Demonstratio Mathematica 2021; 54: 196–211

Research Article

Tareq M. Al-shami, Adnan Tercan, and Abdelwaheb Mhemdi* New soft separation axioms and fixed soft points with respect to total belong and total non-belong relations

https://doi.org/10.1515/dema-2021-0018 received February 17, 2021; accepted May 26, 2021

Abstract: In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with respect to ordinary points, namely tt-soft pre Tii (=0, 1, 2, 3, 4) and tt-soft pre-regular spaces. The motivations to use these relations are, first, cancel the constant shape of soft pre- open and pre-closed subsets of soft pre-regular spaces, and second, generalization of existing comparable properties on classical topology. With the help of examples, we show the relationships between them as well as with soft pre Tii (=0, 1, 2, 3, 4) and soft pre-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a tt-soft pre-regular space and demonstrate that it guarantees the equivalence of tt-soft pre Tii (=0, 1, 2). Furthermore, we investigate the behaviors of these soft separation axioms with the concepts of product and sum of soft spaces. Finally, we introduce a concept of pre-fixed soft point and study its main properties.

Keywords: soft pre-open set, tt-soft pre Ti-space, extended soft topology, soft hyperconnected space, additive property, topological property, soft pre-compact space, pre-fixed soft point

MSC 2020: 47H10, 54A05, 54C15, 54D10, 54D15

1 Introduction

Soft set was established by Molodtsov [1], in 1999, as a new technique to approach real-life problems that suffer from vagueness and uncertainty. He investigated merits of soft sets compared with probability theory and fuzzy set theory. Then, the soft set-theoretic concepts have been introduced and investigated by several researchers, and many applications of soft sets have been given on the disciplines of decision-making problems [2], engineering [3] and medical sciences [4]. In 2011, Shabir and Naz [5] as well as Çağman et al. [6] employed soft sets to introduce the concept of soft topological space. However, they used two different methods to define soft topology. On one hand, Shabir and Naz formulated soft topology on the collection of soft sets over a universal crisp set with a fixed set of parameters. On the other hand, Çağman et al. formulated soft topology on the collection of soft sets over an absolute soft set with different sets of parameters that are subsets of the universal set of parameters. In this paper, we continue studying soft topology using the definition given by Shabir and Naz. Many scholars explored several (crisp) topological concepts via soft topological spaces and examined the validity of some known topological results on soft topological spaces. Soft compactness was defined and

 * Corresponding author: Abdelwaheb Mhemdi, Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia, e-mail: [email protected] Tareq M. Al-shami: Department of Mathematics, Sana’a University, Sana’a, Yemen, e-mail: [email protected] Adnan Tercan: Department of Mathematics, Hacettepe University, Ankara, Turkey, e-mail: [email protected]

Open Access. © 2021 Tareq M. Al-shami et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. New soft separation axioms and fixed soft points  197 studied in [7], in 2012. Then, Hida [8] distinguished between two types of soft compactness depending on the belong relation. Al-shami et al. [9] studied almost soft compact and approximate soft compact spaces as extensions of a soft compact space. Al-shami and El-Shafei [10] utilized soft pre-open sets to generalize soft compactness. In [11], the behavior of soft closed sets in a soft Hausdorff was revised, and in [12] many allege results of soft separation axioms were corrected with the help of interesting counterexamples. Babitha and John explored some notions on soft topological spaces in [13] and introduced some methods to generate soft topology from soft relations in [14]. The concept of soft somewhere dense sets was introduced in [15]. Then, it was applied to define new types of soft maps in [16]. The relations of belong and non-belong given in [5] were utilized in the studies of soft set and soft topology. However, the authors of [17], in 2018, described the relations between an ordinary point and soft set by two new relations, namely partial belong and total non-belong. According to these new relations, many soft topological concepts and notions were reformulated; as an example, see the different types of soft separation axioms given in [17–19]. Wardowski [20] tackled the fixed point in the setup of soft topological spaces. Al-shami and Abo-Tabl [21] generalized Wardowski’s fixed point using the class of soft α-open sets. Soft point is one of the important notions that assists to prove some soft topological properties. Zorlutuna and Çakir [22] gave the first definition of a soft point, then, it was redefined by Das and Samanta [23] as well as by Nazmul and Samanta [24]. Novel kinds of soft topologies, namely enriched and extended soft topology, were introduced in [7] and [24], respectively. The equivalence between these two topologies has been recently proved by Al-shami and Kočinac [25]. The layout of this article is as follows: We recall in Section 2 the main principles to make this work self- contained. Section 3 introduces the concepts of tt-soft pre Tii (=0, 1, 2, 3, 4) and tt-soft pre-regular spaces, which are formulated using total belong and total non-belong relations. The interrelations between them and their essential properties are established with the aid of counterexamples. Section 4 explores a pre- fixed soft point theorem and studies main properties. In particular, we derive the sufficient conditions that keep pre-fixed soft points between a soft topological space and its parametric topological spaces.

2 Preliminaries

We assign this part to mention the necessary concepts and properties to comprehend this manuscript.

2.1 Soft sets

Definition 2.1. [1] For a nonempty set X and a set of parameters M, a map G from M to the power set 2X of

X is said to be a soft set over X. It is denoted by GM. X We describe a soft set GM by ordered pairs: GM ={(mGm,: ( )) m ∈ M and2 Gm ( )∈ }. The family of soft sets defined over X with M as a set of parameters is denoted by S(XM).

∼ Definition 2.2. [26] A soft set GM is said to be a subset of a soft set HM. We write GMM⊆ H , reads as GM is a subset of HM,ifG()⊆(mHm) for each mM∈ . ∼ ∼ We write GMM= H if GMM⊆ H and HGM ⊆ M.

Definition 2.3. [1,17] Let GM be a soft set over X and xX∈ . We write that: (i) x ∈ GM (or x totally belongs to GM or GM totally contains x),ifx ∈(Gm) for each mM∈ . (ii) x ∉ GM (or x does not partially belong to GM or GM does not partially contain x),ifx ∉(Gm) for some mM∈ . (iii) x ⋐ GM (or x partially belongs to GM or GM partially contains x),ifx ∈(Gm) for some mM∈ . (iv) x ⋐̸ GM (or x does not totally belong to GM or GM does not totally contain x),ifx ∉(Gm) for each mM∈ . 198  Tareq M. Al-shami et al.

Definition 2.4. [ ] c c X 27 The complement of a soft set GM is a soft set GM, where G :2M → is a mapping defined by Gc()=⧹(mXGm) for all mM∈ .

Definition 2.5. [17,23,24,28] A soft set ()GM, over X is said to be (i) A null soft set, denoted by Φ,ifG()=∅m for each mM∈ . Its complement is called an absolute soft set. ( ) x ii A soft point Sm if there are mM∈ and xX∈ such that G()={mx} and G(′)=∅m for each mMm′∈ ⧹{ }. x We write that Sm ∈ GM if x ∈(Gm). (iii) A stable soft set if there is a subset A of X such that G()=m A for each mM∈ , and it is denoted by A. In particular, it is denoted by xM if Ax={ }. (iv) An uncountable (resp. infinite) soft set if G(m) is uncountable (resp. infinite) for some mM∈ . Otherwise, it is said to be countable (resp. finite).

Definition 2.6. [27–29] Let GM and HM be two soft sets over X.  X (i) Their intersection, denoted by GMM⋂ H , is a soft set UM, where a mapping U :2E → is given by U()=()⋂(mGmHm).  X (ii) Their union, denoted by GMM⋃ H , is a soft set UM, where a mapping U :2E → is given by U()=()⋃(mGmHm).

(iii) The Cartesian product of GM and HF , denoted by G × HMF× ,isdefined as (GHmfGmHf×)()=()×(, ) for each (mf, )∈ M × F.

Definition 2.7. [4] A soft mapping between S(XA) and S(YB) is a pair ()fϕ, , denoted also by fϕ, of mappings such that f : XY→ , ϕA: → B. Let GA and HB be subsets of S(XA) and S(YB), respectively. Then, the image of GA and pre-image of HB are defined as follows: ( ) 1 i fϕA()=(())GfG ϕ B is a subset of S(YB) such that fϕ()()=⋃Gbaϕ∈()− b fGa (()) for each b ∈ B. ( ) −−11 −1 −1 ii fϕ ()=(())HfHB ϕ A is a subset of S(XA) such that fϕ ()()=Ha f ((()) Hϕa) for each aA∈ .

Definition 2.8. [22] A soft map fϕA: SX()→( SY B) is said to be injective (resp. surjective, bijective) if ϕ and f are injective (resp. surjective, bijective).

2.2 Soft topology

Definition 2.9. [5] A family σ of soft sets over X under a fixed set of parameters M is said to be a soft topo- logy on X if it satisfies the following: (i) X and Φ are members of σ. (ii) The intersection of a finite number of soft sets in σ is a member of σ. (iii) The union of an arbitrary number of soft sets in σ is a member of σ.

The triple ()XσM,, is called a soft topological space. We call a subset of X a soft open set if it is a member in σ and call its complement a soft closed set.

Proposition 2.10. [5] In ()XσM,, , a family σmM={Gm ( ): G ∈ σ} is a classical topology on X for each mM∈ . σm is called a parametric topology, and ()Xσ, m is called a parametric topological space.

 Definition 2.11. [5] Let ()XσM,, be a soft topological space and ∅ ≠⊆Y X.AfamilyσYMM={YGG ⋂: ∈ σ} is called a soft relative topology on Y and the triple ()Yσ,,Y Mis called a soft subspace of ()XσM,, .

∼ Definition 2.12. [30] A subset GM of ()XσM,, is called soft pre-open if GMM⊆(()int cl G ). New soft separation axioms and fixed soft points  199

Theorem 2.13. [30] (i) Every soft open set is soft pre-open. (ii) The family of soft pre-open sets is closed under arbitrary soft union.

pre Definition 2.14. [30] Let GM be a subset of()XσM,, . Then, GM is the intersection of all soft pre-closed sets containing GM.

pre ff   - x pre It is clear that x ∈ GM i GMM⋂≠U Φ for each soft pre open set UM totally containing x; and Sm ∈ GM ff   - x i GMM⋂≠U Φ for each soft pre open set UM containing Sm.

Definition 2.15. [31](X,,σ M) which does not contain disjoint soft open sets is called soft hyperconnected.

Theorem 2.16. [10] The family of soft pre-open sets in a soft hyperconnected space is closed under finite soft intersection.

The above result will assist us to initiate some properties of soft pre-separation axioms and soft pre-compact spaces, see Theorem 3.15 and Proposition 3.22. It means that the class of soft pre-open sets in a soft hyper- connected space ()XσM,, constructs a new soft topology σpre over X containing σ.

Proposition 2.17. [10] Let Y be soft open subset of ()XσM,, . Then, 1. If ()HM, is a soft pre-open and Y is a soft open in ()XσM,, , then (HM,,)⋂( YM) is a soft pre-open subset

of ()Yσ,,Y M.

2. If Y is a soft open in ()XσM,, and ()HM, is a soft pre-open in ()Yσ,,Y M, then ()HM, is a soft pre-open subset of ()XσM,, .

Definition 2.18. [30,32](X,,σ M) is said to be:

(i) Soft pre T0 if there is a soft pre-open set UM for every x ≠∈y X such that x ∈ UM and y ∉ UM;ory ∈ UM and x ∉ UM. (ii) Soft preT1 if there are two soft pre-open setsUM andVM for every x ≠∈y X such that x ∈ UM and y ∉ UM; and y ∈ VM and x ∉ VM. (iii) Soft pre T2 if there are two disjoint soft pre-open sets UM and VM for every x ≠∈y X such that x ∈ GM and y ∈ FM. (iv) Soft pre-regular if for every soft pre-closed set HM and xX∈ such that x ∉ HM, there are two disjoint ∼ soft pre-open sets UM and VM such that HUMM⊆ and x ∈ VM. (v) Soft pre-normal if for every two disjoint soft pre-closed sets HM and FM, there are two disjoint soft pre- ∼ ∼ open sets UM and VM such that HUMM⊆ and FMM⊆ V . (vi) Soft pre T3 (resp. soft pre T4) if it is both soft pre-regular (resp. soft pre-normal) and soft pre T1-space.

Definition 2.19. [ ] - - 10 A family {}GiIiM : ∈ of soft pre open subsets of ()XσM,, is said to be a soft pre open    cover of X if X =⋃iI∈ GiM .

Definition 2.20. [10](X,,σ M) is said to be: ( ) x y  - x i Soft pre T2′ if for every Sm ≠∈SXm′ , there are two disjoint soft pre open sets UM and VM containing Sm y and Sm′, respectively. (ii) Soft pre-compact if every soft pre-open cover of X has a finite subcover.

Proposition 2.21. [10] ( ) - - - i A soft pre compact subset of a soft pre T2′ space is soft pre closed. (ii) A stable soft pre-compact subset of a soft pre T2-space is soft pre-closed. 200  Tareq M. Al-shami et al.

Definition 2.22. ⋆- ( - [ ]) gXσMYτMφ :,,()→( ,,) is called soft pre continuous or soft pre irresolute 10 if the inverse image of each soft pre-open set is soft pre-open.

Proposition 2.23. [10] The soft pre⋆-continuous image of a soft pre-compact set is soft pre-compact.

The definitions of soft pre-continuous, soft pre-open, soft pre-closed and soft pre-homeomorphism map- pings were introduced in [30].

Definition 2.24. (See, [25]) A soft topology σ on X is said to be extended if σ ={GGmσMm:for ( )∈ each mM ∈ }; equivalently, if all soft sets GM such that G()=∅m or X are members of σ.Itisalsocalledanenrichedsoft topology.

For more details of extended (or enriched) soft topologies, see [25].

Theorem 2.25. [25] A subset ()FM, of an extended soft topological space ()XσM,, is soft pre-open if and only if each m-approximate element of ()FM, is pre-open.

Proposition 2.26. [33] Let {()∈XσMii,, : i I} be a family of pairwise disjoint soft topological spaces and X =⋃iI∈ Xi. Then, the collection ∼  σ ={(G,:, M )⊆ X ( G M )⋂ Xiii is a soft open set in ( X ,, σ M ) for every i ∈ I} defines a soft topology (called sum of soft topological spaces) on X with a fixed set of parameters M.

Theorem 2.27. [ ] ∼  - ( - ) 33 A soft set (GM, )⊆⊕iI∈ Xi is soft pre open resp. soft pre closed in ()⊕iI∈ XσMi,, if and  only if all (GM, )⋂ Xi are soft pre-open (resp. soft pre-closed) in ()XσMii,, .

Proposition 2.28. [ ]  n  x 20 Let gXσMXσMφ :,,()→( ,,) be a soft map such that ⋂n∈ gXφ( ) is a soft point Sm. x fi Then, Sm is a unique xed point of gφ.

Theorem 2.29. [22] Let ()XσA,, and ()YτB,, be two soft topological spaces and Ω:={GFGABA × ∈σ and FB ∈ τ}. Then, the family of all arbitrary union of elements of Ω is a soft topology over XY× under a fixed set of parameters AB× .

Lemma 2.30. Let ()GM, and ()HN, be two subsets of ()XσM11,, and ()XσN22,, , respectively. Then, (i) clG(,,,, M )× clHN ( )= clG (( M )×( H N )). (ii) int( G,,,, M )× int ( H N )= int (( G M )×( H N )).

3 Pre-soft separation axioms

Herein, we define new family of soft spaces called tt-soft pre Ti (i = 0, 1, 2, 3, 4) and tt-soft pre-regular spaces, where tt is the abbreviation of the initials of the phrases “total belong” and “total non-belong” used in the definitions of these spaces. We provide several counterexamples to elucidate the relationships between them and validate the obtained findings. Moreover, we explain whether they are hereditary, topological, and additive properties or not.

Definition 3.1. ()XσM,, is said to be:

(i) tt-soft pre T0 if there exists a soft pre-open set UM for every xy≠ in X such that x ∈ UM and y ⋐̸ UM or y ∈ UM and x ⋐̸ UM . New soft separation axioms and fixed soft points  201

(ii) tt-soft pre T1 if there exist soft pre-open sets UM and VM for every xy≠ in X such that x ∈ UM and y ⋐̸ UM; and y ∈ VM and x ⋐̸ VM. (iii) tt-soft preT2 if there exist two disjoint soft pre-open setsUM andVM for every xy≠ in X such that x ∈ UM and y ⋐̸ UM; and y ∈ VM and x ⋐̸ VM. (iv) tt-soft pre-regular if for every soft pre-closed set HM and xX∈ such that x ⋐̸ HM , there exist disjoint soft ∼ pre-open sets UM and VM such that HUMM⊆ and x ∈ VM. (v) tt-soft pre T3 (resp. tt-soft pre T4) if it is both tt-soft pre-regular (resp. soft pre-normal) and tt-soft pre T1.

Remark 3.2. Note that if FM and GM are disjoint soft sets, then the two relations x ∈ FM and x ⋐̸ GM are equivalent. This means that the concepts of tt-soft pre T2 and soft pre T2-spaces are equivalent. We can say that ()XσM,, is tt-soft pre T2 if there exist two disjoint soft pre-open setsUM andVM for every x ≠∈y X such that x ∈ UM and y ∈ VE.

Remark 3.3. The soft pre-regular spaces imply a strict condition on the shape of soft pre-open and soft pre- closed subsets. To clarify this matter, let FM be a soft pre-closed set such that x ∉ HM. Then, we obtain two cases as follows: (i) There are mm, ′∈ Msuch that x ∉(Hm) and x ∈(′Hm). This case is out of the question because there are not two disjoint soft sets UM and VM containing x and HM, respectively. (ii) For all mM∈ , we have x ∉(Hm). This means that HM is stable.

We derive from (i) and (ii) above that all soft pre-closed and soft pre-open subsets of a soft pre-regular space are stable. But, this property does not hold on the tt-soft pre-regular spaces because we replace a partial non-belong relation by a total non-belong relation. Therefore, a tt-soft pre-regular space need not be stable.

Proposition 3.4.

(i) Every tt-soft pre Ti-space is soft pre Ti for i = 0, 1, 4. (ii) Every soft pre-regular space is tt-soft pre-regular.

(iii) Every soft pre T3-space is tt-soft pre T3.

Proof. The proofs of (i) and (ii) are straightforward.

(iii):Itsuffices to prove that a soft preTi-space is tt-soft preTi when ()XσM,, is soft pre-regular. Suppose x ≠∈y X. Then, there exist two soft pre-open sets UM and VM such that x ∈ UM and y ∉ UM; and y ∈ VM and x ∉ VM. Since UM and VM are soft pre-open subsets of a soft pre-regular space, they are stable. So, y ⋐̸ UM and x ⋐̸ VM. Hence, ()XσM,, is tt-soft pre T1, as required. □

We supply the following counterexamples to clarify that the converse of the above proposition fails.

Example 3.5.   Let M ={mm12, }. A family σ ==…{Φ,XG ,iM : i 1, 2, , 5} is a soft topology on X ={xy, }, where

GmxmX11M ={(,,,; { })( 2 )}

GmymX21M ={(,,,; { })( 2 )}

Gm31M ={(,, ∅)( mX 2 ,; )}

Gmmx41M ={(,, ∅)( 2 , { })} and

Gm51M ={(,, ∅)( my 2 , { })} .

It can be checked that a soft subset of ()XσM,, is soft open iff it is soft pre-open. Then, the soft sets {(mx12,,, { }) ( mx { })} and {(my12,,, { }) ( my { })} are not soft pre-open subsets of ()XσM,, . Therefore, it is not tt-soft pre T0. However, it is clear that ()XσM,, is a soft pre T1-space. Also, it is soft pre-normal. Therefore, it is soft pre T4. 202  Tareq M. Al-shami et al.

Example 3.6.   Let M ={mm12, }. A family σ =={Φ,XG ,iM : i 1, 2, 3, 4} is a soft topology over X ={xy, }, where

Gmxmx11M = {(,,, { }) ( 2 { })} ;

Gmymy21M = {(,,, { }) ( 2 { })} ;

GmxmX31M ={(,,, { })( 2 )} and

Gmmy41M ={(,, ∅)( 2 , { })} .

Since there exists an unstable soft pre-open subset of ()XσM,, , ()XσM,, is not soft pre-regular.

Therefore, it is not soft pre T3. However, ()XσM,, is a tt-soft pre-regular space. Also, it is a tt-soft pre T1-space. Hence, it is a tt-soft pre T3-space.

Before we display the relationship between tt-soft pre Ti-spaces, we will prove the following helpful lemma.

Lemma 3.7. ()XσM,, is a tt-soft pre T1-space if and only if xM is soft pre-closed for every xX∈ .

Proof. - Necessity: For each yi ∈{Xx\ }, there is a soft pre open set GiM such that yi ∈ GiM and x ⋐̸ GiM . Therefore,   - X\{}=⋃xGmiI∈ i ( ) and x ⋐⋃̸ iI∈ Gm i ( ) for each mM∈ . Thus, ⋃iI∈ GXxiM ={\ } is soft pre open. Hence, xM is soft pre-closed. ffi - c c Su ciency: Let xy≠ . By hypothesis, xM and yM are soft pre closed sets. Then, xM and yM are soft - c c c c - pre open sets such that x ∈(yM ) and y ∈(xM ). Obviously, y ⋐(̸ yM ) and x ⋐(̸ xM ) . Hence, ()XσM,, is tt soft pre T1. □

Proposition 3.8. For i = 1, 2, 3, 4, a tt-soft pre Ti-space is tt-soft pre Ti−1.

Proof. For i = 3, let xy≠ in a tt-soft pre T3-space ()XσM,, . Then, xM is soft pre-closed. Since y ⋐̸ xM and ∼ ()XσM,, is tt-soft pre-regular, there are disjoint soft pre-open sets GM and FM such that xMM⊆ G and y ∈ FM. Therefore, ()XσM,, is tt-soft pre T2. For i = 4, let HM be a soft pre-closed set such that x ⋐̸ HM , where xX∈ . Since ()XσM,, is tt-soft pre T1,  xM is soft pre-closed. Since xMM⋂=H Φ and ()XσM,, is soft pre-normal, we can find two disjoint soft ∼ ∼ pre-open sets GM and FM satisfying that HGM ⊆ M and xMM⊆ F . Hence, ()XσM,, is tt-soft pre T3. □

The three counterexamples below elucidate that the converse of the above proposition fails.

Example 3.9. Let M ={mm12, }. A family σ ={Φ, Xmxmx , {(12 , {})( , , {})}} is a soft topology on X ={xy, }. - A family of all soft pre open subsets of ()XσM,, is S()⧹XGiMi{ M : = 1,2,3}, where

Gmymy11M = {(,,, { }) ( 2 { })} ;

Gmym21M ={(,,, { })( 2 ∅)} and

Gm31M ={(,, ∅)( my 2 , { })} .

Now, xy≠ . We have x ∈ {(mx12,,, { }) ( mx { })} and y ⋐̸ {(mx12,,, { }) ( mx { })}. Therefore, ()XσM,, is tt-soft pre T0. However, there does not exist a soft pre-open set such that x does not totally belong to it. Hence, ()XσM,, is not tt-soft pre T1.

It is well known that the identification between soft topological and (crisp) topological spaces occurs if E is a singleton. Then, the following counterexample is tt-soft pre T1, but not tt-soft pre T2.

Example 3.10. Let M ={m1}, X =⋃{Yxy, }, where Y is any infinite set and xy, are two distinct points not in Y . Let σ be the family of subsets of X such that (i) A ∈ σ if AY⊆ ; (ii) A ∈ σ if x or y belong to A such that Ac contains only a finite number of Y. New soft separation axioms and fixed soft points  203

Then, σ is a soft topology on X. Now, let a ≠∈b X. Then, we have three cases as follows: (i) a, bY∈ . Then, {}a and {}b are two soft pre-open sets containing a and b, respectively. (ii) aY∈ and bx= or y. Say, bx= . Then,{}a and{xYa}⋃[ ⧹{}]are two soft pre-open sets containinga and b, respectively. Similarly, if b ∈ Y and a = x or y. (iii) a = x and by= . Then, X⧹{a} and X⧹{b} are two soft pre-open sets containing a and b, respectively.

Hence, ()XσM,, is tt-soft pre T1. To prove ()XσM,, is not tt-soft pre T2,itsuffices to show that G ⋂≠∅H for each soft pre-open set G containing x and each soft pre-open set H containing y.We note that the soft pre-openness of G gives that the soft open set int(() cl G ) contains all points of Y except finitely many. So, from int(())⊆( cl G cl G), it follows that clG( ) contains all points of Y except possibly a finite number of points of Y . Next, we see that G is not soft closed. If possible, let G be soft closed. Then, Gc is soft open. Again, the soft closedness together with the soft pre-openness of G gives that G is soft open. Hence, by the definition of σ, Gc containing y contains only a finite number of Y . This contradicts the assertion that Gc is soft open. So, G is not soft closed. Now,[Gy⋃{ }]c being a subset of Y is soft open, by the definition of σ. So, G ⋃{y} is soft closed, and it is the smallest soft closed set containing G. Therefore, clG()= G ⋃{ b} which implies that G ⋃{b} and hence G contains all points of Y except possibly finitely many. Similarly, it can be shown that H contains all points of Y except possibly many. Hence, G ⋂≠∅H .

Example 3.11. To illustrate the remaining cases, it suffices to consider examples that satisfy a (crisp) pre

T2-space (resp. [crisp] pre T3-space) but not (crisp) pre T3 (resp. not [crisp] pre T4).

Now, we establish some findings concerning tt-soft pre Ti and tt-soft pre-regular.

pre Lemma 3.12. Let UM be a subset of ()XσM,, and xX∈ . Then, x ⋐̸ UM iff there exists a soft pre-open set VM  totally containing x such that UMM⋂=V Φ.

pre pre c  Proof. Let x ⋐̸ UM . Then, x ∈(UVM ) = M. So, UMM⋂=V Φ. Conversely, if there exists a soft pre-open   c pre c c set VM totally containing x such that UMM⋂=V Φ, then UM ⊆ VM. Therefore, UM ⊆ VM. Since x ⋐̸ VM , pre x ⋐̸ UM . □

Proposition 3.13. - - pre pre If ()XσM,, is a tt soft pre T0 space, then xM ≠ yM for every x ≠∈y X.

Proof. Let xy≠ in a tt-soft pre T0-space. Then, there is a soft pre-open setUM such that x ∈ UM and y ⋐̸ UM or   pre y ∈ UM and x ⋐̸ UM . Say, x ∈ UM and y ⋐̸ UM. Now, yM ⋂=UM Φ. So, by the above lemma, x ⋐̸ yM . But pre x ∈ xM . Hence, we obtain the desired result. □

Corollary 3.14. - - x pre y pre If ()XσM,, is a tt soft pre T0 space, then Sm ≠ Sm′ for all xy≠ and mm, ′∈ M.

Theorem 3.15. Let M be a finite set and ()XσM,, be soft hyperconnected. Then, ()XσM,, is a tt-soft pre  pre T1-space if and only if xMM=⋂{UxUσ: ∈ M ∈ } for each xX∈ .

Proof. ⇒: Let yX∈ . Then, for each x ∈⧹{Xy}, we have a soft pre-open set UM such that x ∈ UM and y ⋐̸ UM.  ∼ pre Therefore, y ⋐⋂{̸ UxMM: ⊆ U M ∈ σ}, as required. - ⇐: Let xy≠ and ∣M∣=n. Since y ⋐̸ xM, then for each j =…1, 2, , n there is a soft pre open set UiM such n y Ue x U ,,  that ∉(ij) and ∈ iM . Since ()XσMis soft hyperconnected, it follows from Theorem 2.16 that ⋂i=1UiM - n n - is a soft pre open set such that y ⋐⋂̸ i=1UiM and x ∈⋂i=1UiM . Similarly, we can get a soft pre open set VM such that y ∈ VM and x ⋐̸ VM. Thus, ()XσM,, is a tt-soft pre T1-space. □

Theorem 3.16. - - x - x  If ()XσM,, is an extended tt soft pre T1 space, then Sm is soft pre closed for all Sm ∈ X . 204  Tareq M. Al-shami et al.

 Proof. By Lemma 3.7, X\{x} is a soft pre-open set. Since()XσM,, is extended, a soft set HM, where Hm()=∅   and Hm(′)=X for each mm′≠ , is a soft pre-open set. Therefore, X\{}⋃xHM is soft pre-open. Thus,   c x - □ (Xx\{}⋃ HM ) = Sm is soft pre closed.

Corollary 3.17. If ()XσM,, is an extended tt-soft pre T1-space, then the intersection of all soft pre-open sets ∼ containing UM is exactly UM for each UM ⊆ X.

Proof.  x - x c  x - Let UM be a soft subset of X . Since Sm is a soft pre closed set for every Sm ∈ UM, X ⧹Sm is a soft pre   x x c □ open set containing UM. Therefore, UM =⋂{XS ⧹m : Sm ∈ UM}, as required.

Theorem 3.18. Let ()XσM,, be finite soft hyperconnected. Then, ()XσM,, is tt-soft pre T2 if and only if it is tt-soft pre T1.

Proof. By Proposition 3.8 the necessary part holds. Conversely, for each xy≠ , we have xM and yM are soft - fi   - pre closed sets. Since X is nite, ⋃yX∈{}\ x yM and ⋃xX∈{}\ y xM are soft pre closed sets. Since ()XσM,, is soft  c  c - hyperconnected, (⋃)=yX∈{}\ xyxM M and (⋃)=xX∈{}\ yxyM M are soft pre open sets. The disjointness of xM and - □ yM end the proof that ()XσM,, is tt soft pre T2.

Remark 3.19. In Example 3.10, note that xM is not a soft pre-open set for each x ∈ . This clarifies that a soft set xM in a tt-soft pre T1-space need not be soft pre-open if the universal set is infinite.

Theorem 3.20. ()XσM,, is tt-soft pre-regular iff for every soft pre-open subset FM of ()XσM,, totally con- ∼∼pre taining x, there is a soft pre-open set VM such that x ∈⊆VVMM ⊆ FM.

Proof. - c - Let xX∈ and FM be a soft pre open set totally containing x. Then, FM is pre soft closed and  c  - c ∼ xM ⋂=FM Φ. Therefore, there are disjoint soft pre open sets UM and VM such that FM ⊆ UE and x ∈ VM. ∼∼c pre ∼∼c c - Thus, VUFM ⊆⊆M M. Hence, VUFM ⊆⊆M M. Conversely, let FM be a soft pre closed set. Then, for each c - pre ∼ x ⋐̸ FM , we have x ∈ FM. By hypothesis, there is a soft pre open setVM totally containing x such thatVFM ⊆ M. c ∼ pre c  pre c  - - □ Therefore, FM ⊆(VM ) and VVMM⋂( ) =Φ. Thus, ()XσM,, is tt soft pre regular, as required.

Theorem 3.21. If ()XσM,, is a tt-soft pre-regular space, then the concepts of tt-soft pre T2, tt-soft pre T1 and tt-soft pre T0-spaces are equivalent.

Proof. It suffices to prove that a tt-soft preT0-space is tt-soft preT2. To do so, let xy≠ in a tt-soft preT0-space ()XσM,, . Then, there exists a soft pre-open set GM such that x ∈ GM and y ⋐̸ GM or y ∈ GM and x ⋐̸ GM. Say, c c - - x ∈ GM and y ⋐̸ GM. Obviously, x ⋐̸ GM and y ∈ GM. Since ()XσM,, is tt soft pre regular, there exist two - c ∼ - □ disjoint soft pre open setsUM andVM such that x ∈ UM and y ∈⊆GVM M.Hence,()XσM,, is tt soft preT2.

Proposition 3.22. Let ()XσM,, be finite soft hyperconnected. If ()XσM,, is a tt-soft pre T2-space, then it is tt-soft pre-regular.

Proof. Let HM be a soft pre-closed set and xX∈ such that x ⋐̸ HM . Then, xy≠ for each y ⋐ HM. By hypoth- - esis, there are two disjoint soft pre open sets UiM and ViM such that x ∈ UiM and y ∈ ViM . Since {yy: ∈ X} is n a finite set, there is a finite number of soft pre-open sets V such that HV∼  . Since XσM,, is soft iM M ⊆⋃i=1 iM () n  - x hyperconnected, it follows from Theorem 2.16 that ⋂i=1UiM is a soft pre open set containing . Since n n  VU Φ, XσM,, is -soft pre-regular. □ [⋃⋂⋂=i=1 iMM][i=1 i ] ()tt New soft separation axioms and fixed soft points  205

Corollary 3.23. The following properties are equivalent if ()XσM,, is finite soft hyperconnected. (i) A tt-soft pre T3-space. (ii) A tt-soft pre T2-space. (iii) A tt-soft pre T1-space.

Proof. The directions (iii)→( ) and (ii)→( iii) follow from Proposition 3.8. The direction (iii)→( ii) follows from Theorem 3.18. The direction (ii)→( i) follows from Proposition 3.22. □

Theorem 3.24. The property of being a tt-soft pre Ti-space (i = 0, 1, 2, 3) is a soft open hereditary.

Proof. For i = 3.

Let ()Yσ,,Y Mbe a soft open subspace of a tt-soft pre T3-space ()XσM,, . To prove that ()Yσ,,Y Mis tt-soft pre T1, let x ≠∈yY. Since ()XσM,, is tt-soft pre T1, there exist two soft pre-open sets GM and FM such   that x ∈ GM and y ⋐̸ GM; and y ∈ FM and x ⋐̸ FM . Therefore, x ∈=⋂UYGMM and y ∈=⋂VYFMM such that y ⋐̸ UM and x ⋐̸ VM. By Proposition 2.17, UM and VM are soft pre-open subsets of (Yσ,,Y M); thus, ()Yσ,,Y Mis tt-soft pre T1. To prove that ()Yσ,,Y Mis tt-soft pre-regular, let y ∈ Y and FM be a soft pre-closed subset of ()Yσ,,Y M  c  c such that y ⋐̸ FM. Then, FM ⋃ Y is a soft pre-closed subset of ()XσM,, such that y ⋐⋃̸ FYM  . Therefore,  c ∼ there exist disjoint soft pre-open subsets UM and VM of ()XσM,, such that FM ⋃⊆YUM and y ∈ VM . Now,   ∼   UM ⋂ Y andVYM ⋂  are disjoint soft pre-open subsets of ()Yσ,,Y Msuch that FMM⊆⋂UY and y ∈⋂VYM . Thus, ()Yσ,,Y Mis tt-soft pre-regular. Hence, ()Yσ,,Y Mis tt-soft pre T3, as required. □

Theorem 3.25. Let ()XσM,, be extended and i = 0, 1, 2, 3, 4. Then, ()XσM,, is tt-soft preTi iff ()Xσ, m is preTi for each mM∈ .

Proof. For i = 4.

⇒: Let xy≠ in X. Then, there exist two soft pre-open sets UM and VM such that x ∈ UM and y ⋐̸ UM; and y ∈ VM and x ⋐̸ VM. Obviously, x ∈(Um) and y ∉(Um); and y ∈(Vm) and x ∉(Vm). Since ()XσM,, is extended, Theorem 2.25 shows that U(m) and Vm( ) are pre-open subsets of ()Xσ, m for each mM∈ . Thus, ()Xσ, m is a pre T1-space. To prove that ()Xσ, m is pre-normal, let Fm and Hm be two disjoint pre-closed subsets of ()Xσ, m . Let FM and HM be two soft sets given by F()=mFm, Hm()= Hm and F(′)=(′)=∅mHm for each mm′≠ . By Theorem 2.25, we obtain FM and HM are two disjoint soft pre-closed subsets of ()XσM,, .By ∼ ∼ hypothesis, there exist two disjoint soft pre-open sets GM and WM such that FM ⊆ GM and HWMM⊆ . This implies that F()=mFm ⊆ Gm () and Hm()= Hm ⊆ Wm (). Since ()XσM,, is extended, Theorem 2.25 shows that G(m) and Wm( ) are pre-open subsets of ()Xσ, m . Thus, ()Xσ, m is a pre-normal space. Hence, it is a pre T4-space. ⇐: Let xy≠ in X. Then, there exists two pre-open subsets Um and Vm of ()Xσ, m such that x ∈ Um and y ∉ Um; and y ∈ Vm and x ∉ Vm. Let UM and VM be two soft sets given by U()=mUm, Vm()= Vm for each mM∈ . Since ()XσM,, is extended, Theorem 2.25 shows that UM and VM are soft pre-open subsets of ()XσM,, such that x ∈ UM and y ⋐̸ UM; and y ∈ VM and x ⋐̸ VM. Thus, ()XσM,, is a tt-soft pre T1-space. To prove that ()XσM,, is soft pre-normal, let FM and HM be two disjoint soft pre-closed subsets of ()XσM,, . Since ()XσM,, is extended, Theorem 2.25 shows that F(m) and Hm( ) are two disjoint pre-closed subsets of

()Xσ, m . By hypothesis, there exist two disjoint pre-open subsets Gm and Wm of ()Xσ, m such that F()⊆mGm and Hm()⊆ Wm. Let GM and WM be two soft sets given by G()=mGm and Wm()= Wm for each mM∈ . Since ()XσM,, is extended, Theorem 2.25 shows that GM andWM are two disjoint soft pre-open subsets of()XσM,, ∼ ∼ such that FM ⊆ GM and HWMM⊆ . Thus, ()XσM,, is soft pre-normal. Hence, it is a tt-soft pre T4-space. □ 206  Tareq M. Al-shami et al.

In the next counterexamples, we illustrate that there is no relationship between soft topological space and their parametric topological spaces in terms of separation axioms, if an extended condition given in the above theorem does not exist.

Example 3.26. Let ()XσM,, be the same as in Example 3.5. We clarified that ()XσM,, is not tt-soft pre T0.

In contrast, σm1 and σm2 are the discrete topology on X. Hence, ()Xσ, m1 and ()Xσ, m2 are pre T4.

Example 3.27. Let M ={mm12, }. A family σ ={Φ, Xmxmy , {(12 , { })( , , { })}} is a soft topology on X ={xy, }. - A family of all soft pre open subsets of ()XσM,, is S()⧹XGiMi{ M : = 1,2,3}, where

Gmymx11M = {(,,, { }) ( 2 { })} ;

Gmym21M ={(,,, { })( 2 ∅)} and

Gm31M ={(,, ∅)( mx 2 , { })} .

It can be checked that ()XσM,, is tt-soft pre T4. On the other hand, the two parametric topological spaces ()Xσ, m1 and ()Xσ, m2 are not pre T1.

Theorem 3.28. The property of being a tt-soft pre Ti-space (i = 0, 1, 2) is preserved under a finite product soft spaces.

Proof. For i = 2. Let ()XσM11,, 1and ()XσM22,, 2be two tt-soft pre T2-spaces and let (xy,,)≠( x ′ y ′) in X12× X . Then, x ≠ x′ - or y ≠′y . Without loss of generality, let x ≠ x′. Then, there exist two disjoint soft pre open subsets GM1 and   HM1 of ()XσM11,, 1such that x ∈ GM1 and x′⋐̸ GM1; and x′∈HM1 and x ⋐̸ HM1. Obviously, GM1 × X2 and HXM1 × 2 -   are two disjoint soft pre open subsets X12× X such that (xy, )∈ GM1 × X2 and (xy′′)⋐, ̸ GM1 × X2; and   - - □ (xy′′)∈, HM1 × X2 and (xy, )⋐̸ HM1 × X2. Hence, X12× X is a tt soft pre T2 space.

Theorem 3.29. The property of being a tt-soft pre Ti-space (i = 0, 1, 2, 3, 4) is an additive property.

Proof. For i = 2.

Let x ≠∈yX⊕iI∈ i. Then, we have the following two cases: - - 1. There exists i0 ∈ I such that x, yX∈ i0. Since ()XσMii00,, is tt soft pre T2, there exist two disjoint soft pre

open subsets GM and HM of ()XσMii00,, such that x ∈ GM and y ∈ HM. By Theorem 2.27, GM and HM are - disjoint soft pre open subsets of ()⊕iI∈ XσMi,, .   - 2. There exists ij0 ≠∈0 I such that x ∈ Xi0 and y ∈ Xj0.Now, Xi0 and Xj0 are soft pre open subsets of ( )   - ()XσMii00,, and()XσMjj00,, , respectively. By Theorem 2.27 , Xi0 and Xj0 are disjoint soft pre open subsets of ()⊕iI∈ XσMi,, .

- - It follows from the two cases above that ()⊕iI∈ XσMi,, is a tt soft pre T2 space. For i = 3 and i = 4,wesuffice by proving the tt-soft pre-regularity and soft pre-normality. - - - First, we prove the tt soft pre regularity property. Let FM be a soft pre closed subset of ()⊕iI∈ XσMi,,  such that x ⋐̸ FM . It comes from Theorem 2.27 that FMi⋂ X is soft pre-closed in ()XσMii,, for each i ∈ I. Since - x ∈ ⊕iI∈ Xi, there is only i0 ∈ I such that x ∈ Xi0. This implies that there are disjoint soft pre open subsets  ∼  GM and HM of XσMii,, such that FMiM⋂⊆XGand y ∈ HM. Now, GM ⋃ Xi is a soft pre-open subset ()00 0 ii≠ 0 of ()⊕iI∈ XσMi,, containing FM. The disjointness of GMii⋃ ≠ 0 Xi and HM ends the proof that ()⊕iI∈ XσMi,, is a tt-soft pre-regular space.

Second, we prove the soft pre-normality property. Let FM and HM be two disjoint soft pre-closed subsets     - of ()⊕iI∈ XσMi,, . By Theorem 2.27, we obtain FMi⋂ X and HXMi⋂ are soft pre closed in ()XσMii,, for every - - i ∈ I. Since ()XσMii,, is soft pre normal for each i ∈ I, there exist two disjoint soft pre open subsets UiM and V of XσM,, such that F  XU ∼ and HXV  ∼ . This implies that F ∼  U , HV∼  iM ()ii Mii⋂⊆M Mii⋂⊆M M ⊆⋃iI∈ iM M ⊆⋃iI∈ iM           - □ and ⋃⋂⋃=UViMMi Φ. Hence, ()⊕iI∈ XσMi,, is a soft pre normal space.  iI∈∈  iI  New soft separation axioms and fixed soft points  207

In the following, we prove the behaviors of tt-soft pre Ti-spaces under some soft maps.

Definition 3.30. A map fφ :,,()→(XσM YτB ,,) is said to be ⋆- −1 - - 1. Soft pre continuous if fφ (VB) is a soft pre open set for each soft pre open set VB. ⋆ ⋆ 2. Soft pre -open (resp. soft pre -closed) if fφM(U ) is a soft pre-open (resp. soft pre-closed) set for each soft

pre-open (resp. soft pre-closed) set UM. 3. Soft pre⋆-homeomorphism if it is bijective, soft pre⋆-continuous and soft pre⋆-open.

Proposition 3.31. Let fφ :,,()→(XσM YτB ,,) be a soft pre-continuous map such that f is injective. Then, if ()YτB,, is a p-soft Ti-space (i = 0, 1, 2), then ()XσM,, is a tt-soft pre Ti-space.

Proof. For i = 2. Let fφ :,,()→(XσM YτB ,,) be a soft pre-continuous map and a ≠∈b X. Since f is injective, there are two distinct points x and y in Y such that f ()=axand f ()=b y. Since (Yτ,,B) is a p-soft T2-space, there are −1 −1 two disjoint soft open sets GB and FB such that x ∈ GB and y ∈ FB. Now, fφ (GB) and fφ (FB) are two disjoint - −1 −1 - soft pre open subsets of ()XσA,, such that a ∈(fGφ B) and b ∈(fFφ B). Thus, ()XσM,, is a tt soft pre T2-space. □

In a similar way, one can prove the following result.

⋆ Proposition 3.32. Let fφ :,,()→(XσM YτB ,,) be a soft pre -continuous map such that f is injective. Then, if ()YτB,, is a tt-soft pre Ti-space (i = 0, 1, 2), then ()XσM,, is a tt-soft pre Ti-space.

Proposition 3.33. Let fφ :,,()→(XσM YτB ,,) be a bijective soft pre-open map. Then, if ()XσM,, is a p-soft Ti-space, then ()YτB,, is a tt-soft pre Ti-space for i = 0, 1, 2.

Proof. For i = 2. Let fφ :,,()→(XσM YτB ,,) be a soft pre-open map and x ≠∈yY. Since f is bijective, there are two −1 −1 distinct points a and b in X such that a =(fx) and b =(fy). Since (X,,σ M) is a p-soft T2-space, there are two disjoint soft open sets UM and VM such that x ∈ UM and y ∈ VM . Now, fφM(U ) and fφM(V ) are two disjoint soft pre-open subsets of ()YτB,, such that x ∈(fUφM) and y ∈(fVφM). Thus, ()YτB,, is a tt-soft pre T2-space. □

⋆ Proposition 3.34. Let fφ :,,()→(XσM YτB ,,) be a bijective soft pre -open map. Then, if ()XσM,, is a tt- soft pre Ti-space (i = 0, 1, 2), then ()YτB,, is a tt-soft pre Ti-space.

⋆ Proposition 3.35. The property of being tt-soft pre Ti (i = 0, 1, 2, 3, 4) is preserved under a soft pre -homeo- morphism map.

We close this part by studying the relationships between tt-soft pre Ti-spaces (i = 2, 3, 4) and soft pre- compact spaces.

Proposition 3.36. A stable soft pre-compact subset of a tt-soft pre T2-space is soft pre-closed.

Proof. It follows from Proposition 2.21 and Remark 3.2. □

Theorem 3.37. Let HM be a soft pre-compact subset of a soft hyperconnected tt-soft pre T2-space. If x ⋐̸ HM, then there are disjoint soft pre-open sets UM and VM such that x ∈ UM and HVMM⊆ . 208  Tareq M. Al-shami et al.

Proof. Let x ⋐̸ HM . Then, xy≠ for each y ⋐ HM. Since ()XσM,, is a tt-soft pre T2-space, there are disjoint - - soft pre open sets UiM and ViM such that x ∈ UiM and y ∈ ViM . Therefore, {}ViM forms a soft pre open cover n of H . Since H is soft pre-compact, HV . By the soft hyperconnectedness of XσM,, , we obtain M M M ⊆⋃i=1 iM () n - □ ⋂i=1UUiMM = is a soft pre open set. Hence, we obtain the desired result.

Theorem 3.38. A soft hyperconnected, soft pre-compact and tt-soft pre T2-space is tt-soft pre-regular.

Proof. Let HM be a soft pre-closed subset of soft pre-compact and tt-soft pre T2-space ()XσM,, such that x ⋐̸ HM . Then, HM is soft pre-compact. By Theorem 3.37, there exist disjoint soft pre-open sets UM and VM such that x ∈ UM and HVMM⊆ . Thus, ()XσM,, is tt-soft pre-regular. □

Corollary 3.39. A soft hyperconnected, soft pre-compact and tt-soft pre T2-space is tt-soft pre T3.

Lemma 3.40. - - x Let FM be a soft pre open subset of a soft pre regular space. Then, for each Sm ∈ FM, there exists - x pre ∼ a soft pre open set GM such that Sm ∈⊆GFM M.

Proof. - x c - Let FM be a soft pre open set such that Sm ∈ FM. Then, x ∉ FM. Since ()XσM,, is soft pre regular, - c there exist two disjoint soft pre open sets GM and WM totally containing x and FM, respectively. Thus, ∼∼c x ∼∼∼pre c □ x ∈⊆GWFM M ⊆M. Hence, Sm ∈⊆GGMM ⊆ WFM ⊆M.

Theorem 3.41. Let HM be a soft pre-compact subset of a soft pre-regular space and FM be a soft pre-open set ∼∼pre ∼ containing HM . Then, there exists a soft pre-open set GM such that HGGMMM⊆⊆ ⊆ FM.

Proof. fi x x Let the given conditions be satis ed. Then, for each Sm ∈ HM, we have Sm ∈ FM. Therefore, there is - x ∼∼pre x - a soft pre open set WxeM such that Sm ∈⊆WWxeMM xe ⊆ FM. Now, {}WSFxeM : m ∈ M is a soft pre open cover in= in= of H .SinceH is soft pre-compact, HW∼  .PuttingG  W .Thus,HGG∼∼pre ∼ F. M M M ⊆⋃i=1 xeM M =⋃i=1 xeM MMM⊆⊆ ⊆M □

Corollary 3.42. If ()XσM,, is soft pre-compact and soft pre T3, then it is tt-soft pre T4.

Proof. - ∼ c Suppose that F1M and F2M are two disjoint soft pre closed sets. Then, F2M ⊆ F1M . Since ()XσM,, is soft - - - - pre compact, F2M is soft pre compact and since ()XσM,, is soft pre regular, there is a soft pre open set ∼∼pre ∼c ∼ ∼ pre c  pre c  GM such that F2M ⊆⊆GGMM ⊆ F1M . Obviously, F2M ⊆ GM, F1M ⊆(GM ) and GMM⋂(G ) =Φ. Thus, ()XσM,, is soft pre-normal. Since, ()XσM,, is soft pre T3,itistt-soft pre T1. Hence, it is tt-soft pre T4. □

4 Pre-fixed soft points of soft mappings

In this part, we define a pre-fixed soft point property and scrutinize main characterizations. Also, we point out its transmission between soft topology and parametric topologies.

Theorem 4.1. Let {n : n ∈ } be a class of soft sets in a soft pre-compact space ()XσM,, satisfying the following:

(i) n ≠ Φ for each n ∈ ; (ii) n is a soft pre-closed set for each n ∈ ; ∼ (iii) nn+1 ⊆  for each n ∈ .    Then, ⋂n∈ n ≠ Φ. New soft separation axioms and fixed soft points  209

Proof.      c  ( )  c  - Suppose that ⋂n∈ n = Φ. Then, ⋃n∈ n = X . It follows from ii that { n : n ∈ } is a soft pre open cover of X. By hypothesis of soft pre-compactness, there exist ii12,,…∈ , ik , ii12<<…< ik such that  c c c ( )  ∼  c c c  c X =⋃⋃…⋃i12i ik .Itfollowsfrom iii that ik ⊆=X i12 ⋃i ⋃…⋃=ik []ii12 ⋂ ⋂…⋂ ik  c   Φ □ = ik . This yields a contradiction. Thus, we obtain the proof that ⋂n∈ n ≠ .

Proposition 4.2. - - Let ()XσM,, be a soft pre compact and soft pre T2′ space and gXσMXσMφ :,,()→( ,,) be ⋆- x  soft pre continuous. Then, there is a unique soft point Sm ∈ X of gφ.

Proof.     n   ( ) Let { 1 =(gXφ ) and n =(ggXφ n−1 )=(φ ) for each n ∈ } be a family of soft subsets of X,,σ M .  ∼   ⋆-  - It is clear that nn+1 ⊆ for each n ∈ . Since gφ is soft pre continuous, n is a soft pre compact set   -  for each n ∈ and since ()XσM,, is soft pre T2′, n is also a soft pre closed set for each n ∈ .Itfollows   -  n  ∼ from Theorem 4.1 that (HM, )=⋂n∈ n is a non null soft set. Note that gHMφφ()=(⋂())⊆, gn∈ gXφ n+1 ∼ n ∼ x ⋂n∈gXφ ()⊆⋂n∈ gXHMφ ()=(, ). To show that (HM,,)⊆ gφ ( HM), suppose that there is a Sm ∈(HM, ) x  −1 x      ∼    such that Sm ∉(gHMφ , ).Let n =()⋂gSφ m n. Obviously, n ≠ Φ and nn⊆ −1 for each n ∈ .Now, n -  y y −1 x is a soft pre closed set for each n ∈ ; and by Theorem 4.1, there exists a soft point Sm such that Sm ∈(gSφ m)   x y ⋂ n. Therefore, Sm =()∈(gPφ m gHMφ , ). This is a contradiction. Thus, gHMφ()=(,, HM). Hence, the proof is complete. □

Definition 4.3. ( ) -fi ⋆- i ()XσM,, is said to have a pre xed soft point property if every soft pre continuous map gXσMφ :,,( ) → (XσM,, ) has a fixed soft point. (ii) A property that is preserved by any soft pre⋆-homeomorphism map is called a pre⋆-soft topological property.

Proposition 4.4. The property of being a pre-fixed soft point is a pre⋆-soft topological property.

Proof. Let ()XσM,, and ()YτM,, be a soft pre⋆-homeomorphic. Then, there is a bijective soft map −1 ⋆- -fi fφ :,,()→(XσM YτM ,,) such that fφ and fφ are soft pre continuous. Since ()XσM,, has a pre xed ⋆- -fi soft point property, every soft pre continuous map gXσMXσMφ :,,()→( ,,) has a pre xed soft point. ⋆ Now, let hYτMYτMφ :,,()→( ,,) be a soft pre -continuous. Obviously, hfXσMYτMφφ∘(:,, )→( ,,) is ⋆- −1 ⋆- a soft pre continuous. Also, fφ ∘∘hfXσMXσMφφ:,, ( )→( ,,) is a soft pre continuous. Since ()XσM,, -fi −1 x x x  −1 x has a pre xed soft point property, fφ (hfSφφ ( (m ))) = Sm for some Sm ∈ X . Consequently, fφ(fhfSφ (φφ ( (m )))) = x x x x -fi fφ(Sm). This implies that hfSφφ(())=(m fSφ m). Thus, fφ(Sm) is a pre xed soft point of hφ. Hence, ()YτM,, has a pre-fixed soft point property, as required. □

Theorem 4.5. Let σ be an extended soft topology on X. Then, a soft map gXσMYτMφ :,,()→( ,,) is soft ⋆ ⋆ pre -continuous if and only if a map gXσ:,()→(mϕm Yτ ,()) is pre -continuous.

Proof. Necessity: Let U be a pre-open subset of ()Yτ, ϕm(). Then, there exists a soft pre-open subset GM of ⋆- −1 - ()YτM,, such that G(())=ϕm U. Since gφ is a soft pre continuous map, gGϕ ( M) is a soft pre open set. fi ( ) −−11 −1 From De nition 2.7 , it follows that a soft subset gGϕ ()=(())M gGϕ M of ()XσM,, is given by gGmϕ ()()= gGϕm−1((( ))) for each mM∈ . By hypothesis, σ is extended, Theorem 2.25 shows a subset gGϕm−1( ( ( ))) = −1 ⋆ gU( ) of ()Xσ, m is pre-open. Hence, a map g is pre -continuous. Sufficiency: Let GM be a soft pre-open subset of ()YτM,, . According to Definition 2.7, a soft subset −−11 −1 −1 gGϕ ()=(())M gGϕ M of ()XσM,, is given by gGmϕ ()()= gGϕm ((())) for each mM∈ . Since a map g is ⋆ −1 pre -continuous, a subset gGϕm((( ))) of ()Xσ, m is pre-open. By hypothesis, σ is extended, we obtain from −1 - ⋆- Theorem 2.25 that gGϕ ( M) is a soft pre open subset of ()XσM,, . Hence, a soft map gφ is soft pre continuous. □ 210  Tareq M. Al-shami et al.

Definition 4.6. ()Xσ, is said to have a pre-fixed point property if every pre⋆-continuous map gXσ:,()→ ()Xσ, has a fixed point.

Proposition 4.7. ()XσM,, has the property of a pre-fixed soft point iff ()Xσ, m has the property of a pre-fixed point for each mM∈ .

Proof. Necessity: Let ()XσM,, has the property of a pre-fixed soft point. Then, every soft pre⋆- fi x continuous map gXσMXσMφ :,,()→( ,,) has a xed soft point. Say, Sm. The above theorem shows that ⋆- x fi gXσXτm :,()→(mϕm ,()) is pre continuous. Since Sm is a xed soft point of gφ, we obtain gxm()= x. Thus, fi gm has a xed point. ⋆ Sufficiency:Let()Xσ, m has the property of a pre-fixed point for each mM∈ . Then, every pre -continuous fi map gXσXτm :,()→(mϕm ,()) has a xed point. Say, x. The above theorem shows that gXσMφ :,,()→ ⋆- fi x x fi ()XτM,, is soft pre continuous. Since x is a xed point of gm,wegetgSφ()=m Sm.Thus,gφ has a xed soft point. Hence, the proof is complete. □

5 Conclusion

Classification of soft spaces is one of the interesting applications of soft separation axioms. To contribute to this scope, we allocate this paper to explore new families of soft topologies. We have formulated the concepts of tt-soft pre Ti-spaces using the relations of total belong and total non-belong. We have initiated their main properties and elucidated the relationships between them with the aid of several counterex- amples. Also, we have investigated a pre-fixed soft point theorem and established main results. One of the divergences between soft set and crisp set is the diversity of belong and non-belong relations between ordinary points and soft sets. This widely opens the door to establish several classes of separation axiom on soft setting. Therefore, in the upcoming works, we will discuss the presented soft separation axioms using another type of belong and total non-belong relations. Also, we define new types of soft separation axioms using different kinds of generalized soft open sets.

Acknowledgment: The authors would like to thank the editor and the referees for their valuable comments, which help us to improve the manuscript. This publication was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.

Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.

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