## 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 *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*_{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 (*F*, *E*) is called a soft set over *X*, where *F* is a mapping *X* with the parameter set *E* is denoted by

## 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.

A soft set *X* is said to be an empty soft set denoted by *X* is said to be an absolute soft set denoted by

The complement of a soft set

It is clear that

For two soft sets

and$E\subseteq B$ - for all
,$e\in E$ .$F\left(e\right)\subseteq G\left(e\right)$

We write

The union of two soft sets of *X* is the soft set

The intersection *X*, denoted

Let *X* for which *X* and *x* belongs to the soft set

The soft set

We say that

Two soft points

From Definitions 2.6 and 2.7, it is clear that:

is the smallest soft set containing$(x,E)$ *x*.- If
, then$x\tilde{\in}(F,E)$ and${x}_{e}\tilde{\in}(F,E)$ always.${x}_{e}\tilde{\in}(x,E)$ .$(F,E)=\bigsqcup \left\{\right({x}_{e},E):e\in E\}$

[4] Let *X* with a fixed set *E* of parameters. Then,

and$\tilde{\varphi}$ belong to$\tilde{X}$ .$\tilde{\tau}$ - The union of any number of soft sets in
belongs to$\tilde{\tau}$ .$\tilde{\tau}$ - The intersection of any two soft sets in
belongs to$\tilde{\tau}$ .$\tilde{\tau}$

The triplet *X*. The members of

[4] Let

- The soft closure of
is the soft set$(G,E)$ $\tilde{s}cl(G,E)=\sqcap \left\{\right(K,B\left)\tilde{\in}SC\right(\tilde{X}):(G,E)\u2291(K,B\left)\right\}.$ - The soft interior of
is the soft set$(G,E)$

[8] Let

[4] *Let**be a soft subspace of a soft topological space**and**Then,*

*If*$(F,E)$ *is a soft open set in*$\tilde{Y}$ *and* ,$\tilde{Y}\tilde{\in}\tilde{\tau}$ *then* .$(F,E)\tilde{\in}\tilde{\tau}$ $(F,E)$ *is a soft open set in*$\tilde{Y}$ *if and only if*$(F,E)=\tilde{Y}\sqcap (G,E)$ *for some* .$(G,E)\tilde{\in}\tilde{\tau}$ $(F,E)$ *is a soft closed set in*$\tilde{Y}$ *if and only if*$(F,E)=\tilde{Y}\sqcap (H,E)$ *for some soft closed*$(H,E)$ *in* .$\tilde{X}$

[22] A soft subset

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

[21] *A subset**of a soft topological space**is a soft pre-open set if and only if there exists a soft open set**such that*

[21] *Let**where**is a soft topological space and**is a soft pre-open subspace of**Then**if and only if*

[23] *If**is soft open and**is soft pre-open in**, then**is soft pre-open.*

[23] *Let**, where**is a soft topological space and**is a soft subspace of**If**, then*

[25] A soft topological space

- Soft
, if for each pair of distinct soft points${T}_{0}$ *x*, , there exist soft open sets$y\tilde{\in}X$ and$(F,E)$ such that either$(G,E)$ and$x\tilde{\in}(F,E)$ or$y\tilde{\notin}(F,E)$ and$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${T}_{1}$ *x*, , there exist two soft open sets$y\in X$ and$(F,E)$ such that$(G,E)$ but$x\tilde{\in}(F,E)$ and$y\tilde{\notin}(F,E)$ but$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${T}_{2}$ *x*, , there exist two disjoint soft open sets$y\in X$ and$(F,E)$ containing$(G,E)$ *x*and*y*, respectively.

In [11], Bayramov and Aras redefined soft

[11] A soft topological space

- Soft
, if for each pair of distinct soft points${T}_{0}$ ,${x}_{e}$ , there exist soft open sets${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ and$(F,E)$ such that either$(G,E)$ and${x}_{e}\tilde{\in}(F,E)$ or${y}_{e\prime}\tilde{\notin}(F,E)$ and${y}_{e\prime}\tilde{\in}(G,E)$ .${x}_{e}\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${T}_{1}$ ,${x}_{e}$ , there exist two soft open sets${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ and$(F,E)$ such that$(G,E)$ but${x}_{e}\tilde{\in}(F,E)$ and${y}_{e\prime}\tilde{\notin}(F,E)$ but${y}_{e\prime}\tilde{\in}(G,E)$ .${x}_{e}\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${T}_{2}$ ,${x}_{e}$ , there exist two disjoint soft open sets${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ and$(F,E)$ containing$(G,E)$ and${x}_{e}$ , respectively.${y}_{e\prime}$

[11]

*Every soft*${T}_{2}$ *-space*$\Rightarrow $ *soft* -${T}_{1}$ *space*$\Rightarrow $ *soft* -${T}_{0}$ *space*.*A soft topological space*$(X,\tilde{\tau},E)$ *is soft*${T}_{1}$ *if and only if each soft point is soft closed.*

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

[4] If for every *X*, there exist two soft open sets

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

[9] *If for every**and every soft closed set*

[24] A soft pre-open set *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

[21] Let

- The soft pre-closure of
is the soft set$(G,E)$ $\tilde{s}p\text{cl}(G,E)=\sqcap \left\{\right(K,B\left)\tilde{\in}\tilde{s}\text{PC}\right(\tilde{X}):(G,E)\u2291(K,B\left)\right\}.$ - The soft pre-interior of
is the soft set$(G,E)$

[13] Let

- A soft point
is said to be a soft${x}_{e}\tilde{\in}\tilde{X}$ *p*_{c}-limit soft point of a soft set if for every soft$(F,E)$ *p*_{c}-open set containing$(G,E)$ ,${x}_{e}$ .The set of all soft$(G,E)\sqcap \left[\right(F,E)\backslash \{{x}_{e}\left\}\right]\ne \tilde{\varphi}$ *p*_{c}-limit soft points of is called the soft$(F,E)$ *p*_{c}-derived set of and is denoted by$(F,E)$ .$\tilde{s}{p}_{c}D(F,E)$ - The soft
*p*_{c}-closure of is the soft set$(G,E)$ $\tilde{s}{p}_{c}cl(G,E)=\sqcap \left\{\right(K,B\left)\tilde{\in}\tilde{s}{P}_{c}C\right(\tilde{X}):(G,E)\u2291(K,B\left)\right\}.$ - The soft
*p*_{c}-interior of is the soft set$(G,E)$

[24] *If**and**is soft clopen, then**if and only if*

[24] *Let**and**be soft clopen. If**then*

[13] *Let**. If**is soft clopen, then*

[21] A soft topological space

- Soft
, if for each pair of distinct soft points${P}_{0}$ *x*, , there exist soft pre-open sets$y\in X$ and$(F,E)$ such that either$(G,E)$ and$x\tilde{\in}(F,E)$ or$y\tilde{\notin}(F,E)$ and$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${P}_{1}$ *x*, , there exist two soft pre-open sets$y\in X$ and$(F,E)$ such that$(G,E)$ but$x\tilde{\in}(F,E)$ and$y\tilde{\notin}(F,E)$ but$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ - Soft
, if for each pair of distinct soft points${P}_{2}$ *x*, , there exist two disjoint soft pre-open sets$y\in X$ and$(F,E)$ containing$(G,E)$ *x*and*y*, respectively.

## 3 Soft ${p}_{c}-{T}_{i}$ spaces for $(i=0,1,2)$

In this section, we define

A soft topological space

, if for each pair of distinct soft points$\tilde{s}{p}_{c}-{T}_{0}$ ,${x}_{e}$ , there exist${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ such that$(G,E)$ and${x}_{e}\tilde{\in}(F,E)$ or${y}_{e\prime}\tilde{\notin}(F,E)$ and${y}_{e\prime}\tilde{\in}(G,E)$ .${x}_{e}\tilde{\notin}(G,E)$ , if for each pair of distinct soft points$\tilde{s}{p}_{c}-{T}_{1}$ ,${x}_{e}$ , there exist two${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ such that$(G,E)$ but${x}_{e}\tilde{\in}(F,E)$ and${y}_{e\prime}\tilde{\notin}(F,E)$ but${y}_{e\prime}\tilde{\in}(G,E)$ .${x}_{e}\tilde{\notin}(G,E)$ , if for each pair of distinct soft points$\tilde{s}{p}_{c}{T}_{2}$ ,${x}_{e}$ , there exist two disjoint${y}_{e\prime}\tilde{\in}\text{SP}{\left(X\right)}_{E}$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ containing$(G,E)$ and${x}_{e}$ , respectively.${y}_{e\prime}$

*A soft topological space**is**if and only if the**-closure of any two soft points is distinct.*

Let

Conversely, suppose that

*If**is**space, then**, for each pair of distinct soft points*

Let

*Every soft**space is soft*

We shall prove the case when

Let

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

Let

*Every soft**space is**for*

If a soft space is soft

*A soft topological space**is soft**if and only if it is*

Follows directly from Propositions 3.4 and 3.6.□

*A space**is**if and only if every soft point of the space**is an**-closed set.*

Let

Conversely, let

*A soft topological space**is an**space if and only if**, for each*

Let

Conversely, let

*For a soft space**, the following statements are equivalent.*

$\tilde{X}$ *is* ,$\tilde{s}{p}_{c}-{T}_{2}$ *For each*${x}_{e}\tilde{\in}\tilde{X}$ *and each*${x}_{e}\ne {y}_{e\prime}$ *, there exists an*$\tilde{s}{p}_{c}$ *-open set*$(F,E)$ *of*${x}_{e}$ *such that* .${y}_{e\prime}\tilde{\notin}\tilde{s}{p}_{c}\text{Cl}(F,E)$ *For each* ,${x}_{e}\tilde{\in}\tilde{X}$ .$\sqcap \left\{\tilde{s}{p}_{c}cl\right(F,E):{x}_{e}\tilde{\in}(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}=\left\{{x}_{e}\right\}$

(1) ⇒ (2). Since

(2)

(3)

*A soft space**is**if for each pair of distinct soft points**there exists an**-clopen set**containing one of them but not the other.*

Let for each pair of distinct soft points *F*,*E*) containing

*Every soft clopen subspace of an**space is an**space for*

We prove only the case for

*If for each**, there exists a soft clopen set**containing**such that**is* an *subspace of**, then the soft space**is also* an *space for*

We prove only the case for the

## 4 Soft ${p}_{c}-{T}_{i}^{\u204e}$ spaces for ($i=0,1,2$ )

In this section, we define *i* = 0, 1, 2) are discussed.

A soft topological space

, if for each pair of distinct points$\tilde{s}{p}_{c}{T}_{0}^{\u204e}$ , there exist$x,y\in X$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ such that$(G,E)$ and$x\tilde{\in}(F,E)$ or$y\tilde{\notin}(F,E)$ and$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ , if for each pair of distinct points$\tilde{s}{p}_{c}\text{-}{T}_{1}^{\u204e}$ , there exist two$x,y\in X$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ such that$(G,E)$ but$x\tilde{\in}(F,E)$ and$y\tilde{\notin}(F,E)$ but$y\tilde{\in}(G,E)$ .$x\tilde{\notin}(G,E)$ , if for each pair of distinct soft points$\tilde{s}{p}_{c}\text{-}{T}_{2}^{\u204e}$ , there exist two disjoint$x,y\in X$ -open sets$\tilde{s}{p}_{c}$ and$(F,E)$ containing$(G,E)$ *x*and*y*, respectively.

*Every soft**space is a soft**space for*

*If a soft topological space**is a**space, then it is soft*

Let *x* and *y*, respectively. Therefore,

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

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

Let

Then, it can be checked that

Let

Let

Let *X* be any infinite set, *E* consists of infinite parameters and let

*X*and

*x*is

*y*which is disjoint from

From Examples 4.6 and 4.7, we conclude that

Let

Let *y* be two distinct points of *X* and

Let *X* be any infinite set, *X* for each *X*. It can be easily shown that this space is

*Let**be a soft topological space and**such that**. If there exist**-open sets**and**such that**and**or**and**then**is*

Let

In Example 4.6, it can be easily seen that the converse of Proposition 4.10 is not true in general. Because

*A soft topological space**is**if and only if for each pair of distinct points*

Let *X*. There exists an *x* but not *y*. Then, *x* but contains *y*. Since *y*,

Conversely, suppose that *z* be a point of *X* such that *x* belongs to the *y* does not belong to it.□

*For each pair of distinct points**. If a soft topological space**is*

Let *x*, and does not contain the other, which implies that

*If a soft topological space**is**, then it is a soft**(resp., soft**) space due to* [*10*].

Let *X*, there exists an *x*. Since *y* but not *x*. Therefore,

The other proof is similar.□

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

In Example 4.8, we can see that

*Let**be a soft topological space and**such that**. If there exist**-open sets**and**such that**and**then**is*

It is similar to the proof of Proposition 4.10.□

In Example 4.6, it is easy to see that the converse of Proposition 4.16 is not true in general.

*If**is an**space and**, then for each**-open set**with**. The following statements are true:*

.$(x,E)\u2291\sqcap \left\{\right(F,E):x\tilde{\in}(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}$ *For all* ,$y\ne x$ *we have* .$y\tilde{\notin}\sqcap \left\{\right(F,E):x\tilde{\in}(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}$

- (1)Since
, then by Remark 2.8, it is clear that$x\tilde{\in}\sqcap \left\{\right(F,E):(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}$ .$(x,E)\u2291\sqcap \left\{\right(F,E):(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}$ - (2)Let
for$x\ne y$ , then there exists an$x,y\in X$ -open set$\tilde{s}{p}_{c}$ such that$(G,E)$ and$x\tilde{\in}(G,E)$ . This implies that$y\tilde{\notin}(G,E)$ for some$y\notin G\left(e\right)$ , and we have$e\in E$ . Therefore,$y\notin \cap \left\{F\right(e):e\in E\}$ .□

*Every singleton soft set of a soft topological space**is an**-closed set, if and only if**is* an *space.*

Suppose that

Conversely, suppose that *x* but not *y*. Thus,

*If every soft point of a soft topological space**is an**-closed set, then**is* an *space.*

Let *x* ≠ *y*. Now, for *x* ∈ *X*, *y*. Thus,

*A soft topological space**is an**space if and only if**, for each x* ∈ *X*.

Let

A soft topological space *x**X*.

*Let**be a soft topological space and**If**is an**space, then*

Assume that there exist

*The following statements are equivalent for a soft topological space*

$(X,\tilde{\tau},E)$ *is an*$\tilde{s}{p}_{c}-{T}_{2}^{\u204e}$ *space.**For each*$x\in X$ *and each*$y\ne x$ *, there exists an*$\tilde{s}{p}_{c}$ *-open set*(*F*,*E*)*containing x such that* .$y\tilde{\notin}\tilde{s}{p}_{c}cl(F,E)$ *For each* ,$x\in X$ .$\sqcap \left\{\tilde{s}{p}_{c}cl\right(F,E):x\tilde{\in}(F,E\left)\tilde{\in}\tilde{s}{p}_{c}O\right(X\left)\right\}=(x,E)$

(1) ⇒ (2). Since *F*, *E*) and (*G*, *E*) containing *x* and *y*, respectively. Thus,

(2) ⇒ (3). If possible for some *x*, which contradicts (2).

(3) ⇒ (1). Let *x* such that

*A soft space**is**if for each pair of distinct points**there exists an**clopen set**containing one of them but not the other.*

Let for each pair of distinct points *x*, *F*, *E*) containing *x* but not *y*, which implies that *X* is an

## 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

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