A Soft Embedding Lemma for Soft Topological Spaces
Giorgio Nordo
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra (MIFT), Università di Messina, Viale F. Stagno D’Alcontres, 31, Contrada Papardo, Salita Sperone, 98166 Sant’Agata, Messina, Italy; giorgio.nordo@unime.it
Academic Editor: name
Version May 30, 2019 submitted to MDPI
Abstract:In 1999, Molodtsov initiated the theory of soft sets as a new mathematical tool for dealing 1
with uncertainties in many fields of applied sciences. In 2011, Shabir and Naz introduced and studied 2
the notion of soft topological spaces, also defining and investigating many new soft properties as 3
generalization of the classical ones. In this paper, we introduce the notions of soft separation between 4
soft points and soft closed sets in order to obtain a generalization of the well-known Embedding 5
Lemma to the class of soft topological spaces. 6
Keywords:soft set; soft sets theory; soft topology; embedding lemma; soft mapping; soft topological 7
product; soft slab; soft continuous mapping; soft diagonal mapping 8
1. Introduction 9
Almost every branch of sciences and many practical problems in engineering, economics, 10
computer science, physics, meteorology, statistics, medicine, sociology, etc. have its own uncertainties 11
and ambiguities because they depend on the influence of many parameters and, due to the inadequacy 12
of the existing theories of parameterization in dealing with uncertainties, it is not always easy to model 13
such a kind of problems by using classical mathematical methods. In 1999, Molodtsov [1] initiated the 14
novel concept of Soft Sets Theory as a new mathematical tool and a completely different approach for 15
dealing with uncertainties while modelling problems in a large class of applied sciences. 16
In the past few years, the fundamentals of soft set theory have been studied by many researchers. 17
Starting from 2002, Maji, Biswas and Roy [2,3] studied the theory of soft sets initiated by Molodtsov, 18
defining notion as the equality of two soft sets, the subset and super set of a soft set, the complement 19
of a soft set, the union and the intersection of soft sets, the null soft set and absolute soft set, and they 20
gave many examples. In 2005, Pei and Miao [4] and Chen et al. [5] improved the work of Maji. Further 21
contributions to the Soft Sets Theory were given by Yang [6], Ali et al. [7], Fu [8], Qin and Hong [9], 22
Sezgin and Atagün [10], Neog and Sut [11], Ahmad and Kharal [12], Babitha and Sunil [13], Ibrahim 23
and Yosuf [14], Singh and Onyeozili [15], Feng and Li [16], Onyeozili and Gwary [17], Ça ˘gman [18]. 24
In 2011, Shabir and Naz [19] introduced the concept of soft topological spaces, also defining 25
and investigating the notions of soft closed sets, soft closure, soft neighborhood, soft subspace and 26
some separation axioms. Some other properties related to soft topology were studied by Ça ˘gman, 27
Karata¸s and Enginoglu in [20]. In the same year Hussain and Ahmad [21] investigated the properties 28
of soft closed sets, soft neighbourhoods, soft interior, soft exterior and soft boundary, while Kharal 29
and Ahmad [22] defined the notion of a mapping on soft classes and studied several properties of 30
images and inverse images. The notion of soft interior, soft neighborhood and soft continuity were 31
also object of study by Zorlutuna, Akdag, Min and Atmaca in [23]. Some other relations between these 32
notions was proved by Ahmad and Hussain in [24]. The neighbourhood properties of a soft topological 33
space were investigated in 2013 by Nazmul and Samanta [25]. The class of soft Hausdorff spaces was 34
extensively studied by Varol and Aygün in [26]. In 2012, Aygüno ˘glu and Aygün [27] defined and 35
studied the notions of soft continuity and soft product topology. Some years later, Zorlutuna and 36
Çaku [28] gave some new characterizations of soft continuity, soft openness and soft closedness of soft 37
mappings, also generalizing the Pasting Lemma to the soft topological spaces. Soft first countable and 38
soft second countable spaces were instead defined and studied by Rong in [29]. Furthermore, the notion 39
of soft continuity between soft topological spaces was independently introduced and investigated by 40
Hazra, Majumdar and Samanta in [30]. Soft connectedness was also studied in 2015 by Al-Khafaj [31] 41
and Hussain [32]. In the same year, Das and Samanta [33,34] introduced and extensively studied the 42
soft metric spaces. In 2015, Hussain and Ahmad [35] redefined and explored several properties of soft 43
Ti (withi =0, 1, 2, 3, 4) separation axioms and discuss some soft invariance properties namely soft 44
topological property and soft hereditary property. In [36], Xie introduced the concept of soft points 45
and proved that soft sets can be translated into soft points so that they may conveniently dealt as same 46
as ordinary sets. In 2016, Tantawy, El-Sheikh and Hamde [37] continued the study of softTi-spaces 47
(fori = 0, 1, 2, 3, 4, 5) also discussing the hereditary and topological properties for such spaces. In 48
2017, Fu, Fu and You [38] investigated some basic properties concerning the soft topological product 49
space. Further contributions to the theory of soft sets and that of soft topology were added in 2011, by 50
Min [39], in 2012, by Janaki [40], and by Varol, Shostak and Aygün [41], in 2013 and 2014, by Peyghan, 51
Samadi and Tayebi [42], by Wardowski [43], by Nazmul and Samanta [44], by Peyghan [45], and by 52
Georgiou, Megaritis and Petropoulos [46,47], in 2015 by Uluçay, ¸Sahin, Olgun and Kiliçman [48], and 53
by Shi and Pang [49], in 2016 by Wadkar, Bhardwaj, Mishra and Singh [50], by Matejdes [51], and 54
by Fu and Fu [38], in 2017 by Bdaiwi [52], and, more recently, by Bayramov and Aras [53], and by 55
Nordo [54,55]. 56
In the present paper we will present the notions of family of soft mappings soft separating 57
soft points and soft points from soft closed sets in order to give a generalization of the well-known 58
Embedding Lemma for soft topological spaces. 59
2. Preliminaries 60
In this section we present some basic definitions and results on soft sets and suitably exemplify 61
them. Terms and undefined concepts are used as in [56]. 62
Definition 1. [1]LetUbe an initial universe set andEbe a nonempty set of parameters (or abstract attributes) 63
under consideration with respect toUand A ⊆ E, we say that a pair(F,A)is asoft setoverUif F is a 64
set-valued mapping F:A→P(U)which maps every parameter e∈ A to a subset F(e)ofU. 65
In other words, a soft set is not a real (crisp) set but a parameterized family{F(e)}e∈Aof subsets 66
of the universeU. For every parametere ∈ A,F(e)may be considered as the set ofe-approximate 67
elementsof the soft set(F,A). 68
Remark 1. In 2010, Ma, Yang and Hu [57] proved that every soft set(F,A)is equivalent to the soft set(F,E) 69
related to the whole set of parametersE, simply considering empty every approximations of parameters which 70
are missing in A, that is extending in a trivial way its set-valued mapping, i.e. setting F(e) =∅, for every 71
e∈E\A. 72
For such a reason, in this paper we can consider all the soft sets over the same parameter setEas in [58] and we 73
will redefine all the basic operations and relations between soft sets originally introduced in [1–3] as in [25], that 74
is by considering the same parameter set. 75
Definition 2. [23]The set of all the soft sets over a universeUwith respect to a set of parametersEwill be 76
denoted byS S(U)E. 77
Definition 3. [25]Let(F,E),(G,E)∈ S S(U)Ebe two soft sets over a common universeUand a common 78
set of parametersE, we say that(F,E)is asoft subsetof(G,E)and we write(F,E)⊆˜(G,E)if F(e)⊆G(e) 79
for every e∈E. 80
Definition 4. [25]Let (F,E),(G,E) ∈ S S(U)Ebe two soft sets over a common universeU, we say that 81
Remark 2. If (F,E),(G,E) ∈ S S(U)E are two soft sets over U, it is a trivial matter to note that 83
(F,E)=(˜ G,E)if and only if it results F(e) =G(e)for every e∈E. 84
Definition 5. [25]A soft set(F,E)over a universeUis said to be thenull soft setand it is denoted by(∅˜,E) 85
if F(e) =∅for every e∈E. 86
Definition 6. [25]A soft set(F,E)∈ S S(U)Eover a universeUis said to be theabsolute soft setand it is 87
denoted by(U˜,E)if F(e) =Ufor every e∈E. 88
Definition 7. Let(F,E)∈ S S(U)Ebe a soft set over a universeUand V be a nonempty subset of U, the 89
constant soft setof V, denoted by(V˜,E)) (or, sometimes, byV), is the soft set˜ (V,E), where V:E→P(U) 90
is the constant set-valued mapping defined by V(e) =V, for every e∈E. 91
Definition 8. [25]Let (F,E) ∈ S S(U)Ebe a soft set over a universeU, thesoft complement(or more 92
exactly the soft relative complement) of(F,E), denoted by(F,E){, is the soft setF{,Ewhere F{:E→P(U) 93
is the set-valued mapping defined by F{(e) =F(e){=U\F(e), for every e∈E. 94
Definition 9. [25] Let (F,E),(G,E) ∈ S S(U)E be two soft sets over a common universe U, thesoft 95
differenceof(F,E)and(G,E), denoted by(F,E)e\(G,E), is the soft set(F\G,E)where F\G:E→P(U)
96
is the set-valued mapping defined by(F\G)(e) =F(e)\G(e), for every e∈E. 97
Clearly, for every soft set(F,E)∈ S S(U)E, it results(F,E){= (˜ U˜,E)e\(F,E).
98
Definition 10. [25]Let(F,E),(G,E)∈ S S(U)Ebe two soft sets over a universeU, thesoft unionof(F,E) 99
and(G,E), denoted by(F,E)∪˜(G,E), is the soft set(F∪G,E)where F∪G:E→P(U)is the set-valued 100
mapping defined by(F∪G)(e) =F(e)∪G(e), for every e∈E. 101
Definition 11. [25]Let(F,E),(G,E)∈ S S(U)Ebe two soft sets over a universeU, thesoft intersection 102
of(F,E)and(G,E), denoted by(F,E)∩˜(G,E), is the soft set(F∩G,E)where F∩G :E →P(U)is the 103
set-valued mapping defined by(F∩G)(e) =F(e)∩G(e), for every e∈E. 104
Proposition 1. [18]For every soft set(F,E)∈ S S(U)E, the following hold: 105
(1) (F,E)∪˜ (F,E)= (˜ F,E). 106
(2) (F,E)∪˜ (∅˜,E)= (˜ F,E). 107
(3) (F,E)∪˜ (U˜,E)= (˜ U˜,E). 108
(4) (F,E)∩˜ (F,E)= (˜ F,E). 109
(5) (F,E)∩˜ (∅˜,E)= (˜ ∅˜,E). 110
(6) (F,E)∩˜ (U˜,E)= (˜ F,E). 111
Definition 12. [31]Two soft sets(F,E)and(G,E)over a common universeUare said to besoft disjointif 112
their soft intersection is the soft null set, i.e. if(F,E)∩˜(G,E)= (˜ ∅˜,E). If two soft sets are not soft disjoint, we 113
also say that theysoft meeteach other. In particular, if(F,E)∩˜(G,E)6= (˜ ∅˜,E)we say that(F,E)soft meets 114
(G,E). 115
Proposition 2. [19] Let (F,E),(G,E) ∈ S S(U)E be two soft sets over a universe U, we have that 116
(F,E)e\(G,E)=(˜ F,E)∩˜(G,E){.
117
The notions of soft union and intersection admit some obvious generalizations to a family with 118
Definition 13. [25]Let {(Fi,E)}i∈I ⊆ S S(U)E be a nonempty subfamily of soft sets over a universeU, 120
the (generalized)soft unionof{(Fi,E)}i∈I, denoted bye
S
i∈I(Fi,E), is defined by(Si∈IFi,E)whereSi∈IFi: 121
E→P(U)is the set-valued mapping defined by(Si∈IFi) (e) = S
i∈IFi(e), for every e∈E. 122
Definition 14. [25]Let {(Fi,E)}i∈I ⊆ S S(U)E be a nonempty subfamily of soft sets over a universeU, 123
the (generalized)soft intersectionof{(Fi,E)}i∈I, denoted by e
T
i∈I(Fi,E), is defined by(Ti∈IFi,E)where 124
T
i∈IFi:E→P(U)is the set-valued mapping defined by(Ti∈IFi) (e) =Ti∈IFi(e), for every e∈E. 125
Proposition 3. Let{(Fi,E)}i∈I ⊆ S S(U)Ebe a nonempty subfamily of soft sets over a universeU, it results: 126
(1) e
S
i∈I(Fi,E)
{
˜ = e
T
i∈I(Fi,E){. 127
(2) Tei∈I(Fi,E) {
˜
=Sei∈I(Fi,E){.
128
Definition 15. [36]A soft set(F,E)∈ S S(U)Eover a universeUis said to be asoft pointover U if it has 129
only one non-empty approximation which is a singleton, i.e. if there exists some parameterα∈Eand an element
130
p ∈Usuch that F(α) = {p}and F(e) = ∅for every e∈ E\ {α}. Such a soft point is usually denoted by
131
(pα,E). The singleton{p}is called the support set of the soft point andαis called the expressive parameter of
132
(pα,E).
133
Remark 3. In other words, a soft point(pα,E)is a soft set corresponding to the set-valued mapping pα :E→
(U)that, for any e∈E, is defined by
pα(e) =
(
{p} if e=α
∅ if e∈E\ {α} .
Definition 16. [36]The set of all the soft points over a universeUwith respect to a set of parametersEwill be 134
denoted byS P(U)E. 135
Since any soft point is a particular soft set, it is evident thatS P(U)E⊆ S S(U)E. 136
Definition 17. [36]Let (pα,E) ∈ S P(U)E and(F,E) ∈ S S(U)E be a soft point and a soft set over a
137
common universeU, respectively. We say thatthe soft point(pα,E)soft belongs to the soft set(F,E)
138
and we write(pα,E)∈˜(F,E), if the soft point is a soft subset of the soft set, i.e. if(pα,E)⊆˜(F,E)and hence
139
if p∈ F(α). We also say thatthe soft point(pα,E)does not belongs to the soft set(F,E)and we write
140
(pα,E)∈˜/(F,E), if the soft point is not a soft subset of the soft set, i.e. if(pα,E)6⊆˜(F,E)and hence if p∈/ F(α).
141
Definition 18. [33]Let(pα,E),(qβ,E)∈ S P(U)Ebe two soft points over a common universeU, we say that
142
(pα,E)and(qβ,E)aresoft equal, and we write(pα,E)=(˜ qβ,E), if they are equals as soft sets and hence if
143
p=q andα=β.
144
Definition 19. [33] We say that two soft points (pα,E) and (qβ,E) are soft distincts, and we write
145
(pα,E)6=(˜ qβ,E), if and only if p6=q orα6=β.
146
The notion of soft point allows us to express the soft inclusion in a more familiar way. 147
Proposition 4. Let(F,E),(G,E)∈ S S(U)Ebe two soft sets over a common universeUrespect to a parameter 148
setE, then(F,E)⊆˜(G,E)if and only if for every soft point(pα,E)∈˜(F,E)it follows that(pα,E)∈˜(G,E).
149
Proof. Suppose that(F,E)⊆˜(G,E). Then, for every(pα,E)∈˜(F,E), by Definition17, we have that
150
p ∈ F(α). Since, by Definition3, we have in particular that F(α) ⊆ G(α), it follows thatp ∈ G(α),
151
which, by Definition17, is equivalent to say that(pα,E)∈˜(G,E).
Conversely, suppose that for every soft point(pα,E)∈˜(F,E)it follows(pα,E)∈˜(G,E). Then, for every
153
e ∈ Eand any p ∈ F(e), by Definition 17, we have that the soft point(pe,E)∈˜(F,E). So, by our 154
hypotesis, it follows that(pe,E)∈˜(G,E)which is equivalent top∈ G(e). This proves thatF(e)⊆G(e) 155
for everye∈Eand so that(F,E)⊆˜(G,E). 156
Definition 20. [21]Let(F,E)∈ S S(U)Ebe a soft set over a universeUand V be a nonempty subset ofU, 157
thesub soft setof(F,E)over V, is the soft set VF,E
, whereVF : E →P(U)is the set-valued mapping 158
defined byVF(e) =F(e)∩V, for every e∈E. 159
Remark 4. Using Definitions7and11, it is a trivial matter to verify that a sub soft set of(F,E)over V can 160
also be expressed as VF,E
˜
= (F,E)∩˜(V˜,E). 161
Furthermore, it is evident that the sub soft set VF,E
above defined belongs to the set of all the soft sets over V 162
with respect to the set of parametersE, which is contained in the set of all the soft sets over the universeUwith 163
respect toE, that is VF,E
∈ S S(V)E⊆ S S(U)E. 164
Definition 21. [13,59]Let {(Fi,Ei)}i∈I be a family of soft sets over a universe setUiwith respect to a set 165
of parametersEi (with i ∈ I), respectively. Then thesoft product(or, more precisely, thesoft cartesian 166
product) of{(Fi,Ei)}i∈I, denoted by∏ei∈I(Fi,Ei), is the soft set(∏i∈IFi,∏i∈IEi)over the (usual) cartesian
167
product∏i∈IUiand with respect to the set of parameters∏i∈IEi, where∏i∈IFi:∏i∈IEi→P(∏i∈IUi)is 168
the set-valued mapping defined by∏i∈IFi(heiii∈I) =∏i∈IFi(ei), for everyheiii∈I ∈∏i∈IEi. 169
Proposition 5. [60]Let∏ei∈I(Fi,Ei)be the soft product of a family{(Fi,Ei)}i∈Iof soft sets over a universe
170
setUi with respect to a set of parametersEi (with i ∈ I), and let (pα,∏i∈IEi) ∈ S P(∏i∈IUi)∏i∈IEi 171
be a soft point of the product ∏i∈IUi, where p = hpiii∈I ∈ ∏i∈IUi and α = hαiii∈I ∈ ∏i∈IEi, then 172
(pα,∏i∈IEi)∈˜ ∏ei∈I(Fi,Ei)if and only if((pi)αi,Ei)∈˜(Fi,Ei)for every i∈ I. 173
Proof. In fact, by using Definitions21and17,(pα,∏i∈IEi)∈˜∏ei∈I(Fi,Ei)means thatp∈(∏i∈IFi) (α)
174
that ishpiii∈I ∈(∏i∈IFi) (hαiii∈I)which corresponds to say tat pi ∈ Fi(αi)for everyi∈ Iwhich, by 175
Definition17, is equivalent to((pi)αi,Ei)∈˜(Fi,Ei)for everyi∈ I. 176
Corollary 1. [60]The soft product of a family{(Fi,Ei)}i∈I of soft sets over a universe setUiwith respect 177
to a set of parameters Ei (with i ∈ I) is null if and only if at least one of its soft sets is null, that is 178
e
∏i∈I(Fi,Ei)=˜ ∅˜,∏i∈IEi
if f there exists some j∈I such that(Fj,Ej)=(˜ ∅˜,E). 179
Proposition 6. [60]Let{(Fi,Ei)}i∈Iand{(Gi,Ei)}i∈Ibe two families of soft sets over a universe setUiwith 180
respect to a set of parametersEi(with i∈ I), such that(Fi,Ei)⊆˜(Gi,Ei)for every i∈ I, then their respective 181
soft products are such that∏ei∈I(Fi,Ei)⊆˜ ∏ei∈I(Gi,Ei).
182
Proposition 7. [59]Let{(Fi,Ei)}i∈Iand{(Gi,Ei)}i∈Ibe two families of soft sets over a universe setUiwith respect to a set of parametersEi(with i∈ I), then it results:
f
∏
i∈I((Fi,Ei)∩˜(Gi,Ei)) =˜∏
fi∈I(Fi,Ei)∩˜∏
fi∈I(Gi,Ei).According to Remark1the following notions by Kharal and Ahmad have been simplified and 183
slightly modified for soft sets defined on a common parameter set. 184
Definition 22. [22]LetS S(U)EandS S(U0)E0 be two sets of soft open sets over the universe setsUand 185
U0 with respect to the sets of parameters E andE0, respectively. and consider a mapping
ϕ : U → U0
186
between the two universe sets and a mappingψ: E→E0 between the two set of parameters. The mapping
187
ϕψ : S S(U)E → S S(U0)E0 which maps every soft set (F,E) of S S(U)E to a soft set (ϕψ(F),E0) of
188
S S(U0)E0, denoted byϕψ(F,E), whereϕψ(F):E0→P(U0)is the set-valued mapping defined byϕψ(F)(e0) =
S
ϕ(F(e)):e∈ψ−1({e0}) for every e0 ∈ E0, is called a soft mappingfromU toU0 induced by the
190
mappingsϕandψ, while the soft setϕψ(F,E)=(˜ ϕψ(F),E0)is said to be thesoft imageof the soft set(F,E)
191
under the soft mappingϕψ:S S(U)E→ S S(U0)E0.
192
The soft mapping ϕψ : S S(U)E → S S(U0)E0 is saidinjective(respectivelysurjective,bijective) if the 193
mappingsϕ:U→U0andψ:E→E0are both injective (resp. surjective, bijective).
194
Remark 5. In other words a soft mappingϕψ : S S(U)E → S S(U0)E0 matches every set-valued mapping
F:E→P(U)to a set-valued mappingϕψ(F):E0→P(U0)which, for every e0 ∈E0, is defined by
ϕψ(F)(e
0) =
( S
e∈ψ−1({e0})ϕ(F(e)) ifψ
−1({e0})6=∅
∅ otherwise .
In particular, if the soft mappingϕψis bijective, the set-valued mappingϕψ(F):E0 →P(U0)is defined simply
195
byϕψ(F)(e0) =ϕ F ψ−1(e0)
, for every e0∈E0.
196
Let us also note that in some paper (see, for example, [26]) the soft mapping ϕψ is denoted with (ϕ,ψ) :
197
S S(U)E→ S S(U0)E0. 198
It is worth noting that soft mappings between soft sets behaves similarly to usual (crisp) mappings 199
in the sense that they maps soft points to soft points, as proved in the following property. 200
Proposition 8. Letϕψ : S S(U)E → S S(U0)E0 be a soft mapping induced by the mappingsϕ: U→U0
201
andψ :E →E0between the two setsS S(U)E,S S(U0)E0of soft sets. and consider a soft point(pα,E)of 202
S P(U)E. Then the soft imageϕψ(pα,E)of the soft point(pα,E)under the soft mappingϕψis the soft point
203
ϕ(p)ψ(α),E0
, i.e.ϕψ(pα,E)=˜
ϕ(p)ψ(α),E0
. 204
Proof. Let (pα,E) be a soft point of S P(U)E, by Definition 22, its soft image ϕψ(pα,E) under
the soft mapping ϕψ : S S(U)E → S S(U0)E0 is the soft set ϕψ(pα),E0
corresponding to the set-valued mapping ϕψ(pα) : E0 → P(U0) which, for everye0 ∈ E0, is defined by ϕψ(pα)(e0) =
S
ϕ(pα(e)):e∈ψ−1({e0}) . Now, ife0=ψ(α)we have that:
ϕψ(pα) (ψ(α)) =
[n
ϕ(pα(e)):e∈ψ
−1({
ψ(α)})
o
=[{ϕ(pα(e)):e∈E,ψ(e) =ψ(α)}
=[{ϕ(pα(e)):e=α} ∪
[
{ϕ(pα(e)):e∈E\ {α},ψ(e) =ψ(α)}
={ϕ(pα(α))} ∪
[
ϕ(pα(e)) =∅:e∈E\ {α},ψ(e) =e
0
={ϕ(p)} ∪∅
={ϕ(p)}
while, for everye0 ∈E0\ {
ψ(α)}, we have thatψ(α)6=e0and so it follows that:
ϕψ(pα)(e
0) =[n
ϕ(pα(e)):e∈ψ
−1({e0})o
=[
ϕ(pα(e)):e∈E,ψ(e) =e
0
=[
ϕ(pα(e)):e∈E\ {α},ψ(e) =e0
=[
ϕ(pα(e)) =∅:e∈E\ {α},ψ(e) =e
0
=∅.
This proves that the set-valued mapping ϕψ(pα) : E0 → P(U0) sends the parameterψ(α)to the
205
singleton {ϕ(p)} and maps every other parameters of E0\ {ψ(e)} to the empty set, and so, by
Definition15, this means that the soft image ϕψ(pα,E)of the soft point(pα,E) ∈ S P(U)E under
207
the soft mappingϕψ :S S(U)E→ S S(U0)E0is the soft point inS P(U0)E0having{ϕ(p)}as support
208
set andψ(α)as expressive parameter, that is
ϕ(p)ψ(α),E0
. 209
Corollary 2. Let ϕψ :S S(U)E→ S S(U0)E0be a soft mapping induced by the mappingsϕ:U→U0and
210
ψ:E →E0 between the two setsS S(U)E,S S(U0)E0of soft sets, thenϕψ is injective if and only if its soft
211
images of every distinct pair of soft points are distinct too, i.e. if for every(pα,E),(qβ,E)∈ S P(U)Esuch that
212
(pα,E)6=(˜ qβ,E)it follows thatϕψ(pα,E)6=˜ϕψ(qβ,E).
213
Proof. It easily derives from Definitions19and22, and Proposition8. 214
Definition 23. [22]Letϕψ :S S(U)E→ S S(U0)E0be a soft mapping induced by the mappingsϕ:U→U0
215
andψ : E → E0 between the two setsS S(U)E,S S(U0)E0 of soft sets and consider a soft set (G,E0)of 216
S S(U0)E0. Thesoft inverse imageof(G,E0)under the soft mappingϕψ :S S(U)E→ S S(U0)E0, denoted
217
byϕ−ψ1(G,E0)is the soft set(ϕ−ψ1(G),E0)ofS S(U)Ewhereϕ−ψ1(G):E→P(U)is the set-valued mapping 218
defined byϕ−ψ1(G)(e) =ϕ−1(G(ψ(e)))for every e∈E.
219
Proposition 9. [22,23,27]Let ϕψ : S S(U)E → S S(U0)E0 be a soft mapping induced by the mappings 220
ϕ:U→U0andψ:E→E0and let(F,E),(Fi,E)∈ S S(U)Eand(G,E0),(Gi,E0)∈ S S(U0)E0be soft sets 221
overUandU0, respectively, then the following hold: 222
(1) ϕψ ∅˜,E
˜
= ∅˜,E0
. 223
(2) ϕ−ψ1 ∅˜,E0=˜ ∅˜,E.
224
(3) ϕ−ψ1 U˜0,E0=˜ U˜,E.
225
(4) (F,E)⊆˜ ϕψ−1 ϕψ(F,E)
and the soft equality holds whenϕψis injective.
226
(5) ϕψ
ϕ−ψ1(G,E0)
˜
⊆(G,E0)and the soft equality holds whenϕψis surjective.
227
(6) ϕ−ψ1
(G,E0){=˜ ϕ−ψ1(G,E0)
{
. 228
(7) if(F1,E)⊆˜ (F2,E).thenϕψ(F1,E)⊆˜ ϕψ(F2,E). 229
(8) if(G1,E0)⊆˜ (G2,E0).thenϕψ−1(G1,E0)⊆˜ ϕψ−1(G2,E0).
230
(9) ϕψ
e
S
i∈I(Fi,E)
˜ = e
S
i∈Iϕψ(Fi,E).
231
(10) e
T
i∈Iϕψ(Fi,E)⊆˜ ϕψ
e
T
i∈I(Fi,E)
. 232
(11) ϕ−ψ1
e
S
i∈I(Gi,E0)
˜ = e
S
i∈Iϕ−ψ1(Gi,E
0).
233
(12) ϕ−ψ1
e
T
i∈I(Gi,E0)
˜ = e
T
i∈Iϕ−ψ1(Gi,E
0).
234
Proposition 10. [22]Letϕψ :S S(U)E→ S S(U0)E0be a soft mapping induced by the mappingsϕ:U→U0
235
andψ:E→E0and let(F,E),(G,E)∈ S S(U)Eand(F0,E0),(G0,E0)∈ S S(U0)E0be soft sets overUand 236
U0, respectively, then the following hold:
237
(1) (F,E)⊆˜ (G,E)impliesϕψ(F,E)⊆˜ ϕψ(G,E).
238
(2) (F0,E)⊆˜ (G0,E0)impliesϕ−ψ1(F0,E0)⊆˜ ϕψ−1(G0,E0).
239
Corollary 3. Let ϕψ :S S(U)E→ S S(U0)E0be a soft mapping induced by the mappingsϕ:U→U0and
240
ψ :E → E0. If(F,E) ∈ S S(U)Eand(F0,E0) ∈ S S(U0)E0 are soft sets overUandU0, respectively and 241
(pα,E)∈ S P(U)Eand(qβ,E
0)∈ S P(U0)
E0are soft points overUandU0, respectively, then the following 242
hold: 243
(1) (pα,E)∈˜(F,E)impliesϕψ(pα,E)∈˜ϕψ(F,E).
244
(2) (qβ,E0)∈˜(F0,E0)impliesϕ
−1
ψ (qβ,E0)⊆˜ ϕ
−1
ψ (F
0,E0).
Definition 24. Letϕψ :S S(U)E→ S S(U
0)
E0be a bijective soft mapping induced by the mappingsϕ:U→
246
U0and
ψ:E→E0. Thesoft inverse mappingofϕψ, denoted byϕ
−1
ψ , is the soft mappingϕ
−1
ψ = ϕ
−1
ψ−1 :
247
S S(U0)E0 → S S(U)Einduced by the inverse mappingsϕ−1:U0→Uandψ−1:E0 →Eof the mappingsϕ
248
andψ, respectively.
249
Remark 6. Evidently, the soft inverse mapping ϕ−ψ1 : S S(U0)E0 → S S(U)Eof a bijective soft mapping 250
ϕψ:S S(U)E→ S S(U0)E0is also bijective and its soft image of a soft set inS S(U0)E0coincides with the soft 251
inverse image of the corresponding soft set under the soft mappingϕψ.
252
Definition 25. [27]LetS S(U)E,S S(U0)E0andS S(U00)E00 be three sets of soft open sets over the universe 253
setsU,U0,U00with respect to the sets of parametersE,E0,E00, respectively, andϕψ:S S(U)E→ S S(U0)E0,
254
γδ :S S(U)E0 → S S(U0)E00 be two soft mappings between such sets, then thesoft compositionof the soft 255
mappingsϕψandγδ, denoted byγδe◦ϕψis the soft mapping(γ◦ϕ)δ◦ψ :S S(U)E→ S S(U
00)
E00 induced by 256
the compositionsγ◦ϕ:U→U00of the mappingsϕandγbetween the universe sets andδ◦ψ:E→E00of
257
the mappingsψandδbetween the parameter sets.
258
The notion of soft topological spaces as topological spaces defined over a initial universe with a 259
fixed set of parameters was introduced in 2011 by Shabir and Naz [19]. 260
Definition 26. [19]Let X be an initial universe set,Ebe a nonempty set of parameters with respect to X and 261
T ⊆ S S(X)Ebe a family of soft sets over X, we say thatT is asoft topologyon X with respect toEif the 262
following four conditions are satisfied: 263
(i) the null soft set belongs toT, i.e.(∅˜,E)∈ T. 264
(ii) the absolute soft set belongs toT, i.e.(X˜,E)∈ T. 265
(iii) the soft intersection of any two soft sets of T belongs toT, i.e. for every(F,E),(G,E) ∈ T then 266
(F,E)∩˜(G,E)∈ T. 267
(iv) the soft union of any subfamily of soft sets inT belongs toT, i.e. for every{(Fi,E)}i∈I ⊆ T then 268
e
S
i∈I(Fi,E)∈ T. 269
The triplet(X,T,E)is called asoft topological space(or soft space, for short) over X with respect toE. 270
In some case, when it is necessary to better specify the universal set and the set of parameters, the topology will 271
be denoted byT(X,E). 272
Definition 27. [19]Let(X,T,E)be a soft topological space over X with respect toE, then the members ofT 273
are said to besoft open setin X. 274
Definition 28. [30]LetT1andT2be two soft topologies over a common universe set X with respect to a set of 275
paramtersE. We say thatT2isfiner(or stronger) thanT1ifT1⊆ T2where⊆is the usual set-theoretic relation 276
of inclusion between crisp sets. In the same situation, we also say thatT1iscoarser(or weaker) thanT2. 277
Definition 29. [19]Let(X,T,E)be a soft topological space over X and(F,E)be a soft set over X. We say 278
that(F,E)issoft closed setin X if its complement(F,E){is a soft open set, i.e. if(F,E){∈ T. 279
Notation 1. The family of all soft closed sets of a soft topological space(X,T,E)over X with respect toEwill 280
be denoted byσ, or more precisely withσ(X,E)when it is necessary to specify the universal set X and the set of
281
parametersE. 282
Proposition 11. [19]Letσbe the family of soft closed sets of a soft topological space(X,T,E), the following
283 hold: 284
(1) the null soft set is a soft closed set, i.e.(∅˜,E)∈σ.
285
(2) the absolute soft set is a soft closed set, i.e.(X˜,E)∈σ.
(3) the soft union of any two soft closed sets is still a soft closed set, i.e. for every(C,E),(D,E)∈σthen
287
(C,E)∪˜(D,E)∈σ.
288
(4) the soft intersection of any subfamily of soft closed sets is still a soft closed set, i.e. for every{(Ci,E)}i∈I ⊆ 289
σthene
T
i∈I(Ci,E)∈σ.
290
Definition 30. [27]Let(X,T,E)be a soft topological space over X andB ⊆ T be a non-empty subset of 291
soft open sets. We say thatBis asoft open basefor(X,T,E)if every soft open set ofT can be expressed 292
as soft union of a subfamily ofB, i.e. if for every(F,E) ∈ T there exists someA ⊂ B such that(F,E) = 293
e
S
{(A,E): (A,E)∈ A}. 294
Proposition 12. [25]Let(X,T,E)be a soft topological space over X andB ⊆ T be a family of soft open sets 295
of X. ThenBis a soft open base for(X,T,E)if and only if for every soft open set(F,E)∈ T and any soft point 296
(xα,E)∈˜(F,E)there exists some soft open set(B,E)∈ Bsuch that(xα,E)∈˜(B,E)⊆˜(F,E).
297
Definition 31. [23]Let(X,T,E)be a soft topological space,(N,E)∈ S S(X)Ebe a soft set and(xα,E)∈
298
S P(X)Ebe a soft point over a common universe X. We say that(N,E)is asoft neighbourhoodof the soft 299
point(xα,E)if there is some soft open set soft containing the soft point and soft contained in the soft set, that is
300
if there exists some soft open set(A,E)∈ T such that(xα,E)∈˜(A,E)⊆˜(N,E).
301
Notation 2. The family of all soft neighbourhoods (sometimes also called soft neighbourhoods system) of a soft 302
point(xα,E) ∈ S P(X)Ein a soft topological space(X,T,E)will be denoted byN(xα,E)(or more precisely 303
withN(xT
α,E)if it is necessary to specify the topology). 304
Definition 32. [19]Let(X,T,E)be a soft topological space over X and(F,E)be a soft set over X. Then the
soft closureof the soft set(F,E)with respect to the soft space(X,T,E), denoted bys-clX(F,E), is the soft intersection of all soft closed set over X soft containing(F,E), that is
s-clX(F,E)=˜ f
\
(C,E)∈σ(X,E): (F,E)⊆˜(C,E) .
Proposition 13. [19]Let(X,T,E)be a soft topological space over X, and(F,E)be a soft set over X. Then the 305
following hold: 306
(1) s-clX(∅˜,E)= (˜ ∅˜,E). 307
(2) s-clX(X˜,E)= (˜ X˜,E). 308
(3) (F,E)⊆˜ s-clX(F,E). 309
(4) (F,E)is a soft closed set over X if and only ifs-clX(F,E)= (˜ F,E). 310
(5) s-clX(s-clX(F,E))=˜ s-clX(F,E). 311
Proposition 14. [19]Let(X,T,E)be a soft topological space and(F,E),(G,E)∈ S S(X)Ebe two soft sets 312
over a common universe X. Then the following hold: 313
(1) (F,E)⊆˜ (G,E)impliess-clX(F,E)⊆˜ s-clX(G,E). 314
(2) s-clX((F,E)∪˜(G,E))=˜ s-clX(F,E)∪˜ s-clX(G,E). 315
(3) s-clX((F,E)∩˜(G,E))⊆˜ s-clX(F,E)∩˜ s-clX(G,E). 316
Definition 33. [36]Let(X,T,E)be a soft topological space,(F,E)∈ S S(X)Eand(xα,E)∈ S P(X)Ebe a
317
soft set and a soft point over the common universe X with respect to the sets of parametersE, respectively. We 318
say that(xα,E)is asoft adherent point(sometimes also calledsoft closure point) of(F,E)if it soft meets
319
every soft neighbourhood of the soft point, that is if for every(N,E)∈ N(x
α,E),(F,E)∩˜(N,E)6= (˜ ∅˜,E). 320
As in the classical topological space, it is possible to prove that the soft closure coincides with the 321
Proposition 15. [36]Let(X,T,E)be a soft topological space,(F,E)∈ S S(X)Eand(xα,E)∈ S P(X)Ebe
323
a soft set and a soft point over the common universe X with respect to the sets of parametersE, respectively. 324
Then(xα,E)∈˜s-clX(F,E)if and only if(xα,E)is a soft adherent point of(F,E).
325
Having in mind the Definition20we can recall the following proposition. 326
Proposition 16. [21]Let(X,T,E)be a soft topological space over X, and Y be a nonempty subset of X, then the familyTYof all sub soft sets ofT over Y, i.e.
TY=
n
YF,E: (F,E)∈ To
is a soft topology on Y. 327
Definition 34. [21]Let(X,T,E)be a soft topological space over X, and let Y be a nonempty subset of X, the 328
soft topologyTY= Y
F,E
: (F,E)∈ T is said to be thesoft relative topologyofT on Y and(Y,TY,E) 329
is called thesoft topological subspaceof(X,T,E)on Y. 330
Proposition 17. Let(X,T,E)be a soft topological space over X, and(Y,TY,E)be its soft topological subspace over the subset Y ⊆ X, then a soft set (D,E) ∈ S S(Y)Eis a soft closed set respect to the soft subspace
(Y,TY,E)if and only if it is a sub soft set of some soft closed set of the soft space(X,T,E), i.e.
(D,E)∈σ(Y,E) ⇐⇒ ∃(C,E)∈σ(X,E): (D,E)=˜
YC,E.
Proof. It easily follows from Definitions29and34, Remark4, and Proposition2. 331
Proposition 18. [61]Let(X,T,E)be a soft topological space over X,(Y,TY,E)be its soft topological subspace on the subset Y⊆X, and(G,E)∈ S S(Y)Ebe a soft set over Y respect to the set of parameterE. Then the soft closure of(G,E)respect to the soft subspace(Y,TY,E)coincides with the soft intersection of its soft closure respect to the soft space(X,T,E)and of the absolute soft set(Y˜,E)of the subspace, that is
s-clY(G,E)=˜ s-clX(G,E)∩˜ (Y˜,E).
Definition 35. [23]Letϕψ :S S(X)E → S S(X0)E0be a soft mapping between two soft topological spaces
332
(X,T,E)and(X0,T0,E0)induced by the mappingsϕ:X→X0andψ:E→E0and(xα,E)∈ S P(X)Ebe
333
a soft point over X. We say that the soft mappingϕψissoft continuous at the soft point(xα,E)if for each
334
soft neighbourhood(G,E0)ofϕψ(xα,E)in(X
0,T0,E0)there exists some soft neighbourhood(F,E)of(x
α,E)
335
in(X,T,E)such thatϕψ(F,E)⊆˜ (G,E0).
336
Ifϕψis soft continuous at every soft point(xα,E)∈ S P(X)E, thenϕψ:S S(X)E→ S S(X0)E0is calledsoft
337
continuouson X. 338
Proposition 19. [23]Letϕψ :S S(X)E→ S S(X0)E0be a soft mapping between two soft topological spaces 339
(X,T,E)and(X0,T0,E0)induced by the mappings
ϕ:X→X0andψ:E→E0. Then the soft mappingϕψ
340
is soft continuous if and only if every soft inverse image of a soft open set in X0is a soft open set in X, that is, if 341
for each(G,E0)∈ T0we have thatϕψ−1(G,E0)∈ T.
342
Proposition 20. [23]Letϕψ :S S(X)E→ S S(X0)E0be a soft mapping between two soft topological spaces 343
(X,T,E)and(X0,T0,E0)induced by the mappings
ϕ:X→X0andψ:E→E0. Then the soft mappingϕψ
344
is soft continuous if and only if every soft inverse image of a soft closed set in X0is a soft closed set in X, that is, 345
if for each(C,E0)∈σ(X0,E0)we have thatϕ−ψ1(C,E0)∈σ(X,E).
346
Definition 36. [23]Letϕψ :S S(X)E → S S(X0)E0be a soft mapping between two soft topological spaces 347
(X,T,E)and(X0,T0,E0)induced by the mappings
ϕ:X→X0andψ:E→E0, and let Y be a nonempty
subset of X, therestriction of the soft mapping ϕψ to Y, denoted by ϕψ|Y, is the soft mapping(ϕ|Y)ψ :
349
S S(Y)E→ S S(X0)E0induced by the restrictionϕ|Y:Y→X0of the mappingϕbetween the universe sets 350
and by the same mappingψ:E→E0between the parameter sets.
351
Proposition 21. [23]Ifϕψ :S S(X)E→ S S(X0)E0is a soft continuous mapping between two soft topological
352
spaces(X,T,E)and(X0,T0,E0), then its restrictionϕψ|Y:S S(Y)E→ S S(X
0)
E0to a nonempty subset Y 353
of X is soft continuous too. 354
Proposition 22. Ifϕψ :S S(X)E→ S S(X
0)
E0is a soft continuous mapping between two soft topological 355
spaces(X,T,E)and(X0,T0,E0), then its corestriction
ϕψ :S S(X)E→ ϕψ(S S(X)E)is soft continuous
356 too. 357
Proof. It easily follows from Definitions22and23, and Proposition19. 358
Definition 37. [27]Let(X,T,E)be a soft topological space over X andS ⊆ T be a non-empty subset of soft 359
open sets. We say thatS is asoft open subbasefor(X,T,E)if the family of all finite soft intersections of 360
members ofSforms a soft open base for(X,T,E). 361
Proposition 23. [27]LetS ⊆ S S(X)Ebe a family of soft sets over X, containing both the null soft set(∅˜,E) 362
and the absolute soft set(X˜,E). Then the familyT(S)of all soft union of finite soft intersections of soft sets in 363
Sis a soft topology havingS as soft open subbase. 364
Definition 38. [27]LetS ⊆ S S(X)Ebe a a family of soft sets over X respect to a set of parametersEand such 365
that(∅˜,E),(X˜,E)∈ S, then the soft topologyT(S)of the above Proposition23is called thesoft topology 366
generatedby the soft open subbase S over X and (X,T(S),E)is said to be the soft topological space 367
generated bySover X. 368
Definition 39. [27]Let S S(X)E be the set of all the soft sets over a universe set X with respect to a set 369
of parameterEand consider a family of soft topological spaces{(Yi,Ti,Ei)}i∈I and a corresponding family 370
(ϕψ)i i∈Iof soft mappings(ϕψ)i = (ϕi)ψi :S S(X)E→ S S(Yi)Ei induced by the mappingsϕi:X→Yi 371
andψi : E → Ei (with i ∈ I). Then the soft topology T(S) generated by the soft open subbaseS = 372
n (ϕψ)
−1
i (G,Ei): (G,Ei)∈ Ti,i∈I
o
of all soft inverse images of soft open sets ofTiunder the soft mappings 373
(ϕψ)i is called theinitial soft topologyinduced on X by the family of soft mappings(ϕψ)i i∈Iand it is 374
denoted byTini X,E,Yi,Ei,(ϕψ)i;i∈ I
. 375
Proposition 24. [27]The initial soft topologyTini X,E,Yi,Ei,(ϕψ)i;i∈ I
induced on X by the family of 376
soft mappings
(ϕψ)i i∈I is the coarsest soft topology onS S(X)Efor which all the soft mappings(ϕψ)i : 377
S S(X)E→ S S(Yi)E
i (with i∈ I) are soft continuous. 378
Definition 40. [27] Let {(Xi,Ti,Ei)}i∈I be a family of soft topological spaces over the universe sets Xi 379
with respect to the sets of parameters Ei, respectively. For every i ∈ I, the soft mapping (πi)ρi : 380
S S(∏i∈IXi)∏i∈IEi → S S(Xi)Ei induced by the canonical projections πi : ∏i∈IXi → Xi and ρi : 381
∏i∈IEi→Eiis said the i-th soft projection mappingand, by setting(πρ)i = (πi)ρi, it will be denoted by 382
(πρ)i :S S(∏i∈IXi)∏i∈IEi → S S(Xi)Ei. 383
Definition 41. [27] Let {(Xi,Ti,Ei)}i∈I be a family of soft topological spaces and let
(πρ)i i∈I be the 384
corresponding family of soft projection mappings(πρ)i:S S(∏i∈IXi)∏i∈IEi → S S(Xi)Ei (with i∈I). Then, 385
the initial soft topologyTini ∏i∈IXi,E,Xi,Ei,(πρ)i;i∈I
induced on∏i∈IXiby the family of soft projection 386
mappings
(πρ)i i∈Iis called thesoft product topologyof the soft topologiesTi(with i∈ I) and denoted by 387
The triplet (∏i∈IXi,T(∏i∈IXi),∏i∈IEi) will be said the soft topological product space of the soft 389
topological spaces(Xi,Ti,Ei). 390
The following statement easily derives from Definition41and Proposition24. 391
Corollary 4. The soft product topology T(∏i∈IXi)is the coarsest soft topology over S S(∏i∈IXi)∏i∈IEi 392
for which all the soft projection mappings(πρ)i : S S(∏i∈IXi)∏i∈IEi → S S(Xi)Ei (with i ∈ I) are soft 393
continuous. 394
Definition 42. [60] Let (∏i∈IXi,T(∏i∈IXi),∏i∈IEi) be the soft topological product space of the soft 395
topological spaces (Xi,Ti,Ei) (with i ∈ I) and let (πρ)i : S S(∏i∈IXi)∏i∈IEi → S S(Xi)Ei be the i-th 396
the soft projection mapping. The inverse soft image of a soft open set(Fi,Ei) ∈ Tiunder the soft projection 397
mapping(πρ)i, that is(πρ)
−1
i (Fi,Ei)is called asoft slaband it is denoted byh(Fi,Ei)i. 398
Definitions37,39,41and42give immediately the following property. 399
Proposition 25. [60]The familyS ={h(Fi,Ei)i: (Fi,Ei)∈ Ti,i∈I}of all soft slabs of soft open sets ofTi 400
is a soft open subbase of the soft topological product space(∏i∈IXi,T(∏i∈IXi),∏i∈IEi). 401
Proposition 26. [60] Let (∏i∈IXi,T(∏i∈IXi),∏i∈IEi)be the soft topological product space of the soft topological spaces(Xi,Ti,Ei), with i ∈ I and let(Fj,Ej) ∈ Tj be a soft open set of Xj, then its soft slab h(Fj,Ej)icoincides with a soft cartesian product in which only the j-th component is the soft set(Fj,Ej)and the other ones are the absolute soft sets Xi˜,Ei
, that is
h(Fj,Ej)i=˜
∏
fi∈I(Ai,Ei) where (Ai,Ei) =
(
(Fj,Ej) if i= j
˜ Xi,Ei
otherwise .
Proof. By Definitions42and23, we have that
h(Fj,Ej)i= (˜ πρ)
−1
j (Fj,Ej)=˜ (πρ)
−1
j (Fj),
∏
i∈IEi
!
where(πρ)
−1
j (Fj) :∏i∈IEi →P(∏i∈IXi)is the set-valued mapping defined by
(πρ)
−1
j (Fj)
(e) = 402
π−j 1 Fj ρj(e)for everye=heiii∈I∈∏i∈IEi. 403
On the other hand, by Definition21, it results
f
∏
i∈I(Ai,Ei)=˜∏
i∈IAi,
∏
i∈IEi
!
where ∏i∈IAi : ∏i∈IEi → P(∏i∈IXi) is the set-valued mapping defined by (∏i∈IAi)(e) = 404
∏i∈IAi(ei) for every e = heiii∈I ∈ ∏i∈IEi and since Aj(ej) = Fj(ej) and Ai(ei) = Xi for every 405
i ∈ I\ {j}, it follows that(∏i∈IAi)(e) = hFj(ej)i, where the last set is the classical slab of the set 406
Fj(ej)in the usual cartesian product∏i∈IXi. Thus, we also have that(∏i∈IAi)(e) = πj−1(Fj(ej)) = 407
π−j 1(Fj(ρj(e))) =
(πρ)
−1
j (Fj)
(e)for everye∈E, and so, by Remark2, the soft equality holds. 408
Definition 43. [60]The soft intersection of a finite family of slabh(Fi1,Ei1)iof soft open sets(Fik,Eik) ∈ 409
Tik (with k = 1, . . .n), that is e
Tn
k=1h(Fik,Eik)i is said to be a soft n-slab and it is denoted by 410
h(Fi1,Ei1), . . .(Fin,Ein)i. 411
Proposition 27. [60]The family
B=
h(Fi1,Ei1), . . .(Fin,Ein)i: (Fik,Eik)∈ Tik,ik ∈I,n∈N
∗
of all soft n-slabs of soft open sets of Ti is a soft open base of the soft topological product space 413
(∏i∈IXi,T(∏i∈IXi),∏i∈IEi). 414
Proposition 28. [60] Let (∏i∈IXi,T(∏i∈IXi),∏i∈IEi)be the soft topological product space of the soft topological spaces(Xi,Ti,Ei), with i∈I and let(Fik,Eik)∈ Tik be a finite family of soft open sets of Xik, with
k=1, . . .n, respectively, then the soft n-slabh(Fi1,Ei1), . . .(Fin,Ein)icoincides with a soft cartesian product in
which only the ik-th components (with k=1, . . .n) are the soft sets(Fik,Eik)and the other ones are the absolute soft sets Xi˜,Ei
, that is
h(Fi1,Ei1), . . .(Fin,Ein)i=˜
∏
fi∈I(Ai,Ei) where
(Ai,Ei) =
(
(Fik,Eik) if i=ikfor some k=1, . . .n
˜ Xi,Ei
otherwise .
Proof. Similarly to the proof of Proposition26, by applying Definitions43,11,42and23, we have that
h(Fi1,Ei1), . . .(Fin,Ein)i =˜ e
Tn
k=1h(Fik,Eik)i
˜ = e
Tn k=1(πρ)
−1
ik (Fik,Eik)
˜ =Tn
k=1(πρ)
−1
ik (Fik),∏i∈IEi
where Tn k=1(πρ)
−1
ik (Fik) : ∏i∈IEi → P(∏i∈IXi) is the set-valued mapping defined by 415
Tn k=1(πρ)
−1
ik (Fik)
(e) =Tn
k=1πi−k1 Fik ρik(e)
for everye=heiii∈I∈∏i∈IEi. 416
On the other hand, by Definition21, it results
f
∏
i∈I(Ai,Ei)=˜∏
i∈IAi,
∏
i∈IEi
!
where ∏i∈IAi : ∏i∈IEi → P(∏i∈IXi) is the set-valued mapping defined by (∏i∈IAi)(e) = 417
∏i∈IAi(ei) for everye = heiii∈I ∈ ∏i∈IEi and since Aik(eik) = Fik(eik)for every k = 1, . . .nand 418
Ai(ei) =Xifor everyi∈ I\ {i1, . . .in}, it follows that(∏i∈IAi)(e) =hFi1(ei1), . . .Fin(ein)i, where the 419
last set is the classicaln-slab of the setsFik(eik)(fork=1, . . .n) in the usual cartesian product∏i∈IXi. 420
Thus, we also have that(∏i∈IAi)(e) =Tnk=1hFik(eik)i= Tn
k=1π−ik1(Fik(eik)) = Tn
k=1π−ik1(Fik(ρik(e))) = 421
Tn k=1(πρ)
−1
ik (Fik)
(e)for everye∈E, and so, by Remark2, the proposition is proved. 422
Proposition 29. [27]Let {(Xi,Ti,Ei)}i∈I be a family of soft topological spaces, (X,T(X),E) be the soft 423
topological product of such soft spaces induced on the product X=∏i∈IXiof universe sets with respect to the 424
productE =∏i∈IEiof the sets of parameters,(Y,T0,E0)be a soft topological space andϕψ :S S(Y)E0 → 425
S S(X)Ebe a soft mapping induced by the mappingsϕ:Y →X andψ:E0 →E. Then the soft mappings
426
ϕψis soft continuous if and only if, for every i∈ I, the soft compositions(πρ)ie◦ϕψ with the soft projection
427
mappings(πρ)i:S S(X)E→ S S(Xi)Ei are soft continuous mappings. 428
Let us note that the soft cartesian product∏ei∈I(Fi,Ei)of a family{(Fi,Ei)}i∈Iof soft sets over
429
a set Xi with respect to a set of parameters Ei, respectively, as introduced in Definition 21, is a 430
soft set of the soft topological product space(∏i∈IXi,T(∏i∈IXi),∏i∈IEi)i.e. that ∏ei∈I(Fi,Ei) ∈ 431
Proposition 30. [60]Let(∏i∈IXi,T(∏i∈IXi),∏i∈IEi)be the soft topological product space of a family {(Xi,Ti,Ei)}i∈Iof soft topological spaces and let∏ei∈I(Fi,Ei)be the soft product inS S(∏i∈IXi)∏
i∈IEi of a
family{(Fi,Ei)}i∈Iof soft sets ofS S(Xi)E
i, for every i∈ I. Then the soft closure of∏ei∈I(Fi,Ei)in the soft
topological product(∏i∈IXi,T(∏i∈IXi),∏i∈IEi)coincides with the soft product of the corresponding soft closures of the soft sets(Fi,Ei)in the corresponding soft topological spaces(Xi,Ti,Ei), that is:
s-cl∏i∈IXi
f
∏
i∈I(Fi,Ei)
˜ =
∏
fi∈Is-clXi(Fi,Ei).
Proof. Let(xα,∏i∈IEi)be a soft point ofS P(∏i∈IXi)∏i∈IEi, withx =hxiii∈Iandα=hαiii∈I, such that(xα,∏i∈IEi)∈˜ s-cl∏i∈IXi
e
∏i∈I(Fi,Ei)
. For anyj∈ I, let us consider a soft open set(Nj,Ej)∈ Tj
such that(xj)αj,Ej
˜
∈(Nj,Ej). By Proposition25, the soft slabh(Nj,Ej)iis a soft open set of the soft open subbase of the soft topological product space∏i∈IXi. By Proposition26, we know that
h(Nj,Ej)i=˜
∏
fi∈I(Ai,Ei) where (Ai,Ei) =
(
(Nj,Ej) ifi=j
˜ Xj,Ej
otherwise
and so that(xα,∏i∈IEi)∈h˜ (Ni,Ei)i. Thus, by our hypothesis, it follows that
f
∏
i∈I(Fi,Ei)∩˜∏
fi∈I(Ai,Ei)6=˜ ∅˜,∏
i∈IEi
!
which, by Proposition7, is equivalent to
f
∏
i∈I((Fi,Ei)∩˜(Ai,Ei)) 6=˜ ∅˜,∏
i∈IEi
!
and hence, by Corollary1, it follows in particular that
(Fj,Ej)∩˜(Aj,Ej)6=˜ ∅˜,Ej
i.e.
(Fj,Ej)∩˜(Nj,Ej)6=˜ ∅˜,Ej
.
Thus, by Definition33, we have that(xj)αj,Ejis a soft adherent point for the soft set(Fj,Ej)and so, by Proposition15, that
(xj)αj,Ej
˜
∈s-clXj(Fj,Ej), for any fixedj∈ I
that, by Proposition5, is equivalent to say that
xα,
∏
i∈I Ei
!
˜ ∈
∏
fi∈Is-clXi(Fi,Ei)
and, by using Proposition4, this proves that
s-cl∏i∈IXi
f
∏
i∈I(Fi,Ei)
˜ ⊆
∏
fi∈Is-clXi(Fi,Ei).
such that(xα,∏i∈IEi)∈˜(N,∏i∈IEi). By Propositions12and27and Definition30, we have that there exists a finite family of soft open sets(Nik,Eik)∈ Tikwithk=1, . . .nandn∈N
∗such that
xα,
∏
i∈I Ei
!
˜
∈ h(Ni1,Ei1), . . .(Nin,Ein)i⊆˜ N,
∏
i∈I Ei
!
.
Since, by Proposition28, we have that
h(Ni1,Ei1), . . .(Nin,Ein)i=˜
∏
fi∈I(Ai,Ei)where
(Ai,Ei) =
(
(Nik,Eik) ifi=ikfor somek=1, . . .n
˜ Xi,Ei
otherwise ,
it follows that
xα,
∏
i∈I Ei
!
˜ ∈
∏
fi∈I(Ai,Ei)⊆˜ N,
∏
i∈IEi
!
.
Now, we claim that
f
∏
i∈I(Ai,Ei)∩˜∏
fi∈I(Fi,Ei)6=˜ ∅˜,∏
i∈IEi
!
.
In fact, for every k = 1, . . .n, we have that (xik)αik,Eik
∈ Tik and so, being
(xik)αik,Eik
˜
∈s-clXik(Fik,Eik), by Proposition15and Definition33, it follows that
Aik,Eik
˜
∩ Fik,Eik
˜
= Nik,Eik
˜
∩ Fik,Eik
˜
6
= ∅˜,Eik
while, for everyi∈ I\ {i1, . . .in}, by Proposition1(6), it trivially results
(Ai,Ei)∩˜(Fi,Ei)=˜ X˜i,Ei
˜
∩(Fi,Ei)= (˜ Fi,Ei)6=˜ ∅˜,Ei
and so the previous assertion follows from Proposition1. 433
Thus, a fortiori, we have that
N,
∏
i∈IEi
!
˜ ∩
∏
fi∈I(Fi,Ei)6=˜ ∅˜,
∏
i∈IEi
!
which, by Definition33and Proposition15, means that
xα,
∏
i∈I Ei
!
˜
∈ s-cl∏i∈IXi
f
∏
i∈I(Fi,Ei)
and hence, by Proposition4, we have
f
∏
i∈Is-clXi(Fi,Ei)⊆˜ s-cl∏i∈IXif
∏
i∈I(Fi,Ei)
that concludes our proof. 434
3. Soft Embedding Lemma 435
Definition 44. [62]Let (X,T,E)and(X0,T0,E0)be two soft topological spaces over the universe sets X
436
and X0 with respect to the sets of parameters E and E0, respectively. We say that a soft mapping ϕψ :
437
mapping ϕ−ψ1 : S S(X0)E0 → S S(X)E is soft continuous too. In such a case, the soft topological spaces 439
(X,T,E)and(X0,T0,E0)are saidsoft homeomorphicand we write that(X,T,E)˜
≈(X0,T0,E0). 440
Definition 45. Let (X,T,E)and(X0,T0,E0)be two soft topological spaces. We say that a soft mapping 441
ϕψ:S S(X)E→ S S(X0)E0is asoft embeddingif its corestrictionϕψ :S S(X)E→ϕψ(S S(X)E)is a soft
442
homeomorphism. 443
Definition 46. [62]Let(X,T,E)and(X0,T0,E0)be two soft topological spaces. We say that a soft mapping
444
ϕψ:S S(X)E→ S S(X0)E0is asoft closed mappingif the soft image of every soft closed set of(X,T,E)is
445
a soft closed set of(X0,T0,E0), that is if for any(C,E)∈σ(X,E), we haveϕψ(C,E)∈σ(X0,E0).
446
Proposition 31. Let ϕψ : S S(X)E → S S(X0)E0 be a soft mapping between two soft topological spaces 447
(X,T,E) and(X0,T0,E0). If
ϕψ is a soft continuous, injective and soft closed mapping then it is a soft
448
embedding. 449
Proof. If we consider the soft mapping ϕψ : S S(X)E → ϕψ(S S(X)E), by hypothesis and
450
Proposition22, it immediately follows that it is a soft continuous bijective mapping and so we have 451
only to prove that its soft inverse mappingϕ−ψ1= ϕ−1ψ−1 :ϕψ(S S(X)E)→ S S(X)Eis continuous
452
too. In fact, because the bijectiveness of the corestriction and Remark6, for every soft closed set 453
(C,E)∈σ(X,E), the soft inverse image of the(C,E)under the soft inverse mappingϕ−ψ1coincides with
454
the soft image of the same soft set under the soft mappingϕψ, that is
ϕ−ψ1
−1
(C,E)=˜ ϕψ(C,E)and
455
since by hypothesisϕψis soft closed, it follows that
ϕ−ψ1
−1
(C,E)∈σ(X0,E0)which, by Proposition
456
20, proves thatϕ−ψ1:S S(X0)E0 → S S(X)Eis a soft continuous mapping, and so, by Proposition21, 457
we finally have thatϕ−ψ1:ϕψ(S S(X)E)→ S S(X)Eis a soft continuous mapping.
458
Definition 47. Let(X,T,E)be a soft topological space over a universe set X with respect to a set of parameter 459
E, let {(Xi,Ti,Ei)}i∈I be a family of soft topological spaces over a universe set Xi with respect to a set of 460
parametersEi, respectively and consider a family
(ϕψ)i i∈Iof soft mappings(ϕψ)i = (ϕi)ψi :S S(X)E→ 461
S S(Xi)Ei induced by the mappings ϕi : X → Xi andψi : E → Ei (with i ∈ I). Then the soft mapping 462
∆ = ϕψ : S S(X)E → S S(∏i∈IXi)∏i∈IEi induced by the diagonal mappings (in the classical meaning) 463
ϕ=∆i∈Iϕi :X→∏i∈IXi on the universes sets andψ=∆i∈Iψi :E→∏i∈IEion the sets of parameters 464
(respectively defined byϕ(x) =hϕi(x)ii∈Ifor every x∈X and byψ(e) =hψi(e)ii∈Ifor every e∈E) is called 465
thesoft diagonal mappingof the soft mappings(ϕψ)i (with i ∈ I) and it is denoted by∆ = ∆i∈I(ϕψ)i : 466
S S(X)E→ S S(∏i∈IXi)∏i∈IEi. 467
The following proposition establishes a useful relation about the soft image of a soft diagonal 468
mapping. 469
Proposition 32. [60]Let(X,T,E)be a soft topological space over a universe set X with respect to a set of parameterE, let(F,E)∈ S S(X)Ebe a soft set of X, let{(Xi,Ti,Ei)}i∈Ibe a family of soft topological spaces over a universe set Xiwith respect to a set of parametersEi, respectively and let∆=∆i∈I(ϕψ)i :S S(X)E→ S S(∏i∈IXi)∏i∈IEi be the soft diagonal mapping of the soft mappings(ϕψ)i, with i∈I. Then the soft image of
the soft set(F,E)under the soft diagonal mapping∆is soft contained in the soft product of the soft images of the same soft set under the soft mappings(ϕψ)i, that is
∆(F,E)⊆˜
∏
fProof. Setϕ=∆i∈Iϕi:X→∏i∈IXiandψ=∆i∈Iψi :E→∏i∈IEi, by Definition47, we know that
∆=∆i∈I(ϕψ)i =ϕψ. Suppose, by contradiction, that there exists some soft point(xα,E)∈˜(F,E)such
that
∆(xα,E)∈˜/
∏
fi∈I(ϕψ)i(F,E).Set yβ,∏i∈IEi
˜
=∆(xα,E)=˜ ϕψ(xα,E), by Proposition8, it follows that
yβ,
∏
i∈I Ei
!
˜
= ϕ(x)ψ(α),
∏
i∈I Ei
!
where
y=hyiii∈I =ϕ(x) = (∆i∈Iϕi) (x) =hϕi(x)ii∈I
and
β=hβiii∈I =ψ(α) = (∆i∈Iψi) (α) =hψi(α)ii∈I.
So, set(Gi,Ei)= (˜ ϕψ)i(F,E)for everyi∈ I, we have that
yβ,
∏
i∈I Ei
!
˜/ ∈
∏
fi∈I(Gi,Ei)
hence, by Proposition5, it follows that there exists somej∈ Isuch that
(yj)βj,Ej∈˜/(Gj,Ej)
that, by Definition17, means
yj∈/Gj(βj)
i.e.
ϕj(x)∈/Gj ψj(α)
and so, by using again Definition17, we have
ϕj(x)ψj(α),Ej
˜/
∈(Gj,Ej)
that, by Proposition8, is equivalent to
(ϕψ)j(xα,E)∈˜/(Gj,Ej)
which is a contradiction because we know that (xα,E)∈˜(F,E) and by Corollary 3(1) it follows
470
(ϕψ)j(xα,E)∈˜(ϕψ)j(F,E)= (˜ Gj,Ej). 471
Definition 48. Let
(ϕψ)i i∈I be a family of soft mappings(ϕψ)i :S S(X)E → S S(Xi)Ei between a soft 472
topological space(X,T,E)and the members of a family of soft topological spaces{(Xi,Ti,Ei)}i∈I. We say that 473
the family
(ϕψ)i i∈Isoft separates soft pointsof(X,T,E)if for every(xα,E),(yβ,E)∈ S P(X)Esuch
474
that(xα,E)6=(˜ yα,E)there exists some j∈ I such that(ϕψ)j(xα,E)6=(˜ ϕψ)j(yβ,E).
475
Definition 49. Let
(ϕψ)i i∈I be a family of soft mappings (ϕψ)i : S S(X)E → S S(Xi)Ei between a 476
soft topological space(X,T,E)and the members of a family of soft topological spaces{(Xi,Ti,Ei)}i∈I. We 477
say that the family
(ϕψ)i i∈Isoft separates soft points from soft closed setsof(X,T,E)if for every 478
(C,E)∈ σ(X,E)and every(xα,E)∈ S P(X)Esuch that(xα,E)∈˜ X˜,E
e
\(C,E)there exists some j ∈ I 479
such that(ϕψ)j(xα,E)∈˜/s-clXj (ϕψ)j(C,E)
Proposition 33(Soft Embedding Lemma). Let(X,T,E)be a soft topological space,{(Xi,Ti,Ei)}i∈Ibe a 481
family of soft topological spaces and
(ϕψ)i i∈Ibe a family of soft continuous mappings(ϕψ)i :S S(X)E→ 482
S S(Xi)E
i that separates both the soft points and the soft points from the soft closed sets of(X,T,E). Then the 483
soft diagonal mapping∆=∆i∈I(ϕψ)i:S S(X)E→ S S(∏i∈IXi)∏i∈IEi of the soft mappings(ϕψ)iis a soft 484
embedding. 485
Proof. Letϕ =∆i∈Iϕi,ψ =∆i∈Iψiand∆= ∆i∈I(ϕψ)i = ϕψ as in Definition47, for everyi ∈ I, by
using Definition25, we have that every corresponding soft composition is given by
(πρ)ie◦∆= (πi)ρi
e
◦ϕψ= (πi◦ϕ)ρ
i◦ψ = (ϕi)ψi = (ϕψ)i
which, by hypothesis, is a soft continuous mapping. Hence, by Proposition29, it follows that the soft 486
diagonal mapping∆:S S(X)E→ S S(∏i∈IXi)∏i∈IEi is a soft continuous mapping. 487
Now, let(xα,E)and(yβ,E)be two distinct soft points ofS P(X)E. Since, by hypothesis, the
family
(ϕψ)i i∈Iof soft mappings soft separates soft points, by Definition48, we have that there exists somej∈ Isuch that(ϕψ)j(xα,E)6=(˜ ϕψ)j(yβ,E), that is
ϕj
ψj(xα,E) ˜
6
= ϕj
ψj(yβ,E). Hence, by Proposition8, we have that:
ϕj(x)ψ
i(α),Ej
˜
6=ϕj(y)ψ
i(β),Ej
and so, by the Definition19of distinct soft points, it necessarily follows that:
ϕj(x)6=ϕj(y) or ψj(α)6=ψj(β).
Sinceϕ=∆i∈Iϕi :X→∏i∈IXiandψ=∆i∈Iψi:E→∏i∈IEiare usual diagonal mappings, we have that:
ϕ(x)6=ϕ(y) or ψ(α)6=ψ(β)
and, by Definition19, it follows that:
ϕ(x)ψ(α),
∏
i∈I Ei
!
˜
6= ϕ(y)ψ(β),
∏
i∈I Ei
!
hence, applying again Proposition8, we get:
ϕψ(xα,E)=6˜ ϕψ(yβ,E)
that is:
∆i∈I(ϕψ)i(xα,E)=6˜ ∆i∈I(ϕψ)i(yβ,E)
i.e. that∆(xα,E)6=˜ ∆(yβ,E)which, by Corollary2, proves the injectivity of the soft diagonal mapping
488
∆:S S(X)E→ S S(∏i∈IXi)∏i∈IEi. 489
Finally, let(C,E) ∈σ(X,E)be a soft closed set inXand, in order to prove that the soft image
∆(C,E)is a soft closed set ofσ(∏i∈IXi,∏i∈IEi), consider a soft point(xα,E) ∈ S P(X)Esuch that
∆(xα,E)∈˜/∆(C,E)and, hence, by Corollary3(1), such that(xα,E)∈˜/(C,E). Since, by hypothesis, the
family
(ϕψ)i i∈Iof soft mappings soft separates soft points from soft closed sets, by Definition49, we have that there exists somej∈ Isuch that(ϕψ)j(xα,E)∈˜/ s-clXj (ϕψ)j(C,E), that is:
(ϕj)ψj(xα,E)∈˜/ s-clXj (ϕψ)j(C,E)
