not x. Therefore, (X,τ˜,E) is a soft T0-space.

The other proof is similar.□

The next example shows that the converse of Proposition 4.14 is not true in general.

Example 4.15

In Example 4.8, we can see that (X,τ˜,E) is a soft T0-space but it is not a s˜pcT0 space.

Proposition 4.16

Let(X,τ˜,E)be a soft topological space andx,yXsuch thatxy. If there exists˜pc-open sets(F,E)and(G,E)such thatx˜(F,E), y˜X˜\(F,E)andy˜(G,E), x˜X˜\(G,E), then(X,τ˜,E)iss˜pcT1.

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.

Proposition 4.18

If(X,τ˜,E)is ans˜pcT1space andxX, then for eachs˜pc-open set(F,E)withx˜(F,E). The following statements are true:

  1. (x,E){(F,E):x˜(F,E)˜s˜pcO(X)}.
  2. For allyx, we havey˜{(F,E):x˜(F,E)˜s˜pcO(X)}.

Proof

  1. (1)Since x˜{(F,E):(F,E)˜s˜pcO(X)}, then by Remark 2.8, it is clear that (x,E){(F,E):(F,E)˜s˜pcO(X)}.
  2. (2)Let xy for x,yX, then there exists an s˜pc-open set (G,E) such that x˜(G,E) and y˜(G,E). This implies that yG(e) for some eE, and we have y{F(e):eE}. Therefore, y˜{(F,E):x˜(F,E)˜s˜pcO(X)}.□

Proposition 4.19

Every singleton soft set of a soft topological space(X,τ˜,E)is ans˜pc-closed set, if and only if(X,τ˜,E)is an s˜pcT1space.

Proof

Suppose that (X,τ˜,E) is an s˜pcT1 space and xX, we have to show that (x,E) is an s˜pc-closed set or alternatively X\(x,E) is an s˜pc-open set, let y˜X˜\(x,E), then clearly xy, since the space is an s˜pc-T1 space, there must exist s˜pc-open sets (F,E) such that y˜(F,E) but x˜(F,E). Thus, corresponding to each y˜X˜\(x,E) there exists an s˜pc-open (F,E) such that y˜(F,E)X\(x,E), therefore {y;yx}(F,E)X\(x,E). Hence X\(x,E)=(F,E), since (F,E) is an s˜pc-open set, and the union of an arbitrary collection of s˜pc-open sets is an s˜pc-open set. Therefore, (x,E) is an s˜pc-closed set.

Conversely, suppose that (x,E) is an s˜pc-closed set for each xX. Let x,yX such that xy. Now, for xX,X\(x,E) is an s˜pc-open set such that yX\(x,E) and x˜X˜\(x,E). Similarly, X\(y,E) is an s˜pc-open set containing x but not y. Thus, (X,τ˜,E) is an s˜pcT1 space.□

Proposition 4.20

If every soft point of a soft topological space(X,τ˜,E)is ans˜pc-closed set, then(X,τ˜,E)is an s˜pcT1space.

Proof

Let xe be a soft point of (X,τ˜,E) which is an s˜pc-closed. Let x,yX such that xy. Now, for xX, X˜\{xe} is an s˜pc-open set such that y˜X˜\{xe} and x˜X˜\{xe}. Similarly, X\{ye} is an s˜pc-open set containing x but not y. Thus, (X,τ˜,E) is an s˜pcT1 space.□

Proposition 4.21

A soft topological space(X,τ˜,E)is ans˜pcT1space if and only ifs˜pcD(x,E)=ϕ˜, for each xX.

Proof

Let s˜pcD(x,E)=ϕ˜, for each xX. Since, s˜pccl(x,E)=(x,E)s˜pcD(x,E) and s˜pcD(x,E)=ϕ˜, s˜pccl(x,E)=(x,E), which implies that (x,E) is s˜pc-closed and hence by Proposition 4.19, (X,τ˜,E) is an s˜pcT1 space. Conversely, let (X,τ˜,E) be an s˜pcT1 space. If s˜pcD(x,E)ϕ˜, for some xX, then there exists a point, say y˜(x,E), and xy, since (X,τ˜,E) is a s˜pcT1 space, then there exists an s˜pc-open set (F,E) such that y˜(F,E) and x˜(F,E), then (F,E)(x,E)=ϕ˜ and hence y˜s˜pcD(x,E), which is contradiction. Thus, s˜pcD(x,E)=ϕ˜, for each xX.□

Remark 4.22

A soft topological space (X,τ˜,E) is an s˜pcT1 space if and only if s˜pccl(x,E)=(x,E), for each xX.

Proposition 4.23

Let(X,τ˜,E)be a soft topological space andxX. If(X,τ˜,E)is ans˜pcT2space, then(x,E)={(F,E):x˜(F,E)˜s˜pcO(X)}.

Proof

Assume that there exist yX with xy and yF(e) for some eE. Since (X,τ˜,E) is an s˜pcT2space, then there exist s˜pc-open sets (G,E) and (H,E) such that x˜(G,E) and y˜(H,E) with (G,E)(H,E)=ϕ˜ and so (G,E)(y,E)=ϕ˜ and G(α)y(α)=ϕ˜. This contradicts the fact that yF(α) for some eE. Hence proved.□

Proposition 4.24

The following statements are equivalent for a soft topological space(X,τ˜,E):

  1. (X,τ˜,E)is ans˜pcT2space.
  2. For eachxXand eachyx, there exists ans˜pc-open set (F,E) containing x such thaty˜s˜pccl(F,E).
  3. For eachxX, {s˜pccl(F,E):x˜(F,E)˜s˜pcO(X)}=(x,E).

Proof

(1) ⇒ (2). Since (X,τ˜,E) is an s˜pcT2 space, then there exist disjoint s˜pc-open sets (F, E) and (G, E) containing x and y, respectively. Thus, (F,E)X\(G,E). Therefore, s˜pccl(F,E)X\(G,E). Hence, y˜s˜pccl(F,E).

(2) ⇒ (3). If possible for some yx, we have y˜s˜pccl(F,E) for every s˜pc-open set (F,E) containing x, which contradicts (2).

(3) ⇒ (1). Let x,yX and xy. Then, there exists an s˜pc-open set (F,E) containing x such that y˜s˜pccl(F,E). Let (G,E)=X\s˜pccl(F,E), then y˜(G,E) and x˜(F,E) and also (F,E)(G,E)=ϕ˜. Therefore, (X,τ˜,E) is an s˜pcT2 space.□

Proposition 4.25

A soft space(X,τ˜,E)iss˜pcT2if for each pair of distinct pointsx,yX, there exists ans˜pc-clopen set(F,E)containing one of them but not the other.

Proof

Let for each pair of distinct points x, yX, there exists an s˜pc-clopen set (F, E) containing x but not y, which implies that X\(F,E) is also an s˜pc-open set and y˜X˜\(F,E), since (F,E)X\(F,E)=ϕ˜, X is an s˜pcT2 space.□

5 Conclusion

In the last decade, the concept of soft topological spaces has been introduced. After that, several topological concepts are extended to the soft set theory, new definitions, new classes of soft sets and properties for soft continuous mappings between different classes of soft sets are introduced and studied we refer to [14,15,16,26,27]. Also, many types of soft separation axioms are investigated [17,18,28,29]. This paper continues the study of some strong types of soft separation axioms. In Sections 3 and 4, we present the notion of soft pcTi and soft pcTi spaces for i=0,1,2 and we get several characterizations and properties of these spaces. Also, we discuss the relationship among these spaces and other existing soft separation axioms.

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  • [1]

    D. Molodtsov, Soft set theory-first results, Comput. Math. Appl. 37 (1999), 19–31.

    • Crossref
    • Export Citation
  • [2]

    A. Ghareeb, Soft weak Baire spaces, J. Egyptian Math. Soc. 26 (2018), no. 3, 395–405, .

    • Crossref
    • Export Citation
  • [3]

    A. H. Zakari, A. Ghareeb, and S. Omran, On soft weak structures, Soft Comput. 21 (2017), 2553–2559, .

    • Crossref
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