# Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets

Authors: T. M. Al-shami
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

The importance of soft separation axioms comes from their vital role in classifications of soft spaces, and their interesting properties are studied. This article is devoted to introducing the concepts oftt-soft semi-Tii=0,1,2,3,4 and tt-soft semiregular spaces with respect to ordinary points. We formulate them by utilizing the relations of total belong and total nonbelong. The advantages behind using these relations are, first, generalization of existing comparable properties on general topology and, second, eliminating the stability shape of soft open and closed subsets of soft semiregular spaces. By some examples, we show the relationships between them as well as with soft semi-Tii=0,1,2,3,4 and soft semiregular spaces. Also, we explore under what conditions they are kept between soft topology and its parametric topologies. We characterize a tt-soft semiregular space and demonstrate that it guarantees the equivalence of tt-soft semi-Tii=0,1,2. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces, and sum of soft topological spaces. Finally, we define a concept of semifixed soft point and study its main properties.


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## Body

## 1. Introduction

In daily life, human beings face different kinds of uncertainties in fields such as economics, environmental and social sciences, engineering, medicine etc. To tackle such uncertainties, different approaches were proposed like fuzzy and rough set theories. However, they have their own limitations. Molodtsov [1] introduced an effective tool that is free from these limitations to solve uncertainties, namely soft set. This tool does not require specific type of parameters. Instead, it accommodates all types of parameters such as words, numbers, sentences, and functions. After that, Maji et al. [2] established some fundamental operations between two soft sets and then tackled one of their applications in decision-making problems [3]. But the authors of [4] demonstrated some shortcomings of the operations defined in [2]. They also explored novel operations between soft sets which became a goal of many studies on algebraic soft structures. Al-shami and El-Shafei [5] defined new types of soft subset and equality relations and then applied them on soft linear equations.In 2011, Shabir and Naz [6] defined a soft topological space over a family of soft sets. They constructed the fundamental notions of soft topological spaces such as the operators of soft closure and interior, soft subspace, and soft separation axioms. Min [7] continued studying soft separation axioms and corrected some alleged results in [6]. Then, Aygünog˘lu and Aygün [8] scrutinized the concept of compactness on soft setting. By exploiting one of the diverges between soft sets and crisp sets, Hida [9] studied a strong type of soft compactness. Al-shami [10, 11] revised many results given in many studies of soft separation axioms. Then, Al-shami et al. [12] introduced a weak type of soft compactness, namely almost soft compactness. The authors of [13] presented soft maps by using two crisp maps, one of them between the sets of parameters and the second one between the universal sets. However, the authors of [14] introduced soft maps by using the concept of soft points. Some applications of different types of soft maps were the goal of some articles (see [13–15]).Until 2018, the belong and nonbelong relations that utilized in these studies are those given by [6]. In 2018, the authors of [16] came up new relations of belong and nonbelong between an ordinary points and soft set, namely partial belong and total nonbelong relations. In fact, these relations widely open the door to study and redefine many soft topological notions. This leads to obtain many fruitful properties and changes which can be seen significantly on the study of soft separation axioms as it was shown in [16–18].Das and Samanta [19] studied the concept of a soft metric based on the soft real set and soft real numbers given in [20]. Wardowski [14] tackled the fixed point in the setup of soft topological spaces. Abbas et al. [21] presented soft contraction mappings and established a soft Banach fixed point theorem in the framework of soft metric spaces. Recently, many researchers explored fixed point findings in soft metric type spaces (see, for example, [22–25]). Some results related to intuitionistic fuzzy soft sets were investigated in [26, 27]. Recently, Alcantud [28] has discussed the properties of topological separability on soft setting.One of the significant ideas that helps to prove some properties and removes some problems on soft topology is the concept of a soft point. It was first defined by Zorlutuna and Çakir [29] in order to study the interior points of a soft set and soft neighborhood systems. Then, [19, 30] simultaneously redefined soft points to discuss soft metric spaces. In fact, the recent definition of a soft point makes similarity between many set-theoretic properties and their counterparts on soft setting. Two types of soft topologies, namely enriched soft topology and extended soft topology, were studied in [8, 30], respectively. Al-shami and Koc˘inac [31] discussed the equivalence between these two topologies and obtained interesting results. Recently, they [32] have introduced the concept of nearly soft Menger spaces and investigated main characteristics.We organized this paper as follows: After this introduction, Section2 addresses the basic principles about the soft sets and soft topologies. In Section (, we introduce the concepts of tt-soft semi-Tii=0,1,2,3,4 and tt-soft semiregular spaces with respect to ordinary points by using the relations of total belong and total nonbelong. The relationships between them and their main properties are discussed with the help of interesting examples. In Section 4, we explore a semifixed soft point theorem and study some main properties. In particular, we conclude under what conditions semifixed soft points are preserved between a soft topology and its parametric topologies. Finally, conclusions of the paper in Section 5.


## 2. Preliminaries

To well understand the results obtained in this study, we shall mind some essential concepts, definitions, and properties from the literature.


### 2.1. Soft Sets

Definition 1.
[1] A mapGofPinto the power set2Xof a nonempty setXis called a soft set overX, whereXis the universal set andPis a set of parameters.In this study, we use a symbolGP to refer a soft set, and we identify it as ordered pairs GP=p,Gp:p∈PandGp∈2X.A class of all soft sets defined overX with P is symbolized by SXP.Definition 2.
[33] GPis said to be a subset ofHP, symbolized byGP⊆~HP, ifGp⊆Hpfor eachp∈P.We writeGP=HP provided that GP⊆~HP and HP⊆~GP.Definition 3.
[6, 16] For a soft setGPandw∈X, we say that(i)
w∈GP, it is read as follows: w totally belongs to GP, if w∈Gp for each p∈P(ii)
w∈GP, it is read as follows: w does not partially belong to GP, if w∈Gp for some p∈P(iii)
w⋐GP, it is read as follows: w partially belongs to GP, if w∈Gp for some p∈P(iv)
w⋐GP, it is read as follows: w does not totally belong to GP, if w∈Gp for each p∈P.Definition 4.
[4] The relative complement ofGPis another soft setGPc, where a mapGc:P⟶2Xis defined byGcp=X\Gpfor allp∈P.Definition 5.
[2, 16, 19, 30] A soft setGPoverXis said to be(i)
a null soft set (resp., an absolute soft set), symbolized byΦ~ (resp., X~), if Gp is the empty (resp., universal) set for each p∈P(ii)
a soft pointSpx if there are p∈P and x∈X so that Gp=x and Gp′=∅ for each p′∈P\p. We write that Spx∈GP if x∈Gp(iii)
a stable soft set if all p-approximate are equal A and it is symbolized byA~. In particular, we symbolize by xP if A=x(iv)
a countable (resp., finite) soft set ifGp is countable (resp., finite) for each p∈P. Otherwise, it is said to be uncountable (resp., infinite).Definition 6.
[2, 4] The intersection and union ofGPandHPare defined as follows:(i)
Their intersection, symbolized byGP∩~HP, is a soft set UP, where a mapping U:P⟶2X is given by Up=Gp∩Hp(ii)
Their union, symbolized byGP∪~HP, is a soft set UP, where a mapping U:P⟶2X is given by Up=Gp∪Hp.The union and intersection operators were generalized for any number of soft sets in a similar way.Definition 7.
[34] The Cartesian product ofGPandHF, symbolized byG×HP×F, is defined asG×Hp,f=Gp×Hffor eachp,f∈P×F.Definition 8.
[13] A soft mapping ofSXAintoSYB, symbolized byfϕ, is defined by two crisp mapsf:X⟶Yandϕ:A⟶Bso that the image ofGAis a set inSXAand pre-image ofHBis a set inSYBand they are defined as follows.(i)
fϕGA=fϕGB is a set in SYB, where fϕGb=⋃a∈ϕ−1bfGa for each b∈B(ii)
fϕ−1HB=fϕ−1HA is a set in SXA, where fϕ−1Ha=f−1Hϕa for each a∈A.Definition 9.
[29] fϕ:SXA⟶SYBis called aλsoft map provided thatϕandfareλmaps, whereλ∈injective,surjective,andbijective.


### 2.2. Soft Topology

Definition 10.
[6] A familyΩof soft sets overXunder a fixed set of parametersPis said to be a soft topology onXif it satisfies the following:(i)
X~ and Φ~ are elements of Ω(ii)
The intersection of a finite number ofΩ is an element of Ω(iii)
The union of an arbitrary number ofΩ is an element of Ω.We will call a term of soft topological space on the tripleX,Ω,P, where a terminology of soft open is given for every member in Ω and a terminology of soft closed is given for a relative complement of a member of Ω.Proposition 11.
[6] InX,Ω,P, a familyΩp=Gp:GP∈Ωis a classical topology onXfor eachp∈P.We will call a term of parametric topology onΩp and a term of parametric topological space on X,Ωp.Definition 12.
[6] Y,ΩY,Pis called a soft subspace ofX,Ω,P, where∅≠Y⊆XandΩY=Y~∩GP:GP∈Ω.Definition 13.
[35] GPis called a soft semiopen subset ofX,Ω,PifGP⊆~clintGP.Theorem 14.
[35] Every soft open set is soft semiopen, and the union of an arbitrary number of soft semiopen sets is soft semiopen.Definition 15.
[35] The intersection of all soft semiclosed sets inX,Ω,PcontainingGPis the soft preclosure ofGP. It is symbolized byGP¯s.It is clear thatx∈GP¯s iff for each soft semiopen set UP totally containing x we have GP∩~UP≠Φ~; and Spx∈GP¯s iff for each soft semiopen set UP totally containing Spx we have GP∩~UP≠Φ~.Definition 16.
[36] X,Ω,Pwhich does not contain disjoint soft open sets is called soft hyperconnected.The following result will help us establish some properties of soft semiseparation axioms and soft semicompact spaces (see, for example, Theorem42 and Proposition 49). It implies that the family of soft semiopen subsets of a soft hyperconnected space X,Ω,P forms a new soft topology Ωsemi over X that is finer than Ω.Theorem 17.
[37] A class of soft semiopen subsets of a soft hyperconnected space forms a soft topology.Proposition 18.
LetY~be soft open set inX,Ω,P. Then(1)
IfH,P is soft semiopen and Y~ is soft open set in X,Ω,P, then H,P∩~Y,P is a soft semiopen set in Y,ΩY,P(2)
IfY~ is soft open in X,Ω,P and H,P is a soft semiopen in Y,ΩY,P, then H,P is a soft semiopen set in X,Ω,P.Definition 19.
[35] X,Ω,Pis said to be(i)
soft semi-T0 if there is a soft semiopen set UP for each x≠y∈X satisfying x∈UP and y∈UP; or y∈UP and x∈UP(ii)
soft semi-T1 if there are two soft semiopen sets UP and VP for each x≠y∈X satisfying x∈UP and y∈UP; and y∈VP and x∈VP(iii)
soft semi-T2 if there are two disjoint soft semiopen sets UP and VP for each x≠y∈X satisfying x∈UP and y∈VP(iv)
soft semiregular if every soft semiclosed setHP so that x∈HP, Ω contains two disjoint soft semiopen sets UP and VP satisfying HP⊆~UP and x∈VP(v)
soft seminormal if we separate every two disjoint soft semiclosed sets by two disjoint soft semiopen sets(vi)
soft semi-T3 (resp., soft semi-T4) if it is both soft semiregular (resp., soft seminormal) and soft semi-T1-spaceDefinition 20.
[38] A soft semiopen cover ofX,Ω,Pis a family of soft semiopen setsGiP:i∈Iso thatX~=∪~i∈IGiP.Definition 21.
X,Ω,Pis said to be(i)
soft semi-T′2 [37] if for every Spx≠Sp′y∈X~, there are twodisjoint soft semiopen sets UP and VP containing Spx and Sp′y, respectively(ii)
soft semicompact [38] if every soft semiopen cover of X~ has afinite subcover.Proposition 22.
[37] A soft semicompact (resp., stable soft semicompact) set in a soft semi-T2′-space (resp., soft semi-T2-space) is soft semiclosed.To study the properties that preserved under softsemi∗-homeomorphism maps, the concept of a soft semi-irresolute map will be presented in this work under the name of a soft semi∗-continuous map.Definition 23.
[37] gφ:X,Ω,P⟶Y,θ,Pis called softsemi∗−continuousif the inverse image of each soft semiopen set is soft semiopen.Proposition 24.
[38] The softsemi∗−continuousimage of a soft semicompact set is soft semicompact.The definitions of soft semicontinuous, soft semiopen, soft semiclosed, and soft semihomeomorphism maps were given in [35] in a similar way of their counterparts in classical topology.Definition 25.
A soft topologyΩonXis said to be(i)
an enriched soft topology [8] if all soft sets GP so that Gp=∅ or X are members of Ω(ii)
an extended soft topology [30] if Ω=GP:Gp∈Ωpforeachp∈P, where Ωp is a parametric topology on X.Al-shami and Kočinac [31] proved that extended and enriched soft topologies are identical and obtained some useful results that help to discover the interrelations between soft topology and its parametric topologies.Theorem 26.
[31] A subsetF,Pof an extended soft topological spaceX,Ω,Pis soft semiopen if and only if each p-approximate element ofF,Pis semiopen.Proposition 27.
[39] LetXi,Ωi,P:i∈Ibe a class of mutually disjoint soft topological spaces andX=⋃i∈IXi. Then the classΩ=G,P⊆~X~:G,P∩~Xi~isasoftopensetinXi,Ωi,Pforeveryi∈I.
forms a soft topology onX with P.Definition 28.
[39] X,Ω,Pgiven in the above proposition is said to be the sum of soft topological spaces and is symbolized by⊕i∈IXi,Ω,P.Theorem 29.
[39] A soft setG,P⊆~⊕i∈IXi~is soft semiopen (resp., soft semiclosed) in⊕i∈IXi,Ω,Pif and only if allG,P∩~Xi~are soft semiopen (resp., soft semiclosed) inXi,Ωi,P.Proposition 30.
[14] Letgφ:X,Ω,P⟶X,Ω,Pbe a soft map so that∩~n∈ℕgφnX~is a soft pointSpx. Then,Spxis a unique fixed point ofgφ.Theorem 31.
[29] LetΩ=GA×FB:GA∈ΩandFB∈θ, whereX,Ω,AandY,θ,Bare two soft topological spaces. A class of all arbitrary union of members ofΩdefines a soft topology overX×Yunder a fixed set of parametersA×B.Lemma 32.
LetG,PandH,Pbe two subsets ofX,Ω,PandY,θ,P, respectively. Then(i)
clG,P×clH,P=clG,P×H,P(ii)
intG,P×intH,P=intG,P×H,P.


## 2.1. Soft Sets

Definition 1.
[1] A mapGofPinto the power set2Xof a nonempty setXis called a soft set overX, whereXis the universal set andPis a set of parameters.In this study, we use a symbolGP to refer a soft set, and we identify it as ordered pairs GP=p,Gp:p∈PandGp∈2X.A class of all soft sets defined overX with P is symbolized by SXP.Definition 2.
[33] GPis said to be a subset ofHP, symbolized byGP⊆~HP, ifGp⊆Hpfor eachp∈P.We writeGP=HP provided that GP⊆~HP and HP⊆~GP.Definition 3.
[6, 16] For a soft setGPandw∈X, we say that(i)
w∈GP, it is read as follows: w totally belongs to GP, if w∈Gp for each p∈P(ii)
w∈GP, it is read as follows: w does not partially belong to GP, if w∈Gp for some p∈P(iii)
w⋐GP, it is read as follows: w partially belongs to GP, if w∈Gp for some p∈P(iv)
w⋐GP, it is read as follows: w does not totally belong to GP, if w∈Gp for each p∈P.Definition 4.
[4] The relative complement ofGPis another soft setGPc, where a mapGc:P⟶2Xis defined byGcp=X\Gpfor allp∈P.Definition 5.
[2, 16, 19, 30] A soft setGPoverXis said to be(i)
a null soft set (resp., an absolute soft set), symbolized byΦ~ (resp., X~), if Gp is the empty (resp., universal) set for each p∈P(ii)
a soft pointSpx if there are p∈P and x∈X so that Gp=x and Gp′=∅ for each p′∈P\p. We write that Spx∈GP if x∈Gp(iii)
a stable soft set if all p-approximate are equal A and it is symbolized byA~. In particular, we symbolize by xP if A=x(iv)
a countable (resp., finite) soft set ifGp is countable (resp., finite) for each p∈P. Otherwise, it is said to be uncountable (resp., infinite).Definition 6.
[2, 4] The intersection and union ofGPandHPare defined as follows:(i)
Their intersection, symbolized byGP∩~HP, is a soft set UP, where a mapping U:P⟶2X is given by Up=Gp∩Hp(ii)
Their union, symbolized byGP∪~HP, is a soft set UP, where a mapping U:P⟶2X is given by Up=Gp∪Hp.The union and intersection operators were generalized for any number of soft sets in a similar way.Definition 7.
[34] The Cartesian product ofGPandHF, symbolized byG×HP×F, is defined asG×Hp,f=Gp×Hffor eachp,f∈P×F.Definition 8.
[13] A soft mapping ofSXAintoSYB, symbolized byfϕ, is defined by two crisp mapsf:X⟶Yandϕ:A⟶Bso that the image ofGAis a set inSXAand pre-image ofHBis a set inSYBand they are defined as follows.(i)
fϕGA=fϕGB is a set in SYB, where fϕGb=⋃a∈ϕ−1bfGa for each b∈B(ii)
fϕ−1HB=fϕ−1HA is a set in SXA, where fϕ−1Ha=f−1Hϕa for each a∈A.Definition 9.
[29] fϕ:SXA⟶SYBis called aλsoft map provided thatϕandfareλmaps, whereλ∈injective,surjective,andbijective.


## 2.2. Soft Topology

Definition 10.
[6] A familyΩof soft sets overXunder a fixed set of parametersPis said to be a soft topology onXif it satisfies the following:(i)
X~ and Φ~ are elements of Ω(ii)
The intersection of a finite number ofΩ is an element of Ω(iii)
The union of an arbitrary number ofΩ is an element of Ω.We will call a term of soft topological space on the tripleX,Ω,P, where a terminology of soft open is given for every member in Ω and a terminology of soft closed is given for a relative complement of a member of Ω.Proposition 11.
[6] InX,Ω,P, a familyΩp=Gp:GP∈Ωis a classical topology onXfor eachp∈P.We will call a term of parametric topology onΩp and a term of parametric topological space on X,Ωp.Definition 12.
[6] Y,ΩY,Pis called a soft subspace ofX,Ω,P, where∅≠Y⊆XandΩY=Y~∩GP:GP∈Ω.Definition 13.
[35] GPis called a soft semiopen subset ofX,Ω,PifGP⊆~clintGP.Theorem 14.
[35] Every soft open set is soft semiopen, and the union of an arbitrary number of soft semiopen sets is soft semiopen.Definition 15.
[35] The intersection of all soft semiclosed sets inX,Ω,PcontainingGPis the soft preclosure ofGP. It is symbolized byGP¯s.It is clear thatx∈GP¯s iff for each soft semiopen set UP totally containing x we have GP∩~UP≠Φ~; and Spx∈GP¯s iff for each soft semiopen set UP totally containing Spx we have GP∩~UP≠Φ~.Definition 16.
[36] X,Ω,Pwhich does not contain disjoint soft open sets is called soft hyperconnected.The following result will help us establish some properties of soft semiseparation axioms and soft semicompact spaces (see, for example, Theorem42 and Proposition 49). It implies that the family of soft semiopen subsets of a soft hyperconnected space X,Ω,P forms a new soft topology Ωsemi over X that is finer than Ω.Theorem 17.
[37] A class of soft semiopen subsets of a soft hyperconnected space forms a soft topology.Proposition 18.
LetY~be soft open set inX,Ω,P. Then(1)
IfH,P is soft semiopen and Y~ is soft open set in X,Ω,P, then H,P∩~Y,P is a soft semiopen set in Y,ΩY,P(2)
IfY~ is soft open in X,Ω,P and H,P is a soft semiopen in Y,ΩY,P, then H,P is a soft semiopen set in X,Ω,P.Definition 19.
[35] X,Ω,Pis said to be(i)
soft semi-T0 if there is a soft semiopen set UP for each x≠y∈X satisfying x∈UP and y∈UP; or y∈UP and x∈UP(ii)
soft semi-T1 if there are two soft semiopen sets UP and VP for each x≠y∈X satisfying x∈UP and y∈UP; and y∈VP and x∈VP(iii)
soft semi-T2 if there are two disjoint soft semiopen sets UP and VP for each x≠y∈X satisfying x∈UP and y∈VP(iv)
soft semiregular if every soft semiclosed setHP so that x∈HP, Ω contains two disjoint soft semiopen sets UP and VP satisfying HP⊆~UP and x∈VP(v)
soft seminormal if we separate every two disjoint soft semiclosed sets by two disjoint soft semiopen sets(vi)
soft semi-T3 (resp., soft semi-T4) if it is both soft semiregular (resp., soft seminormal) and soft semi-T1-spaceDefinition 20.
[38] A soft semiopen cover ofX,Ω,Pis a family of soft semiopen setsGiP:i∈Iso thatX~=∪~i∈IGiP.Definition 21.
X,Ω,Pis said to be(i)
soft semi-T′2 [37] if for every Spx≠Sp′y∈X~, there are twodisjoint soft semiopen sets UP and VP containing Spx and Sp′y, respectively(ii)
soft semicompact [38] if every soft semiopen cover of X~ has afinite subcover.Proposition 22.
[37] A soft semicompact (resp., stable soft semicompact) set in a soft semi-T2′-space (resp., soft semi-T2-space) is soft semiclosed.To study the properties that preserved under softsemi∗-homeomorphism maps, the concept of a soft semi-irresolute map will be presented in this work under the name of a soft semi∗-continuous map.Definition 23.
[37] gφ:X,Ω,P⟶Y,θ,Pis called softsemi∗−continuousif the inverse image of each soft semiopen set is soft semiopen.Proposition 24.
[38] The softsemi∗−continuousimage of a soft semicompact set is soft semicompact.The definitions of soft semicontinuous, soft semiopen, soft semiclosed, and soft semihomeomorphism maps were given in [35] in a similar way of their counterparts in classical topology.Definition 25.
A soft topologyΩonXis said to be(i)
an enriched soft topology [8] if all soft sets GP so that Gp=∅ or X are members of Ω(ii)
an extended soft topology [30] if Ω=GP:Gp∈Ωpforeachp∈P, where Ωp is a parametric topology on X.Al-shami and Kočinac [31] proved that extended and enriched soft topologies are identical and obtained some useful results that help to discover the interrelations between soft topology and its parametric topologies.Theorem 26.
[31] A subsetF,Pof an extended soft topological spaceX,Ω,Pis soft semiopen if and only if each p-approximate element ofF,Pis semiopen.Proposition 27.
[39] LetXi,Ωi,P:i∈Ibe a class of mutually disjoint soft topological spaces andX=⋃i∈IXi. Then the classΩ=G,P⊆~X~:G,P∩~Xi~isasoftopensetinXi,Ωi,Pforeveryi∈I.
forms a soft topology onX with P.Definition 28.
[39] X,Ω,Pgiven in the above proposition is said to be the sum of soft topological spaces and is symbolized by⊕i∈IXi,Ω,P.Theorem 29.
[39] A soft setG,P⊆~⊕i∈IXi~is soft semiopen (resp., soft semiclosed) in⊕i∈IXi,Ω,Pif and only if allG,P∩~Xi~are soft semiopen (resp., soft semiclosed) inXi,Ωi,P.Proposition 30.
[14] Letgφ:X,Ω,P⟶X,Ω,Pbe a soft map so that∩~n∈ℕgφnX~is a soft pointSpx. Then,Spxis a unique fixed point ofgφ.Theorem 31.
[29] LetΩ=GA×FB:GA∈ΩandFB∈θ, whereX,Ω,AandY,θ,Bare two soft topological spaces. A class of all arbitrary union of members ofΩdefines a soft topology overX×Yunder a fixed set of parametersA×B.Lemma 32.
LetG,PandH,Pbe two subsets ofX,Ω,PandY,θ,P, respectively. Then(i)
clG,P×clH,P=clG,P×H,P(ii)
intG,P×intH,P=intG,P×H,P.


## 3. Semisoft Separation Axioms

In this section, the concepts oftt-soft semi-Tii=0,1,2,3,4 and tt-soft semiregular spaces are introduced, where tt denotes the total belong and total nonbelong relations that are utilized in the definitions of these concepts. The relationships between them are shown, and their main features are studied. In addition, their behaviours with the concepts of hereditary, topological, and additive properties are investigated. Some examples are provided to elucidate the obtained results.First of all, we see that it is necessary to classify containment into several categories as it is shown in remark below. Factually, this classification will play a vital role in redefining many soft theoretic-set and soft topological concepts, in particular, the concepts of soft interior and closure operators, soft compactness, and soft separation axioms.Remark 33.
For a soft setGPandx∈X, we have the following notes:(i)
GP totally contains x if x∈GP(ii)
GP does not partially contain x if x∈GP(iii)
GP partially contains x if x⋐GP(iv)
GP does not totally contain x if x⋐GP.Definition 34.
X,Ω,Pis said to be(i)
tt-soft semi-T0 if there exists a soft semiopen set UP for every x≠y∈X satisfying x∈UP and y⋐UP or y∈UP and x⋐UP(ii)
tt-soft semi-T1 if there exist soft semiopen sets UP and VP for every x≠y∈X satisfying x∈UP and y⋐UP; and y∈VP and x⋐VP(iii)
tt-soft semi-T2 if there exist two disjoint soft semiopen sets UP and VP for every x≠y∈X satisfying x∈UP and y⋐UP; and y∈VP and x⋐VP(iv)
tt-soft semiregular if for every soft semiclosed set HP so that x⋐HP, there exist disjoint soft semiopen sets UP and VP so that HP⊆~UP and x∈VP(v)
tt-soft semi-T3 (resp., tt-soft semi-T4) if it is both tt-soft semiregular (resp., soft seminormal) and tt-soft semi-T1.Remark 35.
It can be noted that ifFPandGPare disjoint soft set, thenx∈FPiffx⋐GP. This implies thatX,Ω,Pis att-soft semi-T2-space iff is a soft semi-T2-space. That is, the concepts of att-soft semi-T2-space and a soft semi-T2-space are equivalent.We can say thatX,Ω,P is tt-soft semi-T2 if there exist two disjoint soft semiopen sets UP and VP for every x≠y∈X so that UP and VP totally contain x and y, respectively.Remark 36.
The soft semiregular spaces imply a strict condition on the shape of soft semiopen and soft semiclosed subsets. To explain this matter, letFPbe a soft semiclosed set so thatx∈HP. Then, we have two cases:(i)
There arep,p′∈P so that x∈Hp and x∈Hp′. This case is impossible because there do not exist two disjoint soft sets UP and VP containing x and HP, respectively(ii)
For eachp∈P, x∈Hp. This implies that HP must be stable.As a direct consequence, we infer that every soft semiclosed and soft semiopen subsets of a soft semiregular space must be stable. However, this matter does not hold on thett-soft semiregular spaces because we replace a partial nonbelong relation by a total nonbelong relation. Therefore a tt-soft semiregular space needs not be stable.Proposition 37.
(i)
Att-soft semi-Ti-space is always soft semi-Ti for i=0,1,4(ii)
A soft semiregular space is alwaystt-soft semiregular(iii)
A soft semi-T3-space is always tt-soft semi-T3.Proof.
Sincex⋐GP implies x∈GP, then the proofs of (i) and (ii) follow.
To prove (iii), it suffices to prove that a soft semi-Ti-space is tt-soft semi-Ti when X,Ω,P is soft semiregular. Suppose x≠y∈X. Then, there exist two soft semiopen sets UP and VP so that x∈UP and y∈UP; and y∈VP and x∈VP. Since UP and VP are soft semiopen subsets of a soft semiregular space, then they are stable. So y⋐UP and x⋐VP. Thus, X,Ω,P is tt-soft semi-T1. Hence, we obtain the desired result.
The following examples clarifies that the converse of the above proposition is not always true.Example 1.
LetP=p1,p2. A familyΩ=Φ~,X~,GiP:i=1,2,⋯,6is a soft topology onX=v,w, where(1)G1P=p1,v,p2,XG2P=p1,X,p2,wG3P=p1,v,p2,wG4P=p1,w,p2,∅G5P=p1,∅,p2,vG6P=p1,w,p2,v.Since the soft setsp1,v,p2,v and p1,w,p2,w are not soft semiopen subsets of X,Ω,P, then it is not tt-soft semi-T0. However, it is clear that X,Ω,P is a soft semi-T1-space. Also, it is soft seminormal. Therefore, it is soft semi-T4.Example 2.
LetP=p1,p2. A familyΩ=Φ~,X~,GiP:i=1,2,3,4is a soft topology onX=v,w, where(2)G1P=p1,v,p2,vG2P=p1,w,p2,wG3P=p1,v,p2,XG4P=p1,∅,p2,w.It can be checked that a set inX,Ω,P is soft open iff it is soft semiopen. Since there exists an unstable soft semiopen set in X,Ω,P, then X,Ω,P is not soft semiregular. Therefore, it is not soft semi-T3. However, X,Ω,P is a tt-soft semiregular space. Also, it is a tt-soft semi-T1-space. Hence, it is a tt-soft semi-T3-space.Before we show the relationship betweentt-soft semi-Ti-spaces, we need to prove the next useful lemma.Lemma 3.
X,Ω,Pis att-soft semi-T1-space iffxPis soft semiclosed for everyx∈X.Proof.
Necessity: for any distinct point yi∈X than x, Ω contains a soft semiopen set and GiP totally contains yi so that x⋐GiP. So ⋃i∈IGip=X\x and x⋐⋃i∈IGip for each p∈P. Thus, ∪~i∈IGiP=X\x~ is soft semiopen. Hence, xP is soft semiclosed.
Sufficiency: let x≠y. By hypothesis, xP and yP are soft semiclosed sets. Then xPc and yPc are soft semiopen sets so that x∈yPc and y∈xPc. Obviously, y⋐yPc and x⋐xPc. Hence, X,Ω,P is tt-soft semi-T1.Proposition 38.
Everytt-soft semi-Ti-space istt-soft semi-Ti−1fori=1,2,3,4.Proof.
We suffice by proving the cases ofi=3,4.
Fori=3, let x≠y in a tt-soft semi-T3-space X,Ω,P. Then, xP is soft semiclosed. Since y⋐xP and X,Ω,P is tt-soft semiregular, then Ω contains two disjoint soft semiopen sets GP and FP satisfying xP⊆~GP and y∈FP. Therefore, X,Ω,P is tt-soft semi-T2.
Fori=4, let HP be a soft semiclosed set so that x⋐HP. Since X,Ω,P is tt-soft semi-T1, then xP is soft semiclosed. Since xP∩~HP=Φ~ and X,Ω,P is soft seminormal, then there are disjoint soft semiopen sets GP and FP so that HP⊆~GP and xP⊆~FP. Hence, X,Ω,P is tt-soft semi-T3.We explain that the converse of proposition above is not always true by presenting the next example.Example 1.
LetP=p1,p2. A familyΩ=Φ~,X~,p1,x,p2,∅is a soft topology onX=v,w. The following soft sets withΩare all soft semiopen subsets ofX,Ω,P.
(3)G1P=p1,v,p2,vG2P=p1,v,p2,wG3P=p1,v,p2,XG4P=p1,X,p2,wG5P=p1,X,p2,w.Since the soft setsp1,v,p2,v are a soft semiopen set in X,Ω,P, then it is tt-soft semi-T0. However, p1,w,p2,w is not a soft semiopen set in X,Ω,P. Hence, it is not tt-soft semi-T1.Example 2.
LetΩ=Φ~,GP⊆~ℕ:GPcisfinitebe a soft topology on the set of natural numbersℕ, wherePis a set of parameters. By a simple check, one notes that a class of soft semiopen and a class of soft open sets inℕ,Ω,Pcoincide. For eachv≠w∈ℕ, we haveℕ\w~andℕ\v~are soft semiopen sets so thatv∈ℕ\w~andw⋐ℕ\w~; andw∈ℕ\v~andv⋐ℕ\v~. Therefore,ℕ,Ω,Pistt-soft semi-T1. In contrast, all nonnull proper semiopen sets have nonnull soft intersection. Hence,ℕ,Ω,Pis nottt-soft semi-T2.Example 3.
As we know, the coincidence of soft topological and classical topological spaces occurs ifEis a singleton. Then, it suffices to consider examples that satisfy a semi-T2-space but not semi-T3and satisfy a semi-T3-space but not semi-T4.In what follows, we found some properties oftt-soft semi-Ti and tt-soft semiregular.Lemma 39.
LetUPbe a subset ofX,Ω,Pandx∈X. Thenx⋐UP¯siff there is a soft semiopen setVPtotally containingxso thatUP∩~VP=Φ~.Proof.
Letx⋐UP¯s. Then, x∈UP¯sc=VP. So, UP∩~VP=Φ~. Conversely, if there exists a soft semiopen set VP totally containing x so that UP∩~VP=Φ~, then UP⊆VPc. Therefore, UP¯s⊆VPc. Since x⋐VPc, then x⋐UP¯s.Proposition 40.
IfX,Ω,Pis att-soft semi-T0-space, thenxP¯s≠yP¯sfor everyx≠y∈X.Proof.
Letx≠y in a tt-soft semi-T0-space. Then, there is a soft semiopen set UP so that x∈UP and y⋐UP or y∈UP and x⋐UP. Say, x∈UP and y⋐UP. Now, yP∩~UP=Φ~. So, by the above lemma, x⋐yP¯s. But x∈xP¯s. Hence, we obtain the desired result.Corollary 41.
IfX,Ω,Pis att-soft semi-T0-space, thenSpx¯s≠Sp′y¯sfor allx≠yandp,p′∈P.Theorem 42.
LetPbe a finite set andX,Ω,Pbe soft hyperconnected. ThenX,Ω,Pis att-soft semi-T1-space if and only ifxP=∩~UP:x∈UP∈Ωsfor eachx∈X.Proof.
To prove the “if” part, let y∈X. Then for each x∈X\y, we have a soft semiopen set UP so that x∈UP and y⋐UP. Therefore y⋐∩~UP:xP⊆~UP∈Ωs. Since y is chosen arbitrary, then the desired result is proved.
To prove the “only if” part, let the given conditions be satisfied and let x≠y. Let ∣E∣=m. Since y⋐xP, then for each j=1,2,⋯,m there is a soft semiopen set UiP so that y∈Uiej and x∈UiP. Since X,Ω,P is soft hyperconnected, then it follows from Theorem 17 that ∩~i=1i=mUiP is a soft semiopen set so that y⋐∩~i=1i=mUiP and x∈∩~i=1i=mUiP. Similarly, we can get a soft semiopen set VP so that y∈VP and x⋐VP. Thus, X,Ω,P is a tt-soft semi-T1-space.Theorem 43.
IfX,Ω,Pis an extendedtt-soft semi-T1-space, thenSpxis soft semiclosed for allSpx∈X~.Proof.
By Lemma3, we have X\x~ is a soft semiopen set. Since X,Ω,P is extended, then a soft set HP, where Hp=∅ and Hp′=X for each p′≠p, is a soft semiopen set. Therefore, X\x~∪~HP is soft semiopen. Thus, X\x~∪~HPc=Spx is soft semiclosed.Corollary 44.
IfX,Ω,Pis an extendedtt-soft semi-T1-space, then the intersection of all soft semiopen sets containingUPis exactlyUPfor eachUP⊆~X~.Proof.
LetUP be a soft subset of X~. Since Spx is a soft semiclosed set for every Spx∈UPc, then X~\Spx is a soft semiopen set containing UP. Therefore, UP=∩~X~Spx:Spx∈UPc, as required.Theorem 45.
LetX,Ω,Pbe finite soft hyperconnected. Then,X,Ω,Pistt-soft semi-T2if and only if it istt-soft semi-T1.Proof.
Necessity: it is obtained from Proposition 38.
Sufficiency: for each x≠y, we have xP and yP are soft semiclosed sets. Since X is finite, then ∪~y∈X\xyP and ∪~x∈X\yxP are soft semiclosed sets. Since X,Ω,P is soft hyperconnected, then ∪~y∈X\xyPc=xP and ∪~x∈X\yxPc=yP are soft semiopen sets. The disjointness of xP and yP ends the proof that X,Ω,P is tt-soft semi-T2.Remark 46.
In Example2, note thatxPis not a soft semiopen set for eachx∈ℕ. This clarifies that a soft setxPin att-soft semi-T1-space need not be soft semiopen if the universal set is infinite.Theorem 47.
X,Ω,Pistt-soft semiregular iff for every soft semiopen subsetFPofX,Ω,Ptotally containingx, there is a soft semiopen setVPso thatx∈VP⊆~VP¯s⊆~FP.Proof.
LetFP be a soft semiopen set totally containing x. Then FPc is semisoft closed and xP∩~FPc=Φ~. So Ω contains two disjoint soft semiopen sets UP and VP satisfying FPc⊆~UE and x∈VP. Thus, VP⊆~UPc⊆~FP. Hence, VP¯s⊆~UPc⊆~FP. Conversely, let FPc be a soft semiclosed set. Then, for each x⋐FPc, we have x∈FP. By hypothesis, there is a soft semiopen set VP totally containing x so that VP¯s⊆~FP. Therefore, FPc⊆~VP¯sc and VP∩~VP¯sc=Φ~. Thus, X,Ω,P is tt-soft semiregular, as required.Theorem 48.
The three concepts below are identical ifX,Ω,Pis att-soft semiregular space.(i)
att-soft semi-T2-space(ii)
att-soft semi-T1-space(iii)
att-soft semi-T0-space.Proof.
The directionsi⟶ii and ii⟶iii are obvious.
To proveiii⟶i, let x≠y in a tt-soft semi-T0-space X,Ω,P. Then, there exists a soft semiopen set GP so that x∈GP and y⋐GP, or y∈GP and x⋐GP. Say, x∈GP and y⋐GP. Obviously, x⋐GPc and y∈GPc. Since X,Ω,P is tt-soft semiregular, then there exist two disjoint soft semiopen sets UP and VP so that x∈UP and y∈GPc⊆~VP. Hence, X,Ω,P is tt-soft semi-T2.Proposition 49.
LetX,Ω,Pbe finite soft hyperconnected. IfX,Ω,Pis att-soft semi-T2-space, then it itt-soft semiregular.Proof.
LetHP be a soft semiclosed set and x∈X so that x⋐HP. Then x≠y for each y⋐HP. By hypothesis, there are two disjoint soft semiopen sets UiP and ViP so that x∈UiP and y∈ViP. Since y:y∈X is a finite set, then there is a finite number of soft semiopen sets ViP so that HP⊆~∪~i=1i=mViP. Since X,Ω,P is soft hyperconnected, then it follows from Theorem 17 that ∩~i=1i=mUiP is a soft semiopen set containing x. Since ∪~i=1i=mViP∩~∩~i=1i=mUiP=Φ~, then X,Ω,P is tt-soft semiregular.Corollary 50.
The following concepts are identical ifX,Ω,Pis finite soft hyperconnected.(i)
att-soft semi-T3-space(ii)
att-soft semi-T2-space(iii)
att-soft semi-T1-space.Proof.
We obtain the directionsi⟶ii and ii⟶iii from Proposition 38.
We obtain the directioniii⟶ii from Theorem 45.
We obtain the directionii⟶i from Proposition 49.Theorem 51.
The property of being att-soft semi-Ti-spacei=0,1,2,3is a soft open hereditary.Proof.
We suffice by proving the cases ofi=3.
LetY,ΩY,P be a soft open subspace of a tt-soft semi-T3-space X,Ω,P. To prove that Y,ΩY,P is tt-soft semi-T1, let x≠y∈Y. Since X,Ω,P is a tt-soft semi-T1-space, then there exist two soft semiopen sets GP and FP so that x∈GP and y⋐GP; and y∈FP and x⋐FP. Therefore, x∈UP=Y~∩~GP and y∈VP=Y~∩~FP so that y⋐UP and x⋐VP. It follows from Proposition 18 that UP and VP are soft semiopen subsets of Y,ΩY,P, so that Y,ΩY,P is tt-soft semi-T1.
To prove thatY,ΩY,P is tt-soft semiregular, let y∈Y and FP be a soft semiclosed subset of Y,ΩY,P so that y⋐FP. Then, FP∪~Yc~ is a soft semiclosed subset of X,Ω,P so that y⋐FP∪~Yc~. Therefore, there exist disjoint soft semiopen subsets UP and VP of X,Ω,P so that FP∪~Yc~⊆~UP and y∈VP. Now, UP∩~Y~ and VP∩~Y~ are disjoint soft semiopen subsets of Y,ΩY,P so that FP⊆~UP∩~Y~ and y∈VP∩~Y~. Thus, Y,ΩY,P is tt-soft semiregular.
Hence,Y,ΩY,P is tt-soft semi-T3, as required.Theorem 52.
LetX,Ω,Pbe extended andi=0,1,2,3,4. Then,X,Ω,Pistt-soft semi-TiiffX,Ωpis semi-Tifor eachp∈P.Proof.
We prove the theorem in the case ofi=4, and one can similarly prove the other cases.
Necessity: let x≠y in X. Then, there exist two soft semiopen sets UP and VP so that x∈UP and y⋐UP; and y∈VP and x⋐VP. Obviously, x∈Up and y∈Up; and y∈Vp and x∈Vp. Since X,Ω,P is extended, then it follows from Theorem 26 that Up and Vp are semiopen subsets of X,Ωp for each p∈P. Thus, X,Ωp is a semi-T1-space. To prove that X,Ωp is seminormal, let Fp and Hp be two disjoint semiclosed subsets of X,Ωp. Let FP and HP be two soft sets given by Fp=Fp, Hp=Hp, and Fp′=Hp′=∅ for each p′≠p. It follows from Theorem 26 that FP and HP are two disjoint soft semiclosed subsets of X,Ω,P. By hypothesis, there exist two disjoint soft semiopen sets GP and WP so that FP⊆~GP and HP⊆~WP. This implies that Fp=Fp⊆Gp and Hp=Hp⊆Wp. Since X,Ω,P is extended, then it follows from Theorem 26 that Gp and Wp are semiopen subsets of X,Ωp. Thus, X,Ωp is a seminormal space. Hence, it is a semi-T4-space.
Sufficiency: let x≠y in X. Then there exists two semiopen subsets Up and Vp of X,Ωp so that x∈Up and y∈Up; and y∈Vp and x∈Vp. Let UP and VP be two soft sets given by Up=Up and Vp=Vp for each p∈P. Since X,Ω,P is extended, then it follows from Theorem 26 that UP and VP are soft semiopen subsets of X,Ω,P so that x∈UP and y⋐UP; and y∈VP and x⋐VP. Thus, X,Ω,P is a tt-soft semi-T1-space. To prove that X,Ω,P is soft seminormal, let FP and HP be two disjoint soft semiclosed subsets of X,Ω,P. Since X,Ω,P is extended, then it follows from Theorem 26 that Fp and Hp are two disjoint semiclosed subsets of X,Ωp. By hypothesis, there exist two disjoint semiopen subsets Gp and Wp of X,Ωp so that Fp⊆Gp and Hp⊆Wp. Let GP and WP be two soft sets given by Gp=Gp and Wp=Wp for each p∈P. Since X,Ω,P is extended, then it follows from Theorem 26 that GP and WP are two disjoint soft semiopen subsets of X,Ω,P so that FP⊆~GP and HP⊆~WP. Thus, X,Ω,P is soft seminormal. Hence, it is a tt-soft semi-T4-space.In the following examples, we show that a condition of an extended soft topology given in the above theorem is not superfluous.Example 1.
LetX,Ω,Pbe the same as in Example1. We showed thatX,Ω,Pis nottt-soft semi-T0. On the other hand,Ωp1andΩp2are the discrete topology onX. Hence, the two parametric topological spacesX,Ωp1andX,Ωp2are semi-T4.Theorem 53.
The property of being att-soft semi-Ti-spacei=0,1,2is preserved under a finite product soft spaces.Proof.
We prove the theorem in case ofi=2. The other cases follow similar lines.
LetX1,Ω1,E1 and X2,Ω2,E2 be two tt-soft semi-T2-spaces and let x1,y1≠x2,y2 in X1×X2. Then, x1≠x2 or y1≠y2. Without loss of generality, let x1≠x2. Then there exist two disjoint soft semiopen subsets GE1 and HE1 of X1,Ω1,E1 so that x1∈GE1 and x2⋐GE1; and x2∈HE1 and x1⋐HE1. Obviously, GE1×X2~ and HE1×X2~ are two disjoint soft semiopen subsets X1×X2 so that x1,y1∈GE1×X2~ and x2,y2⋐GE1×X2~; and x2,y2∈HE1×X2~ and x1,y1⋐HE1×X2~. Hence, X1×X2 is a tt-soft semi-T2-space.For the sake of brevity, we omit the proof of the following result.Theorem 54.
The property of being att-soft semi-Ti-space is an additive property fori=0,1,2,3,4.We complete this section by discussing some interrelations betweentt-soft semi-Ti-spaces i=2,3,4 and soft semicompact spaces.Proposition 55.
A stable soft semicompact set in att-soft semi-T2-space is soft semiclosed.Proof.
It follows from Proposition22 and Remark 35.Theorem 56.
LetHPbe a soft semicompact subset of a soft hyperconnectedtt-soft semi-T2-space. Ifx⋐HP, then there are disjoint soft semiopen setsUPandVPso thatx∈UPandHP⊆VP.Proof.
Letx⋐HP. Then x≠y for each y⋐HP. Since X,Ω,P is a tt-soft semi-T2-space, then there exist disjoint soft semiopen sets UiP and ViP so that x∈UiP and y∈ViP. Therefore, ViP forms a soft semiopen cover of HP. Since HP is soft semicompact, then HP⊆∪~i=1i=nViP. By the soft hyperconnectedness of X,Ω,P, we obtain ⋂~i=1i=nUiP=UP is a soft semiopen set. Hence, we obtain the desired result.Theorem 57.
Every soft hyperconnected, soft semicompact, andtt-soft semi-T2-space istt-soft semiregular.Proof.
LetHP be a soft semiclosed subset of soft semicompact and tt-soft semi-T2-space X,Ω,P so that x⋐HP. Then, HP is soft semicompact. By Theorem 56, there exist disjoint soft semiopen sets UP and VP so that x∈UP and HP⊆VP. Thus, X,Ω,P is tt-soft semiregular.Corollary 58.
Every soft hyperconnected, soft semicompact, andtt-soft semi-T2-space istt-soft semi-T3.Lemma 59.
LetFPbe a soft semiopen subset of a soft semiregular space. Then, for eachSpx∈FP, there exists a soft semiopen setGPso thatSpx∈GP¯s⊆~FP.Proof.
LetFP be a soft semiopen set so that Spx∈FP. Then, x∈FPc. Since X,Ω,P is soft semiregular, then there exist two disjoint soft semiopen sets GP and WP totally containing x and FPc, respectively. Thus, x∈GP⊆~WPc⊆~FP. Hence, Spx∈GP⊆~GP¯s⊆~WPc⊆~FP.Theorem 60.
LetHPbe a soft semicompact subset of a soft semiregular space andFPbe a soft semiopen set containinHP. Then, there exists a soft semiopen setGPso thatHP⊆~GP⊆~GP¯s⊆~FP.Proof.
Let the given conditions be satisfied. Then, for eachSpx∈HP, we have Spx∈FP. Therefore, there is a soft semiopen set WxeP so that Spx∈WxeP⊆~WxeP¯s⊆~FP. Now, WxeP:Spx∈FP is a soft semiopen cover of HP. Since HP is soft semicompact, then HP⊆~∪~i=1i=nWxeP. Putting GP=∪~i=1i=nWxeP. Thus, HP⊆~GP⊆~GP¯s⊆~FP.Corollary 61.
IfX,Ω,Pis soft semicompact and soft semi-T3, then it istt-soft semi-T4.Proof.
Suppose thatF1P and F2P are two disjoint soft semiclosed sets. Then, F2P⊆~F1Pc. Since X,Ω,P is soft semicompact, then F2P is soft semicompact and since X,Ω,P is soft semiregular, then there is a soft semiopen set GP so that F2P⊆~GP⊆~GP¯s⊆~F1Pc. Obviously, F2P⊆~GP,F1P⊆~GP¯sc and GP∩~GP¯sc=Φ~. Thus, X,Ω,P is soft seminormal. Since X,Ω,P is soft semi-T3, then it is tt-soft semi-T1. Hence, it is tt-soft semi-T4.


## 4. Semifixed Soft Points of Soft Mappings

In this section, we introduce a semifixed soft point property and investigate some main features, in particular, those related to parametric topological spaces.Theorem 62.
LetBn:n∈ℕbe a collection of soft subsets of a soft semicompact spaceX,Ω,Psatisfying the following:(i)
Bn≠Φ~ for each n∈ℕ(ii)
Bn is a soft semiclosed set for each n∈ℕ(iii)
Bn+1⊆~Bn for each n∈ℕ.Then,∩~n∈ℕBn≠Φ~.Proof.
Suppose that∩~n∈ℕBn=Φ~. Then, ∪~n∈ℕBnc=X~. It follows from (ii) that Bnc:n∈ℕ is a soft semiopen cover of X~. By hypothesis of soft semicompactness, there exist i1,i2,⋯,ik∈ℕ, i1<i2<⋯<ik so that X~=Bi1c∪~Bi2c∪~⋯∪~Bikc. It follows from (iii) that Bik⊆~X~=Bi1c∪~Bi2c∪~⋯∪~Bikc=Bi1∩~Bi2∩~⋯∩~Bikc=Bikc. This yields a contradiction. Thus, we obtain the proof that ∩~n∈ℕBn≠Φ~.Proposition 63.
LetX,Ω,Pbe a soft semicompact and soft semi-T′2-space andgφ:X,Ω,P⟶X,Ω,Pbe a softsemi∗-continuous map. Then there exists a unique soft pointSpx∈X~ofgφ.Proof.
LetB1=gφX~andBn=gφBn−1=gφnX~foreachn∈ℕ be a family of soft subsets of X,Ω,P. It is clear that Bn+1⊆~Bn for each n∈ℕ. Since gφ is soft semi∗-continuous, then Bn is a soft semicompact set for each n∈ℕ and since X,Ω,P is soft semi-T′2, then Bn is also a soft semiclosed set for each n∈ℕ. It follows from Theorem 62 that H,P=∩~n∈ℕBn is a nonnull soft set. Note that gφH,P=gφ∩~n∈ℕgφnX~⊆~∩~n∈ℕgφn+1X~⊆~∩~n∈ℕgφnX~=H,P. To show that H,P⊆~gφH,P, suppose that there is a Spx∈H,P so that Spx∈gφH,P. Let Cn=gφ−1Spx∩~Bn. Obviously, Cn≠Φ~ and Cn⊆~Cn−1 for each n∈ℕ. Now, gφ−1Spx is a soft compact subset of a soft semi-T′2-space, then gφ−1Spx is soft closed; therefore, Cn is a soft semiclosed set for each n∈ℕ. By Theorem 62, there exists a soft point Pmy so that Pmy∈gφ−1Spx∩~Bn. Therefore, Spx=gφPmy∈gφH,P. This is a contradiction. Thus, gφH,P=H,P. Hence, the proof is complete.Definition 64.
(i)
X,Ω,P is said to have a semifixed soft point property if every soft semi∗-continuous map gφ:X,Ω,P⟶X,Ω,P has a fixed soft point(ii)
A property is said to be asemi∗-soft topological property if the property is preserved by soft semi∗-homeomorphism maps.Proposition 65.
The property of being a semifixed soft point is asemi∗-soft topological property.Proof.
LetX,Ω,P and Y,θ,P be a soft semi∗-homeomorphic. Then, there is a bijective soft map fφ:X,Ω,P⟶Y,θ,P so that fφ and fφ−1 are soft semi∗-continuous. Since X,Ω,P has a semifixed soft point property, then every soft semi∗-continuous map gφ:X,Ω,P⟶X,Ω,P has a semifixed soft point. Now, let hφ:Y,θ,P⟶Y,θ,P be a soft semi∗-continuous. Obviously, hφ∘fφ:X,Ω,P⟶Y,θ,P is a soft semi∗-continuous. Also, fφ−1∘hφ∘fφ:X,Ω,P⟶X,Ω,P is a soft semi∗-continuous. Since X,Ω,P has a semifixed soft point property, then fφ−1hφfφSpx=Spx for some Spx∈X~. consequently, fφfφ−1hφfφSpx=fφSpx. This implies that hφfφSpx=fφSpx. Thus, fφSpx is a semifixed soft point of hφ. Hence, Y,θ,P has a semifixed soft point property, as required.Before we investigate a relationship between soft topological space and their parametric topological spaces in terms of possessing a fixed (soft) point, we need to prove the following result.Theorem 66.
IfΩis extended, thengφ:X,Ω,P⟶Y,θ,Pis softsemi∗-continuous iffg:X,Ωp⟶Y,θϕpissemi∗-continuous.Proof.
Necessity: let U be a semiopen subset of Y,θϕp. Then θ contains a soft semiopen set GP satisfying Gϕp=U. Since gφ is a soft semi∗-continuous map, then gϕ−1GP is a soft semiopen set. Definition 8 implies that a soft subset gϕ−1GP=gϕ−1GP of X,Ω,P is given by gϕ−1Gp=g−1Gϕp for each p∈P. By hypothesis, Ω is extended; we obtain from Theorem 26 that a subset g−1Gϕp=g−1U of X,Ωp is semiopen. Hence, a map g is semi∗-continuous.
Sufficiency: let GP be a soft semiopen subset of Y,θ,P. According to Definition 8, a soft subset gϕ−1GP=gϕ−1GP of X,Ω,P is given by gϕ−1Gp=g−1Gϕp for each p∈P. Since a map g is semi∗-continuous, then a subset g−1Gϕp of X,Ωp is semiopen. By hypothesis, Ω is extended; we obtain from Theorem 26 that gϕ−1GP is a soft semiopen subset of X,Ω,P. Hence, a soft map gφ is soft semi∗-continuous.Definition 67.
X,Ωis said to have a semifixed point property if every semi∗-continuous mapg:X,Ω⟶X,Ωhas a fixed point.Proposition 68.
X,Ω,Phas the property of a semifixed soft point iffX,Ωphas the property of a semifixed point for eachp∈P.Proof.
Necessity: let X,Ω,P has the property of a semifixed soft point. Then every soft semi∗-continuous map gφ:X,Ω,P⟶X,Ω,P has a fixed soft point. Say, Spx. It follows from the above theorem that gp:X,Ωp⟶X,θϕp is semi∗-continuous. Since Spx is a fixed soft point of gφ, then it must be that gpx=x. Thus, gp has a fixed point. Hence, we obtain the desired result.
Sufficiency: let X,Ωp has the property of a semifixed point for each p∈P. Then, every semi∗-continuous map gp:X,Ωp⟶X,θϕp has a fixed point. Say, x. It follows from the above theorem that gφ:X,Ω,P⟶X,θ,P is soft semi∗-continuous. Since x is a fixed point of gp, then it must be that gφSpx=Spx. Thus, gφ has a fixed soft point. Hence, we obtain the desired result.


## 5. Conclusion

This work is a contribution of studying new classes to soft topology. First, we have introduced new soft separation axioms with respect to ordinary points by using total belong and total nonbelong relations. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft semiregular spaces. In general, we have studied their main properties and showed the interrelations between them with help of interesting examples. Second, we have defined semifixed soft point and investigated its basic properties.As future works, we shall study them on the contents of supra soft topological spaces, minimal soft topological spaces, and soft weak structures. Finally, we hope that the concepts initiated herein will find their applications in many fields soon.



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