Beyond the realm of soft topology, soft continuity can aid in the creation of digital images and computational topological applications. This paper investigates soft almost weakly continuous, a novel family of generalized soft continuous functions. The soft pre-continuous and soft weakly continuous function classes are included in this class. We obtain many characterizations of soft almost weakly continuous functions. Furthermore, we investigate the link between soft almost weakly continuous functions and their general topology counterparts. We present adequate conditions for a soft almost weakly continuous function to become soft weakly continuous (soft pre-continuous). We also present various results of soft composition, restriction, preservation, product, and soft graph theorems in terms of soft almost weakly continuous functions.
Citation: Samer Al-Ghour, Jawaher Al-Mufarrij. Soft almost weakly continuous functions and soft Hausdorff spaces[J]. AIMS Mathematics, 2024, 9(12): 35218-35237. doi: 10.3934/math.20241673
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Beyond the realm of soft topology, soft continuity can aid in the creation of digital images and computational topological applications. This paper investigates soft almost weakly continuous, a novel family of generalized soft continuous functions. The soft pre-continuous and soft weakly continuous function classes are included in this class. We obtain many characterizations of soft almost weakly continuous functions. Furthermore, we investigate the link between soft almost weakly continuous functions and their general topology counterparts. We present adequate conditions for a soft almost weakly continuous function to become soft weakly continuous (soft pre-continuous). We also present various results of soft composition, restriction, preservation, product, and soft graph theorems in terms of soft almost weakly continuous functions.
Mathematical modeling of uncertainty in economics, engineering, social sciences, environmental science, and health is necessary to solve complex problems. Despite their shortcomings, other theories like fuzzy sets [1] and rough sets could be useful in managing ambiguity and uncertainty. Digital image processing combines stochastic and cognitive uncertainties. Stochastic uncertainty can be reduced using statistical models and deep learning algorithms, while cognitive uncertainty is handled via fuzzy logic and uncertainty propagation techniques. Uncertainty modeling improves the robustness and accuracy of image processing in complex applications such as medical imaging, autonomous driving, and remote sensing [2]. Particularly in numerical and computational topological applications, soft set theory is a more straightforward and comprehensive method than fuzzy or rough sets. In this framework, soft continuity is a useful idea for handling ambiguous, partial, or incomplete data. The parameters in a manner that deviates from the strict frameworks of rough or fuzzy set theory. The requirement for additional parametrization tools is one of the main areas where this mathematical technique needs to be improved. Molodtsov [3] created the soft set theory in response to criticisms of previous uncertainty management strategies. Soft sets, or parameterized universe possibilities, were considered. Uncertainty in set modeling was initially demonstrated in [4] and enhanced in [5]. This consistent construction also has a wide range of applications. Soft set theory has been used in finance [6], medicine [7,8,9], statistics [10,11], decision-making [12,13,14], and forecasting [15] …. These real-world applications have proved the framework's problem-solving abilities while also confirming its applicability and efficacy. Several academics have examined and investigated the key concepts and principles of soft set theory [16,17,18].
To construct a soft topology for a given set of parameters, Shabir and Naz [19] defined a soft topology spanning a family of soft sets. More research in this field was prompted by their work, which clarified the relationships between concepts in soft topology and classical topology. Since the inception of soft topology, several contributions have been made to the study of topological concepts in soft contexts, including soft semi-compact spaces [20], soft nodec spaces [21], soft weakly quasi-continuous functions [22], generalized soft open sets [23,24,25], soft Q-sets [26,27], soft separation axioms [28,29,30], sum of soft topological spaces [31], and generated soft topological spaces [32].
The authors [33] examined soft set mappings and their potential applications in medical diagnosis. The concept of soft mapping with characterizations was first presented in [34]. Soft continuity for soft mappings was introduced in [35]. Several modifications, such as soft α-continuity [36], soft β-continuity [37], soft continuity [38,39], soft SD-continuity [40], soft ωp-continuity [41], soft ω0-continuity [42], soft bi-continuity [43], and soft semi ω-continuity [44] appeared. The smooth transition of a function between its values at adjacent places is characterized by this mathematical idea, which is explored in depth in these publications.
Numerous studies in soft topology and other branches of mathematics have focused on soft continuity. Soft continuity has an important role in many fields, such as engineering, science, business, economics, and soft topological models. Scientists have taken an interest in this subject. Applying soft set theory in fields like computational topology, where handling imprecision and uncertainty is a significant difficulty, can result from an understanding of its significance. In image processing, uncertainty is a natural challenge. Smooth transition patterns can be the framework, particularly in applications like edge detection and image extraction where blurred areas at white margins and smooth transitions are highly realistic. Applying models to handle stochastic and perceptual uncertainty can lead to more resilient systems for dealing with uncertainty. Basic simulation models enhance the accuracy and reliability of digital imaging and help bridge theoretical and practical applications, especially in areas such as medical imaging, self-driving cars, and surveillance. Soft topology provides a flexible way to model data by allowing partial membership in sets, which helps manage ambiguity in ambiguous or incomplete data. It is particularly useful in fields such as bioinformatics, machine learning, and image processing, where data is sparse or noisy. This topology enhances the flexibility of the model by handling ambiguity, making it valuable for applications such as sensor data analysis, pattern recognition, and clustering, further raising readers' interest in its benefits in managing uncertainty. This inspired us to write this paper.
This paper looks into soft almost weakly continuous functions, which are a new class of generalized soft continuous functions. This class includes both the soft pre-continuous and soft weakly continuous function classes. We find several characterizations of soft almost weakly continuous functions. Furthermore, we look at the relationship between soft nearly weakly continuous functions and their general topology equivalents. We provide sufficient requirements for a soft almost weakly continuous function to become soft weakly continuous (soft pre-continuous). We also offer findings from the soft composition, limitation, preservation, product, and soft graph theorems in terms of soft nearly weakly continuous functions.
Our findings have possible important implications for the development of digital image processing and computational topological applications, indicating soft topology's ability to expand beyond its traditional boundaries. As the subject of soft topology advances, this study sets the door for future investigation of soft continuous functions and their applications in a variety of disciplines.
Let M and T be two non-empty sets, with M being a set of parameters. A soft set over T relative to M is a function from M to T's powerset. SS(T,M) refers to the collection of all soft sets over T relative to M. Assume that K is in SS(T,M). K is denoted by CU if K(a)=U for all a∈M. C∅ will be denoted by 0M, and CT by 1M. If K(a)=U and K(d)=∅ for every d∈M−{a}, then aU denotes K. For convenience, a{x} shall be denoted as ax and will be referred to as a soft point for every a∈M and x∈T. The collection of all soft points over T with respect to M is denoted by SP(T,M). ax∈SP(T,M) is considered to belong to K∈SS(T,M) (notation: ax˜∈K) if x∈K(a). Let S,R∈SS(T,M). Then S is a soft subset of R, denoted by S˜⊆R, if S(a)⊆R(a) for each a∈M. The soft union (resp. intersection, difference) of S and R is denoted by S˜∪R (resp. S˜∩R, S−R) and defined by (S˜∪R)(a)=S(a)∪R(a) (resp. (S˜∩R)(a)=S(a)∩R(a), (S−R)(a)=S(a)−R(a)) for each a∈M. For any sub-collection R⊆SS(T,M), the soft union (resp. soft intersection) of the members of R are denoted by ˜∪R∈RR (resp. ˜∩R∈RR) and defined by (˜∪R∈RR)(a)=∪R∈RR(a) (resp. (˜∩R∈RR)(a)=∩R∈RR(a)) for each a∈M. Let SS(T,M) and SS(W,N) be two families of soft sets, and s:T⟶W, v:M⟶N be two functions. Then a soft mapping fsv:SS(T,M)⟶SS(W,N) is defined as follows: For each H∈SS(T,M) and K∈SS(W,N), (fsv(H))(b)=∅ if v−1(b)=∅, (fsv(H))(b)=∪a∈v−1(b)s(H(a)) if v−1(b)≠∅, and (f−1sv(K))(a)=s−1(K(v(a))). A sub-collection σ⊆SS(T,M) is called a soft topology on T relative to M, and the triplet (T,σ,M) is called a soft topological space if {0M,1M}⊆σ, S˜∩R∈σ for any {S,R}⊆σ, and ˜∪R∈RR for any R⊆σ. Let (T,σ,M) be a soft topological space and let R∈SS(T,M). Then R is called a soft open set in (T,σ,M) if R∈σ and R is called a soft closed set in (T,σ,M) if 1M−R∈σ.
To be clear, throughout this work, we will make reference to concepts and terms from [45,46]. The acronyms TS and STS stand for topological space and soft topological space, respectively.
Now, let us review some of the main concepts that will be applied in the follow-up.
Definition 1.1. [47] A function p:(G,ℑ)⟶(H,ℵ) between TSs is called almost weakly continuous (a.w.c) if p−1(V)˜⊆Intℑ(Clℑ(p−1(Clℵ(V)))) for each V∈ℵ.
In this paper, we would like mainly to extend the concept of almost weak continuity in classical topology to include STSs.
Definition 1.2. Let (G,Ψ,L) be a STS and let H∈SS(G,L). Then
(a) H is a soft semi-open [48] (resp. soft pre-open [49], soft α -open [44], soft regular-open [50]) set in (G,Ψ,L) if H˜⊆ClΨ(IntΨ(H))
(resp. H˜⊆IntΨ(IntΨ(H)), H˜⊆IntΨ(ClΨ(IntΨ(H))), H=IntΨ(ClΨ(H))). The family of all soft semi-open sets (resp. soft pre-open sets, soft α-open sets, soft regular-open sets) in (G,Ψ,L) will be denoted by SO(Ψ) (resp. PO(Ψ), α(Ψ), RO(Ψ)).
(b) H is called a soft pre-closed [49] (resp. soft regular-closed [50]) set in (G,Ψ,L) if 1L−H∈PO(Ψ) (resp. 1L−H∈RO(Ψ)). The family of all soft pre-closed sets (resp. soft regular-closed sets) in (G,Ψ,L) will be denoted by PC(Ψ) (resp. RC(Ψ)).
Definition 1.3. [49] Let (G,Ψ,L) be a STS and let H∈SS(G,L). Then
(a) pIntΨ(H) represents the soft pre-interior of H in (G,Ψ,L) and is defined by
pIntΨ(H)=˜∪{R:R∈PO(Ψ) and R˜⊆H}. |
(b) pClΨ(H) represents the soft pre-closure of H in (G,Ψ,L) and is defined by
pClΨ(H)=˜∩{D:D∈PC(Ψ) and H˜⊆D}. |
Definition 1.4. [51] Let (G,Ψ,L) be a STS and let H∈SS(G,L). The soft θ-closure of H in (G,Ψ,L) is denoted by ClθΨ(H), where θClφ(K)∈SS(G,L) and defined as follows:
ax˜∈ClθΨ(H) iff for each K∈Ψ such that ax˜∈K, H˜∩ClΨ(K)≠0L.
Definition 1.5. A soft function fsv:(G,Ψ,L)⟶(H,Φ,M) is called
(a) soft semi-continuous [52] if f−1sv(Y)∈SO(Ψ) for every Y∈Φ.
(b) soft pre-continuous [53] if f−1sv(Y)∈PO(Ψ) for every Y∈Φ.
(c) soft weakly continuous [54] if for every dx∈SP(G,L) and every R∈Φ such that fsv(dx)˜∈R, there exists T∈Ψ such that dx˜∈T and fsv(T)˜⊆ClΦ(R).
(d) soft almost α-continuous [55] if for every dx∈SP(G,L) and every R∈RO(Φ) such that fsv(dx)˜∈R, we find T∈α(Ψ) such that dx˜∈T and fsv(T)˜⊆R.
Definition 1.6. A STS (G,Ψ,L) is called
(a) [56] soft Hausdorff if for each dx,ey∈SP(G,L) such that dx≠ey, we find T,R∈Ψ such that dx˜∈T, ey˜∈R, and T˜∩R=0L.
(b) [56] soft regular if for each dx∈SP(G,L) and every T∈Ψ such that dx˜∈T, we find R∈Ψ such that dx˜∈R˜⊆ClΨ(R)˜⊆T.
(c) [56] soft Urysohn if for each dx,ey∈SP(G,L) such that dx≠ey, we find T,R∈Ψ such that dx˜∈T, ey˜∈R, and ClΨ(T)˜∩ClΨ(R)=0L.
(d) [57] soft pre-T2 if for each dx,ey∈SP(G,L) such that dx≠ey, we find T,R∈PO(Ψ) such that dx˜∈T, ey˜∈R, and T˜∩R=0L.
(e) [58] soft submaximal if {M:ClΨ(M)=1L}⊆Ψ.
For a soft function fsv:SP(G,A)⟶SP(H,B), the soft set
˜∪{(a,v(a))(x,s(x)):a∈A and x∈G} is represented by Graph(fsv) and is called the soft graph of fsv. So, (d,e)(x,y)˜∈Graph(fsv) iff fsv(dx)=ey iff s(x)=y and v(d)=e.
Definition 1.7. [22] Let fsv:(G,Ψ,L)⟶(H,Φ,M) be a soft function. Then Graph(fsv) is said to be soft strongly closed with respect to (G×H,pr(Ψ×Φ),L×M) if for each (d,e)(x,y)˜∈1L×M−Graph(fsv), there exist T∈Ψ and R∈Φ such that dx˜∈T, ey˜∈R, and (T×ClΦ(R))˜∩Graph(fsv)=0L×M.
Definition 2.1. A soft function fsv:(G,Ψ,L)⟶(H,Φ,M) is called soft almost weakly continuous (soft a.w.c) if f−1sv(Y)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(Y)))) for each Y∈Φ.
Theorem 2.2. Let {(G,φa):a∈L} and {(H,σb):b∈M} be two collections of TSs. Consider the functions s:G⟶H and v:L⟶M, where v is a bijection. Then fsv:(G,⊕a∈Lφa,L)⟶(H,⊕b∈Mσb,M) is soft a.w.c iff s:(G,φa)⟶(H,σv(a)) is a.w.c for all a∈L.
Proof. Necessity. Let fsv:(G,⊕a∈Lφa,L)⟶(H,⊕b∈Mσb,M) be soft a.w.c. Let a∈L. Let U∈σw(a). Then, (v(a))U∈⊕b∈Mσb. So, f−1sv((v(a))U)˜⊆Int⊕a∈Lφa(Cl⊕a∈Lφa(f−1sv(Cl⊕b∈Mσb((v(a))U)))). Thus,
(f−1sv((v(a))U))(a)⊆(Int⊕a∈Lφa(Cl⊕a∈Pφa(f−1sv(Cl⊕b∈Mσb((v(a))U)))))(a). Since v:L⟶M is injective, f−1sv((v(a))U)=as−1(U) and so, (f−1sv((v(a))U))(a)=(as−1(U))(a)=s−1(U). By Lemma 4.9 of [59],
(Int⊕a∈Lφa(Cl⊕a∈Lφa(f−1sv(Cl⊕b∈Mσb((v(a))U)))))(a)=Intφa(Clφa((f−1sv(Cl⊕b∈Mσb((v(a))U)))(a))).
Furthermore, it is not difficult to check that (f−1sv(Cl⊕b∈Mσb((v(a))U)))(a)=s−1(Clσv(a)(U)). Therefore, s−1(U)⊆Intφa(Clφa(s−1(Clσv(a)(U)))). This shows that s:(G,φa)⟶(H,σv(a)) is a.w.c.
Sufficiency.Let s:(G,φa)⟶(H,σv(a)) be a.w.c for all a∈L. Let K∈⊕b∈Mσb. Then, K(b)∈σb for all b∈M. For every b∈M, s:(G,φv−1(b))⟶(H,σb) is a.w.c, and so
s−1(K(b))⊆Intφv−1(b)(Clφv−1(b)(s−1(Clσb(K(b))))).
Claim. f−1sv(K)˜⊆Int⊕a∈Lφa(Cl⊕a∈Lφa(f−1sv(Cl⊕b∈Mσb(K)))) which ends the proof.
Proof of Claim. Let ax˜∈f−1sv(K). Then fsv(ax)=(v(a))s(x)˜∈K. So, s(x)∈K(v(a)) and thus, x∈s−1(K(v(a)))⊆Intφa(Clφa(s−1(Clσv(a)(K(v(a)))))). It is not difficult to check that s−1(Clσv(a)(K(v(a)))=(f−1sv(Cl⊕b∈Mσb(K)))(a). Thus, x∈Intφa(Clφa((f−1sv(Cl⊕b∈Mσb(K)))(a)))). Furthermore, by Lemma 4.9 of [59],
Intφa(Clφa((f−1sv(Cl⊕b∈Mσb(K)))(a))))=(Int⊕a∈Lφa(Cl⊕a∈Lφa(f−1sv(Cl⊕b∈Mσb(K)))))(a).
This shows that ax˜∈Int⊕a∈Lφa(Cl⊕a∈Lφa(f−1sv(Cl⊕b∈Mσb(K)))).
Corollary 2.3. Consider the functions s:(G,ℑ)⟶(H,ℵ) and v:L⟶M, where v is a bijection. Then s:(G,ℑ)⟶(H,ℵ) is a.w.c iff fsv:(G,τ(ℑ),L)⟶(H,τ(ℵ),M) is soft a.w.c.
Proof. For each a∈L and b∈M, put φa=ℑ and σb=ℵ. Then τ(ℑ)=⊕a∈Lφa and τ(ℵ)=⊕b∈Mσb. Theorem 2.2 ends the proof.
Theorem 2.4. Soft weakly continuous functions are soft a.w.c.
Proof. Let fsv:(G,Ψ,L)⟶(H,Φ,M) be soft weakly continuous. Let Y∈Φ. Then by Theorem 5.1 of [59],
f−1sv(Y)˜⊆IntΨ(f−1sv(ClΦ(Y)))˜⊆IntΨ(ClΨ(f−1sv(ClΦ(Y)))).
It follows that fsv is soft a.w.c.
Theorem 2.5. Soft pre-continuous functions are soft a.w.c.
Proof. Let fsv:(G,Ψ,L)⟶(H,Φ,M) be soft pre-continuous. Let Y∈Φ. Then, f−1sv(Y)˜⊆IntΨ(ClΨ(f−1sv(Y)))˜⊆IntΨ(ClΨ(f−1sv(ClΦ(Y)))). It follows that fsv is soft a.w.c.
The following two examples show that the implications in Theorems 2.4 and 2.5 are not reversible in general:
Example 2.6. Let ℑ and ℵ be the indiscrete and discrete topologies on R. Consider the identity functions s:(R,ℑ)⟶(R,ℵ) and v:N⟶N. Then fsv:(R,τ(ℑ),N)⟶(R,τ(ℵ),N) is soft a.w.c but not soft weakly continuous.
Example 2.7. Let G={1,2,3,4}, ℑ={∅,G,{2},{3},{2,3},{1,2},{1,2,3},{2,3,4}}, and L=(0,1). Define s:(G,ℑ)⟶(G,ℑ) and v:L⟶L by s(1)=3, s(2)=4, s(3)=2, s(4)=1, and v(a)=a for all a∈L. Then fsv:(R,τ(ℑ),L)⟶(R,τ(ℵ),L) is soft a.w.c but not soft pre-continuous.
Theorem 2.8. A soft function fsv:(G,Ψ,A)⟶(H,Φ,B) is soft a.w.c iff fsv(ClΨ(K))˜⊆ClθΦ(fsv(K)) for every K∈Ψ.
Proof. Necessity. Suppose that fsv is soft a.w.c. Let K∈Ψ and suppose to the contrary that there exists by˜∈fsv(ClΨ(K))−ClθΦ(fsv(K)). Since by˜∉ClθΦ(fsv(K)), then there exists T∈Φ such that by˜∈T and ClΦ(T)˜∩fsv(K)=0B. This implies that f−1sv(ClΦ(T))˜∩K=0A and consequently,
IntΨ(ClΨ(f−1sv(ClΦ(T))))˜∩ClΨ(K)=0A.
Since fsv is soft a.w.c, then f−1sv(T)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T)))). Thus, f−1sv(T)˜∩ClΨ(K)=0A and hence, T˜∩fsv(ClΨ(K))=0B. This implies that by˜∉fsv(ClΨ(K)), which is a contradiction.
Sufficiency. Suppose that fsv(ClΨ(K))˜⊆ClθΦ(fsv(K)) for every K∈Ψ. Let K∈Ψ. Then
fsv(1A−IntΨ(ClΨ(f−1sv(ClΦ(K)))))=fsv(ClΨ(1A−ClΨ(f−1sv(ClΦ(K)))))=fsv(ClΨ(IntΨ(1A−f−1sv(ClΦ(K)))))=fsv(ClΨ(IntΨ(f−1sv(1B−ClΦ(K))))).
Since IntΨ(f−1sv(1B−ClΦ(K)))∈Φ, then by assumption,
fsv(ClΨ(IntΨ(f−1sv(1B−ClΦ(K)))))˜⊆ClθΦ(fsv(IntΨ(f−1sv(1B−ClΦ(K)))))˜⊆ClθΦ(fsv((f−1sv(1B−ClΦ(K)))))˜⊆ClθΦ(1B−ClΦ(K)).
Since 1B−ClΦ(K)∈Φ, then ClθΦ(1B−ClΦ(K))=ClΦ(1B−ClΦ(K)). Since 1B−ClΦ(K)˜⊆1B−K and 1B−K∈Φc, then ClΦ(1B−ClΦ(K))˜⊆1B−K. Therefore,
fsv(1A−IntΨ(ClΨ(f−1sv(ClΦ(K)))))=fsv(ClΨ(IntΨ(f−1sv(1B−ClΦ(K)))))˜⊆ClθΦ(1B−ClΦ(K))˜⊆1B−K,
and thus,
1A−IntΨ(ClΨ(f−1sv(ClΦ(K))))˜⊆f−1sv(fsv(1A−IntΨ(ClΨ(f−1sv(ClΦ(K))))))˜⊆f−1sv(1B−K)=1A−f−1sv(K).
This implies that f−1sv(K)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(K)))). Hence, fsv is soft a.w.c.
Theorem 2.9. If fsv:(G,Ψ,A)⟶(H,Φ,B) is soft a.w.c and (H,Φ,B) is soft regular, then fsv is soft pre-continuous.
Proof. Let fsv:(G,Ψ,L)⟶(H,Φ,M) be soft a.w.c. Let K∈Ψ. Then, by Theorem 2.8, fsv(ClΨ(K))˜⊆ClθΦ(fsv(K)). Since (H,Φ,B) is soft regular, then ClθΦ(fsv(K))=ClΦ(fsv(K)). Thus, fsv(ClΨ(K))˜⊆ClΦ(fsv(K)). Therefore, by Theorem 18 (d) of [61], fsv is soft pre-continuous.
Theorem 2.10. For a soft function fsv:(G,Ψ,A)⟶(H,Φ,B), the next are equivalent:
(a) fsv is soft a.w.c.
(b) ClΨ(IntΨ(f−1sv(K)))˜⊆f−1sv(ClΦ(K)) for every K∈Φ.
(c) For each ax∈SP(G,A) and each K∈Φ such that fsv(ax)˜∈K, ClΨ(f−1sv(ClΦ(K))) is a soft neighborhood of ax.
Proof. (a) ⟶ (b): Let K∈Φ. Then 1B−ClΦ(K)∈Φ, and by (a),
1A−f−1sv(ClΦ(K))=f−1sv(1B−ClΦ(K))˜⊆IntΨ(ClΨ(f−1sv(ClΦ(1B−ClΦ(K)))))=IntΨ(ClΨ(f−1sv(1B−IntΦ(ClΦ(K)))))=IntΨ(ClΨ(1A−f−1sv(IntΦ(ClΦ(K)))))=IntΨ(1A−IntΨ(f−1sv(IntΦ(ClΦ(K)))))=1A−ClΨ(IntΨ(f−1sv(IntΦ(ClΦ(K)))))˜⊆1A−ClΨ(IntΨ(f−1sv((K)))).
It follows that ClΨ(IntΨ(f−1sv(K)))˜⊆f−1sv(ClΦ(K)).
(b) ⟶ (c): Let ax∈SP(G,A) and let K∈Φ such that fsv(ax)˜∈K. Since 1B−ClΦ(K)∈Φ, then by (b),
ClΨ(IntΨ(f−1sv(1B−ClΦ(K))))˜⊆f−1sv(ClΦ(1B−ClΦ(K))).
Thus,
1A−IntΨ(ClΨ(f−1sv(ClΦ(K))))=ClΨ(1A−ClΨ(f−1sv(ClΦ(K))))=ClΨ(IntΨ(1A−f−1sv(ClΦ(K))))=ClΨ(IntΨ(f−1sv(1B−ClΦ(K))))˜⊆f−1sv(ClΦ(1B−ClΦ(K)))=f−1sv(1B−IntΦ(ClΦ(K)))˜⊆f−1sv(1B−K)=1A−f−1sv(K).
Therefore, we obtain ax˜∈f−1sv(K)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(K)))), and hence ClΨ(f−1sv(ClΦ(K))) is a soft neighborhood of ax.
(c) ⟶ (a): Let K∈Φ. To show that f−1sv(K)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(K)))), let ax˜∈f−1sv(K). Then fsv(ax)˜∈K and by (c), ClΨ(f−1sv(ClΦ(K))) is a soft neighborhood of ax. Therefore, ax˜∈IntΨ(ClΨ(f−1sv(ClΦ(K)))).
Definition 2.11. A soft function fsv:(G,Ψ,L)⟶(H,Φ,M) is called soft pre-open if fsv(K)˜⊆IntΦ(ClΦ(fsv(K))) for each K∈Ψ.
Theorem 2.12. Every soft open function is soft pre-open.
Proof. Let fsv:(G,Ψ,A)⟶(H,Φ,B) be soft open. Let K∈Ψ. Then fsv(K)∈Φ and so, fsv(K)˜⊆IntΦ(ClΦ(fsv(K))). It follows that fsv is soft pre-open.
Soft pre-open functions are not soft open in general.
Example 2.13. Let G=R, ℑ={∅,G,Q}, and ℵ be the usual topology on R. Let L={a,b}. Consider the identity functions s:(G,ℑ)⟶(G,ℵ) and v:L⟶L. Then fsv:(G,τ(ℑ),L)⟶(G,τ(ℵ),L) is soft pre-open but not soft open.
Theorem 2.14. A soft function fsv:(G,Ψ,A)⟶(H,Φ,B) is soft pre-open iff f−1sv(ClΦ(T))˜⊆ClΨ(f−1sv(T)) for every T∈Φ.
Proof. Necessity. Let fsv be soft pre-open. Let T∈Φ. Let ax˜∈f−1sv(ClΦ(T)) and let K∈Ψ such that ax˜∈K. Since fsv is soft pre-open, then fsv(K)˜⊆IntΦ(ClΦ(fsv(K))). Since ax˜∈K, then fsv(ax)˜∈fsv(K)˜⊆IntΦ(ClΦ(fsv(K))) and so, fsv(ax)˜∈IntΦ(ClΦ(fsv(K)))∈Φ. Since ax˜∈f−1sv(ClΦ(T)), then fsv(ax)˜∈ClΦ(T). Thus, T˜∩IntΦ(ClΦ(fsv(K)))≠0B and hence, T˜∩ClΦ(fsv(K))≠0B. Consequently, T˜∩fsv(K)≠0B. Choose by˜∈K such that fsv(by)˜∈T. Then, by˜∈K˜∩f−1sv(T). Hence, ax˜∈ClΨ(f−1sv(T)).
Sufficiency. Suppose that f−1sv(ClΦ(T))˜⊆ClΨ(f−1sv(T)) for every T∈Φ. Let K∈Ψ. Suppose to the contrary that there exists ax˜∈K such that fsv(ax)˜∉IntΦ(ClΦ(fsv(K))). Let T=1B−ClΦ(fsv(K)). Then, T∈Φ and by assumption, f−1sv(ClΦ(T))˜⊆ClΨ(f−1sv(T)). Since
f−1sv(ClΦ(T))=f−1sv(ClΦ(1B−ClΦ(fsv(K))))=f−1sv(1B−IntΦ(ClΦ(fsv(K))))=1A−f−1sv(IntΦ(ClΦ(fsv(K)))),
and ax˜∈1A−f−1sv(IntΦ(ClΦ(fsv(K)))), then ax˜∈f−1sv(ClΦ(T))˜⊆ClΨ(f−1sv(T)) and so,
ax˜∈ClΨ(f−1sv(T))=ClΨ(f−1sv(1B−ClΦ(fsv(K))))=ClΨ(1A−f−1sv(ClΦ(fsv(K)))).
Since ax˜∈K∈Ψ, then K˜∩(1A−f−1sv(ClΦ(fsv(K))))≠0A.
Since K˜⊆f−1sv(fsv(T))˜⊆f−1sv(ClΦ(fsv(K))), then 1A−f−1sv(ClΦ(fsv(K)))˜⊆1A−K.
Thus, K˜∩(1A−K)≠0. This is a contradiction.
Theorem 2.15. Soft a.w.c soft pre-open functions are soft pre-continuous.
Proof. Let fsv:(G,Ψ,A)⟶(H,Φ,B) be soft a.w.c and soft pre-open. Let T∈Φ. Since fsv is soft a.w.c, then f−1sv(T)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T)))). Since fsv is soft pre-open, then by Theorem 2.14, f−1sv(ClΦ(T))˜⊆ClΨ(f−1sv(T)). Therefore, we have f−1sv(T)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T))))˜⊆IntΨ(ClΨ(f−1sv(T))). This shows that fsv is soft pre-continuous.
Corollary 2.16. Soft weakly continuous soft pre-open functions are soft pre-continuous.
Theorem 2.17. Soft a.w.c soft semi-continuous functions are soft weakly continuous.
Proof. Let fsv:(G,Ψ,A)⟶(H,Φ,B) be soft a.w.c and soft semi-continuous. Let T∈Φ. Since fsv is soft semi-continuous, then f−1sv(T)∈SO(Ψ). So, by Lemma 1 of [22], ClΨ(f−1sv(T))=ClΨ(IntΨ((f−1sv(T)))). Furthermore, by Theorem 2.10, ClΨ(IntΨ(f−1sv(T)))˜⊆f−1sv(ClΦ(T)). Hence, ClΨ(f−1sv(T))˜⊆f−1sv(ClΦ(T)). It follows from Theorem 5.1 of [59] that fsv is soft weakly continuous.
Lemma 2.18. Let (G,Ψ,A) be a STS and let D∈SS(G,A). Then
(a) If K∈Ψ, then K˜∩ClΨ(D)˜⊆ClΨ(K˜∩D).
(b) If S∈Ψc, then IntΨ(D˜∪S)˜⊆IntΨ(D)˜∪S.
Proof. (a) Let ax˜∈K˜∩ClΨ(D) and let H∈Ψ such that ax˜∈H. Then, we have ax˜∈K˜∩H∈Ψ. Since ax˜∈ClΨ(D), then (K˜∩D)˜∩H=(K˜∩H)˜∩D≠0A. Thus, ax˜∈ClΨ(K˜∩D).
(b) Since S∈Ψc, then 1A−S∈Ψ and by (a),
(1A−S)˜∩ClΨ(1A−D)˜⊆ClΨ((1A−S)˜∩(1A−D)).
Since ClΨ(1A−D)=1A−IntΨ(D), then
(1A−S)˜∩ClΨ(1A−D)=(1A−S)˜∩(1A−IntΨ(D))=1A−(S˜∪IntΨ(D)).
Furthermore,
ClΨ((1A−S)˜∩(1A−D))=ClΨ(1A−(S˜∪D))=1A−(IntΨ(S˜∪D)).
Therefore, 1A−(S˜∪IntΨ(D))˜⊆1A−(IntΨ(S˜∪D)) and hence, IntΨ(D˜∪S)˜⊆IntΨ(D)˜∪S.
Theorem 2.19. Let (G,Ψ,A) be a STS and let D∈SS(G,A). Then pClΨ(D)=D˜∪ClΨ(IntΨ(D)).
Proof. Since ClΨ(IntΨ(D))∈Ψc, then by Lemma 2.18 (b),
IntΨ(D˜∪ClΨ(IntΨ(D)))˜⊆IntΨ(D)˜∪ClΨ(IntΨ(D)).
So,
ClΨ(IntΨ(D˜∪ClΨ(IntΨ(D))))˜⊆ClΨ(IntΨ(D)˜∪ClΨ(IntΨ(D)))˜⊆ClΨ(IntΨ(D))˜⊆D˜∪ClΨ(IntΨ(D)).
Hence, D˜∪ClΨ(IntΨ(D))∈PC(Ψ) and thus, pClΨ(D)˜⊆D˜∪ClΨ(IntΨ(D)). Furthermore, since pClΨ(D)∈PC(Ψ), then ClΨ(IntΨ(D))˜⊆ClΨ(IntΨ(pClΨ(D)))˜⊆pClΨ(D) and hence, D˜∪ClΨ(IntΨ(D))˜⊆pClΨ(D).
Theorem 2.20. For a soft function fsv:(G,Ψ,A)⟶(H,Φ,B), the following are equivalent:
(a) fsv is soft a.w.c.
(b) pClΨ(f−1sv(T))˜⊆f−1sv(ClΦ(T)) for every T∈Φ.
(c) f−1sv(T)˜⊆pIntΨ(f−1sv(ClΦ(T))) for every T∈Φ.
(d) For each ax∈SP(G,A) and each T∈Φ such that fsv(ax)˜∈T, there exists K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΦ(T).
Proof. (a) ⟶ (b): Let T∈Φ. Then by Theorem 2.10, ClΨ(IntΨ(f−1sv(T)))˜⊆f−1sv(ClΦ(T)). So,
f−1sv(T)˜∪ClΨ(IntΨ(f−1sv(T)))˜⊆f−1sv(T)˜∪f−1sv(ClΦ(T))=f−1sv(ClΦ(T)).
Furthermore, by Lemma 2.19, pClΨ(f−1sv(T))=f−1sv(T)˜∪ClΨ(IntΨ(f−1sv(T))). It follows that pClΨ(f−1sv(T))˜⊆f−1sv(ClΦ(T)).
(b) ⟶ (c): Let T∈Φ. Then 1B−ClΦ(T)∈Φ and by (b),
pClΨ(f−1sv(1B−ClΦ(T)))˜⊆f−1sv(ClΦ(1B−ClΦ(T))).
Now,
1A−pIntΨ(f−1sv(ClΦ(T)))=pClΨ(1A−f−1sv(ClΦ(T)))=pClΨ(f−1sv(1B−ClΦ(T))).
Furthermore, we have
f−1sv(1B−IntΦ(ClΦ(T)))=1A−f−1sv(IntΦ(ClΦ(T)))˜⊆1A−f−1sv(T).
Therefore, 1A−pIntΨ(f−1sv(ClΦ(T)))˜⊆1A−f−1sv(T). Hence, f−1sv(T)˜⊆pIntΨ(f−1sv(ClΦ(T))).
(c) ⟶ (d): Let ax∈SP(G,A) and let T∈Φ such that fsv(ax)˜∈T. Then by (c), ax˜∈f−1sv(T)˜⊆pIntΨ(f−1sv(ClΦ(T))). Let K=pIntΨ(f−1sv(ClΦ(T))). Then ax˜∈K∈PO(Ψ) and
fsv(K)=fsv(pIntΨ(f−1sv(ClΦ(T))))˜⊆fsv(f−1sv(ClΦ(T)))˜⊆ClΦ(T).
(d) ⟶ (a): Let T∈Φ. To show that f−1sv(T)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T)))), let ax˜∈f−1sv(T). Then fsv(ax)˜∈T∈Φ. So, by (d), there exists K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΦ(T). Thus, K˜⊆f−1sv(ClΦ(T)), and so, IntΨ(ClΨ(K))˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T)))). Furthermore, since K∈PO(Ψ), then K˜⊆IntΨ(ClΨ(K)). It follows that ax˜∈IntΨ(ClΨ(f−1sv(ClΦ(T)))).
Lemma 2.21. Let (G,Ψ,A) be a STS and let T∈PO(Ψ). Then ClΦ(T)=ClΦ(IntΦ(ClΦ(T))).
Proof. Since T∈PO(Ψ), then T˜⊆IntΦ(ClΦ(T)) and so ClΦ(T)˜⊆IntΦ(ClΦ(T)). Furthermore, since IntΦ(ClΦ(T))˜⊆ClΦ(T), then ClΦ(IntΦ(ClΦ(T)))˜⊆ClΦ(ClΦ(T))=ClΦ(T).
Theorem 2.22. For a soft function fsv:(G,Ψ,A)⟶(H,Φ,B), the following are equivalent:
(a) fsv is soft a.w.c.
(b) fsv(pClΨ(D))˜⊆ClθΦ(fsv(D)) for every D∈SS(G,A).
(c) pClΨ(f−1sv(M))˜⊆f−1sv(ClθΦ(M)) for every M∈SS(H,B).
(d) pClΨ(f−1sv(IntΦ(ClθΦ(M))))˜⊆f−1sv(ClθΦ(M)) for every M∈SS(H,B).
(e) pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)) for every T∈Φ.
(f) pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)) for every T∈PO(Φ).
(g) pClΨ(f−1sv(IntΦ(R)))˜⊆f−1sv(R) for every R∈RC(Φ).
Proof. (a) ⟶ (b): Let D∈SS(G,A). To see that fsv(pClΨ(D))˜⊆ClθΦ(fsv(D)), let ax˜∈pClΨ(D) and let T∈Φ such that fsv(ax)˜∈T. Since fsv is soft a.w.c, then by Theorem 2.20 (d), there exists K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΦ(T). Since ax˜∈pClΨ(D) and ax˜∈K∈PO(Ψ), then K˜∩D≠0A and hence, 0B≠fsv(K˜∩D)˜⊆fsv(K)˜∩fsv(D)˜⊆ClΦ(T)˜∩fsv(D). Therefore, we obtain fsv(ax)˜∈ClθΦ(fsv(D)).
(b) ⟶ (c): Let M∈SS(H,B). Then by (b),
fsv(pClΨ(f−1sv(M))˜⊆ClθΦ(fsv(f−1sv(M))˜⊆ClθΦ(M)
And so,
pClΨ(f−1sv(M))˜⊆f−1sv(fsv(pClΨ(f−1sv(M)))˜⊆f−1sv(ClθΦ(M)).
(c) ⟶ (d): Let M∈SS(H,B). Then by (c),
pClΨ(f−1sv(IntΦ(ClθΦ(M))))˜⊆f−1sv(ClθΦ(IntΦ(ClθΦ(M)))).
Since IntΦ(ClθΦ(M))∈Φ, then ClθΦ(IntΦ(ClθΦ(M)))=ClΦ(IntΦ(ClθΦ(M))). Since ClΦ(IntΦ(ClθΦ(M)))˜⊆ClΦ(ClθΦ(M)) and ClθΦ(M)∈Φc, then ClΦ(ClθΦ(M))=ClθΦ(M). Therefore, we have
pClΨ(f−1sv(IntΦ(ClθΦ(M)))˜⊆f−1sv(ClθΦ(IntΦ(ClθΦ(M))))=f−1sv(ClΦ(IntΦ(ClθΦ(M))))˜⊆f−1sv(ClΦ(ClθΦ(M)))=f−1sv(ClθΦ(M)).
(d) ⟶ (e): Let T∈Φ. Then, by (d), pClΨ(f−1sv(IntΦ(ClθΦ(T))))˜⊆f−1sv(ClθΦ(T)). Furthermore, since T∈Φ, then ClθΦ(T)=ClΦ(T). Therefore,
pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)).
(e) ⟶ (f): Let T∈PO(Φ). Then IntΦ(ClΦ(T))∈Φ and by (e), pClΨ(f−1sv(IntΦ(ClΦ(IntΦ(ClΦ(T))))))˜⊆f−1sv(ClΦ(IntΦ(ClΦ(T)))). Furthermore, since T∈PO(Φ), then by Lemma 2.21, pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)).
(f) ⟶ (g): Let R∈RC(Φ). Then R=ClΦ(IntΦ(R)), and so, IntΦ(R)=IntΦ(ClΦ(IntΦ(R))). Since IntΦ(R)∈PO(Φ), then by (f),
pClΨ(f−1sv(IntΦ(R)))=pClΨ(f−1sv(IntΦ(ClΦ(IntΦ(R))))˜⊆f−1sv(ClΦ(IntΦ(R)))=f−1sv(R).
(g) ⟶ (a): We will apply Theorem 2.20 (b). Let T∈Φ. Then ClΦ(T)∈RC(Φ) and by (g), pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)). Since T∈Φ, then T˜⊆IntΦ(ClΦ(T)), and so, pClΨ(f−1sv(T))˜⊆pClΨ(f−1sv(IntΦ(ClΦ(T))))˜⊆f−1sv(ClΦ(T)).
Theorem 2.23. Let (H,Φ,B) be a soft regular STS. The following are equivalent for a soft function fsv:(G,Ψ,A)⟶(H,Φ,B):
(a) fsv is soft pre-continuous.
(b) f−1sv(ClθΦ(M))∈PC(Ψ) for every M∈SS(H,B).
(c) f−1sv(S)∈PC(Ψ) for every S∈(Φθ)c.
(d) f−1sv(T)∈PO(Ψ) for every T∈Φθ.
(e) fsv is soft a.w.c.
Proof. (a) ⟶ (b): Let M∈SS(H,B). Since ClΦ(M)∈Ψc, then by (a), f−1sv(ClΦ(M))∈PC(Ψ). Furthermore, since (H,Φ,B) is soft regular, then ClθΦ(M)=ClΦ(M). Hence, f−1sv(ClθΦ(M))=f−1sv(ClΦ(M))∈PC(Ψ).
(b) ⟶ (c): Let S∈(Φθ)c. Then S∈Φc and so ClΦ(S)=S. Since (H,Φ,B) is soft regular, then S=ClΦ(S)=ClθΦ(S). Therefore, by (b), f−1sv(ClθΦ(S))=f−1sv(S)∈PC(Ψ).
(c) ⟶ (d): Let T∈Φθ. Then 1B−T∈(Φθ)c and by (c), f−1sv(1B−T)=1A−f−1sv(T)∈PC(Ψ). It follows that f−1sv(T)∈PO(Ψ).
(d) ⟶ (e): Since (H,Φ,B) is soft regular, then Φθ=Φ. Thus, fsv is soft pre-continuous, and by Theorem 2.5, it is soft a.w.c.
(e) ⟶ (a): Follows from Theorem 2.9.
Theorem 2.24. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c function and X⊆G such that CX∈Ψ−{0A}, then (fsv)|CX:(X,ΨX,A)⟶(H,Φ,B) is soft a.w.c.
Proof. Let T∈Φ. Since fsv is soft a.w.c, then f−1sv(T)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(T)))). Since ((frw)|CX)−1(T)=CX˜∩f−1sv(T), then ((frw)|CX)−1(T)˜⊆CX˜∩IntΨ(ClΨ(f−1sv(ClΦ(T)))). Since CX∈Ψ, then
CX˜∩IntΨ(ClΨ(f−1sv(ClΦ(T))))=IntΨX(CX˜∩ClΨ(f−1sv(ClΦ(T))))˜⊆IntΨX(CX˜∩ClΨ(CX˜∩f−1sv(ClΦ(T))))=IntΨX(ClΨX(((frw)|CX)−1(T))).
Therefore, ((frw)|CX)−1(T)˜⊆IntΨX(ClΨX(((frw)|CX)−1(T))). Hence, (fsv)|CX:(X,ΨX,A)⟶(H,Φ,B) is soft a.w.c.
In Theorem 2.24, the condition 'CX∈Ψ−{0A}' cannot be dropped:
Example 2.25. Let ℑ be the usual topology R. Consider the functions s:(R,ℑ)⟶(R,ℑ) and v:{a}⟶{a}, where s(x)=1 if x∈Q, s(x)=−1 if x∈R−Q, and v(a)=a. Let X=([0,∞)∩Q)∪((−∞,0)∩(R−Q)) Then fsv:(R,τ(ℑ),{a})⟶(R,τ(ℑ),{a}) is soft a.w.c while (fsv)|CX:(X,τ(ℑ),{a})⟶(R,τ(ℑ),{a}) is not soft a.w.c.
Theorem 2.26. If fs1v1:(G,Ψ,A)⟶(H,Φ,B) is soft a.w.c and fs2v2:(H,Φ,B)⟶(M,Π,L) is soft continuous, then f(s2∘s1)(v2∘v1):(G,φ,A)⟶(M,Π,L) is soft a.w.c.
Proof. Let fs1v1 be soft a.w.c and fs2v2 be soft continuous. Let T∈Π. Since fs2v2 is soft continuous, then f−1s2v2(T)∈Φ and ClΦ(f−1s2v2(T))˜⊆f−1s2v2(ClΠ(T)). Since fs1v1 is soft a.w.c, then f−1s1v1(f−1s2v2(T))˜⊆IntΨ(ClΨ(f−1s1v1(ClΦ(f−1s2v2(T))))). Thus,
f−1(s2∘s1)(v2∘v1)(T)=f−1s1v1(f−1s2v2(T))˜⊆IntΨ(ClΨ(f−1s1v1(ClΦ(f−1s2v2(T)))))˜⊆IntΨ(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T)))))=IntΨ(ClΨ(f−1(s2∘s1)(v2∘v1)(ClΠ(T)))).
It follows that f(s2∘s1)(v2∘v1) is soft a.w.c.
It is not necessary for the soft composition of two soft a.w.c. functions to be soft a.w.c:
Example 2.27. Let G={a,b}, H=M={a,b,c}, A={1,2}, ℑ={∅,G,{b}}, ℵ={∅,H,{a,c},{b,c},{c}}, and ℘={∅,M,{a},{b},{a,b}}. Let s1:(G,ℑ)⟶(H,ℵ), s2:(H,ℵ)⟶(M,℘) be the inclusion functions and v1,v2:A⟶A be the identity functions. Then fs1v1:(G,τ(ℑ),A)⟶(H,τ(ℵ),A) and fs2v2:(H,τ(ℵ),A)⟶(M,τ(℘),A) are soft a.w.c, while f(s2∘s1)(v2∘v1):(G,τ(ℑ),A)⟶(M,τ(℘),A) is not soft a.w.c.
The composition f(s2∘s1)(v2∘v1) of a soft continuous function fs1v1:(G,Ψ,A)⟶(H,Φ,B) and a soft pre-continuous function fs2v2:(H,Φ,B)⟶(M,Π,L) is not necessarily soft a.w.c:
Example 2.28. Let ℑ, ℵ, and ℘ be the usual, indiscrete, and discrete topologies on R, respectively. Let A={a,b}. Let s1:(R,ℑ)⟶(R,ℵ), s2:(R,ℵ)⟶(R,℘) and v1,v2:A⟶A be the identity functions. Then fs1v1:(R,τ(ℑ),A)⟶(R,τ(ℵ),A) is soft continuous and fs2v2:(R,τ(ℵ),A)⟶(R,τ(℘),A) is soft pre-continuous, while f(s2∘s1)(v2∘v1):(R,τ(ℑ),A)⟶(R,τ(℘),A) is not soft a.w.c.
Let G and H be two non-empty sets. The projection functions h:G×H⟶G and g:G×H⟶H defined by h(x,y)=x and g(x,y)=y for all (x,y)∈G×H will be denoted by πG and πH, respectively.
Theorem 2.29. Let (G,Ψ,A), (H,Φ,B), and (M,Π,L) be three STSs. If fsv:(G,Ψ,A)⟶(H×M,pr(Φ×Π),B×L) is soft a.w.c, then f(πH∘s)(πB∘v):(G,Ψ,A)⟶(H,Φ,B) and f(πM∘s)(πL∘v):(G,Ψ,A)⟶(M,Π,L) are soft a.w.c.
Proof. Let fsv be soft a.w.c. Since f(πH)(πB):(H×M,pr(Φ×Π),B×L)⟶(H,Φ,B) and f(πM)(πL):(H×M,pr(Φ×Π),B×L)⟶(M,Π,L) are always soft continuous, then by Theorem 2.26, f(πH∘s)(πB∘v):(G,Ψ,A)⟶(H,Φ,B) and f(πM∘s)(πL∘v):(G,Ψ,A)⟶(M,Π,L) are soft a.w.c.
For every function p:G⟶H, the function h:G⟶G×H defined by h(x)=(x,p(x)) is represented by p#.
Theorem 2.30. Let fsv:(G,Ψ,A)⟶(H,Φ,B) be a soft function. Then fs#v#:(G,Ψ,A)⟶(G×H,pr(Ψ×Φ),A×B) is soft a.w.c iff fsv is soft a.w.c.
Proof. Necessity.Let fs#v# be soft a.w.c. Then, by Theorem 2.29, fsv=f(πH∘s#)(πB∘v#):(G,Ψ,A)⟶(H,Φ,B) is soft a.w.c.
Sufficiency. Let fsv be soft a.w.c. We will apply Theorem 2.10 (c). Let ax∈SP(G,A) and let R∈pr(Ψ×Φ) such that fs#v#(ax)˜∈R. Choose U∈Ψ and W∈Φ such that fs#v#(ax)=(a,v(a))(x,s(x))˜∈U×W˜⊆R. Since fsv is soft a.w.c and fsv(ax)˜∈W∈Φ, then by Theorem 2.10 (c), ClΨ(f−1sv(ClΦ(W))) is a soft neighborhood of ax and by Lemma 1.18 (a), U˜∩ClΨ(f−1sv(ClΦ(W)))˜⊆ClΨ(U˜∩f−1sv(ClΦ(W))). Furthermore, we have
U˜∩f−1sv(ClΦ(W))˜⊆f−1s#v#(U×ClΦ(W))˜⊆f−1s#v#(Clpr(Ψ×Φ)(R)).
Therefore, ClΨ(f−1s#v#(Clpr(Ψ×Φ)(R))) is a soft neighborhood of ax. Hence, by Theorem 2.10 (c), fs#v# is soft a.w.c.
Theorem 2.31. Let fs1v1:(G,Ψ,A)⟶(H,Φ,B) be a soft open continuous surjection and fs2v2:(H,Φ,B)⟶(M,Π,L) be a soft function. If f(s2∘s1)(v2∘v1):(G,Ψ,A)⟶(M,Π,L) is soft a.w.c, then fs2v2 is soft a.w.c.
Proof. Let T∈Π. Since f(s2∘s1)(v2∘v1) is soft a.w.c, then
f−1s1v1(f−1s2v2(T))=f−1(s2∘s1)(v2∘v1)(T)˜⊆IntΨ(ClΨ(f−1(s2∘s1)(v2∘v1)(ClΠ(T))))=IntΨ(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T))))).
Since fs1v1 is soft continuous, then ClΨ(f−1s1v1(f−1s2v2(ClΠ(T))))˜⊆f−1s1v1(ClΨ((f−1s2v2(ClΠ(T))))). Therefore, we have
f−1s1v1(f−1s2v2(T))˜⊆IntΨ(f−1s1v1(ClΨ((f−1s2v2(ClΠ(T)))))), and so
fs1v1(f−1s1v1(f−1s2v2(T)))˜⊆fs1v1(IntΨ(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T)))))).
Since fs1v1 is surjective, then fs1v1(f−1s1v1(f−1s2v2(T)))=f−1s2v2(T).
Since fs1v1 is soft open, then
fs1v1(IntΨ(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T))))))˜⊆IntΦ(fs1v1(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T)))))).
Since fs1v1 is soft continuous, then
fs1v1(ClΨ(f−1s1v1(f−1s2v2(ClΠ(T)))))˜⊆ClΦ(fs1v1(f−1s1v1(f−1s2v2(ClΠ(T)))))˜⊆ClΦ(f−1s2v2(ClΠ(T))).
Therefore, we have f−1s2v2(T)˜⊆IntΦ(ClΦ(f−1s2v2(ClΠ(T)))) and hence, fs2v2 is soft a.w.c.
Theorem 2.32. Let fs1v1:(G,Ψ,A)⟶(H,Φ,B) and fs2v2:(M,Γ,L)⟶(N,Π,F) be two soft functions. Let s∗:G×M⟶H×N and v∗:A×L⟶B×F be the functions defined by s∗(x,z)=(s1(x),s2(z)) and v∗(a,l)=(v1(a),v2(l)). Then fs∗v∗:(G×M,pr(Ψ×Γ),A×L)⟶(H×N,pr(Φ×Π),B×F) is soft a.w.c iff fs1v1 and fs2v2 are soft a.w.c.
Proof. Necessity. Let T∈Φ−{0B} and R∈F−{0F}. Then T×R∈pr(Φ×Π) and so,
f−1s1v1(T)×f−1s2v2(R)=f−1s∗v∗(T×R)˜⊆Intpr(Ψ×Γ)(Clpr(Ψ×Γ)(f−1s∗v∗(Clpr(Φ×Π)(T×R))))=Intpr(Ψ×Γ)(Clpr(Ψ×Γ)(f−1s∗v∗(ClΦ(T)×ClΠ(R))))=Intpr(Ψ×Γ)Clpr(Ψ×Γ)(f−1s1v1(ClΦ(T))×f−1s2v2(ClΠ(R)))=Intpr(Ψ×Γ)(ClΨ(f−1s1v1(ClΦ(T)))×ClΓ(f−1s2v2(ClΠ(R))))=IntΨ(ClΨ(f−1s1v1(ClΦ(T))))×IntΨ(ClΓ(f−1s2v2(ClΠ(R)))).
This implies that f−1s1v1(T)˜⊆IntΨ(ClΨ(f−1s1v1(ClΦ(T)))) and
f−1s2v2(R)˜⊆IntΓ(ClΓ(f−1s2v2(ClΠ(R)))). It follows that fs1v1 and fs2v2 are soft a.w.c.
Sufficiency. Let fs1v1 and fs2v2 be soft a.w.c. We will apply Theorem 2.10 (c). Let (a,b)(x,y)∈SP(G×M,A×L) and let Z∈pr(Φ×Π) such that fs∗v∗((a,b)(x,y))˜∈Z. Choose U∈Ψ and W∈Φ such that fs∗v∗((a,b)(x,y))=(v1(a),v2(b))(s1(x),s2(y))˜∈U×W˜⊆Z. Then (v1(a))s1(x)˜∈U and (v2(b))s2(y)˜∈W. Since fs1v1 and fs2v2 are soft a.w.c, then ClΨ(f−1s1v1(ClΦ(U))) is a soft neighborhood of (v1(a))s1(x) and ClΓ(f−1s2v2(ClΠ(W))) is a soft neighborhood of (v2(b))s2(y). Then we have ClΨ(f−1s1v1(ClΦ(U)))×ClΓ(f−1s2v2(ClΠ(W))) is a soft neighborhood of (v1(a),v2(b))(s1(x),s2(y)). But
ClΨ(f−1s1v1(ClΦ(U)))×ClΓ(f−1s2v2(ClΠ(W)))=Clpr(Ψ×Γ)(f−1s1v1(ClΦ(U))×f−1s2v2(ClΠ(W)))=Clpr(Ψ×Γ)(f−1s∗v∗(ClΦ(U)×ClΠ(W)))=Clpr(Ψ×Γ)(f−1s∗v∗(Clpr(Φ×Π)(U×W)))˜⊆Clpr(Ψ×Γ)(f−1s∗v∗(Clpr(Φ×Π)(Z))).
This shows that Clpr(Ψ×Γ)(f−1s∗v∗(Clpr(Φ×Π)(Z))) is a soft neighborhood of (v1(a),v2(b))(s1(x),s2(y)). Hence, fs∗v∗ is soft a.w.c.
Theorem 3.1. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c function and (H,Φ,B) is soft Hausdorff, then Graph(fsv)∈PC(pr(Ψ×Φ)).
Proof. We will show that 1A×B−Graph(fsv)˜⊆1A×B−pClΨ×Φ(Graph(fsv)). Let (d,e)(x,y)˜∈1A×B−Graph(fsv). Then fsv(dx)≠ey. Since (H,Φ,B) is soft Hausdorff, then there are T,R∈Φ such that fsv(dx)˜∈T, ey˜∈R, and T˜∩R=0B; hence ClΦ(T)˜∩R=0B. Since fsv is soft a.w.c, by Theorem 2.20 (d), there exists K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΦ(T). Thus, fsv(K)˜∩R=0B. Therefore, we have (d,e)(x,y)˜∈K×R∈PO(pr(Ψ×Φ)) and (K×R)˜∩Graph(fsv)=0A×B. This implies that (d,e)(x,y)˜∈1A×B−pClΨ×Φ(Graph(fsv)).
Corollary 3.2. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft weakly continuous function and (H,Φ,B) is soft Hausdorff, then Graph(fsv)∈PC(pr(Ψ×Φ)).
Proof. Theorems 2.4 and 3.1 provide the proof.
Corollary 3.3. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft pre-continuous function and (H,Φ,B) is soft Hausdorff, then Graph(fsv)∈PC(pr(Ψ×Φ)).
Proof. Theorems 2.5 and 3.1 provide the proof.
Lemma 3.4. Let (G,Ψ,A) be a STS. If X∈α(Ψ) and Y∈PO(Ψ), then X˜∩Y∈PO(Ψ).
Proof. Let X∈α(Ψ) and Y∈PO(Ψ). Then X˜⊆IntΨ(ClΨ(IntΨ(X))) and Y˜⊆IntΨ(ClΨ(Y)). By Lemma 2.18 (a),
X˜∩Y˜⊆IntΨ(ClΨ(IntΨ(X)))˜∩IntΨ(ClΨ(Y))=IntΨ(ClΨ(IntΨ(X))˜∩IntΨ(ClΨ(Y)))˜⊆IntΨ(ClΨ(IntΨ(X)˜∩IntΨ(ClΨ(Y))))˜⊆IntΨ(ClΨ(IntΨ(X)˜∩(ClΨ(Y))))˜⊆IntΨ(ClΨ(ClΨ(IntΨ(X)˜∩Y)))=IntΨ(ClΨ(IntΨ(X)˜∩Y))˜⊆IntΨ(ClΨ(X˜∩Y)).
Therefore, X˜∩Y∈PO(Ψ).
Theorem 3.5. Let fsv,fpu:(G,Ψ,A)⟶(H,Φ,B) be soft functions, and (H,Φ,B) be soft Hausdorff. If fsv is soft almost α-continuous, fpu is soft a.w.c, and Z=˜∪{ax∈SP(G,A):fsv(ax)=fpu(ax)}, then Z∈PC(Ψ).
Proof. We will show that 1A−Z˜⊆1A−pClΨ(Z). Let ax˜∈1A−Z. Then fsv(ax)≠fpu(ax). Since (H,Φ,B) is soft Hausdorff, then there are U,V∈Φ such that fsv(ax)˜∈U, fpu(ax)˜∈V, and U˜∩V=0B; hence IntΦ(ClΦ(U))˜∩ClΦ(V)=0B. Since fsv is soft almost α-continuous, and IntΦ(ClΦ(U))∈RO(Φ), then there exists Y∈α(Ψ) such that ax˜∈Y and fsv(Y)˜⊆IntΦ(ClΦ(U)). Since fpu is soft a.w.c, then by Theorem 2.20 (d), there exists K∈PO(Ψ) such that ax˜∈K and fpu(K)˜⊆ClΦ(V). We have fsv(Y)˜∩fpu(K)=0B and thus, (K˜∩Y)˜∩Z=0A. Furthermore, by Lemma 3.4, K˜∩Y∈PO(Ψ). This shows that ax˜∈1A−pClΨ(Z).
Corollary 3.6. Let fsv,fpu:(G,Ψ,A)⟶(H,Φ,B) be soft functions, and (H,Φ,B) is soft Hausdorff. If fsv is soft almost α-continuous, fpu is soft weakly continuous, and Z=˜∪{ax∈SP(G,A):fsv(ax)=fpu(ax)}, then Z∈PC(Ψ).
Proof. Theorems 2.4 and 3.5 provide the proof.
Corollary 3.7. Let fsv,fpu:(G,Ψ,A)⟶(H,Φ,B) be soft functions, and (H,Φ,B) is soft Hausdorff. If fsv is soft almost α-continuous, fpu is soft pre-continuous, and Z=˜∪{ax∈SP(G,A):fsv(ax)=fpu(ax)}, then Z∈PC(Ψ).
Proof. Theorems 2.5 and 3.5 provide the proof.
Theorem 3.8. Let (G,Ψ,A) be soft Hausdorff, and let Y be a non-empty subset of G. If there is a soft a.w.c function fsv:(G,Ψ,A)⟶(Y,ΨY,A) such that fsv(ax)=ax for all ax∈SP(G,A), then CY∈PC(Ψ).
Proof. Suppose, on the contrary, there exists ax˜∈pClΨ(CY)−CY. Then fsv(ax)≠ax. Since (G,Ψ,A) is soft Hausdorff, there exist U,V∈Ψ such that ax˜∈U, fsv(ax)˜∈V, and U˜∩V=0A; hence U˜∩ClΨ(V)=0A. Since V˜∩CY∈ΨY and fsv is soft a.w.c, then by Theorem 2.20 (d), there exists K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΨY(V˜∩CY)˜⊆ClΨ(V). Since ax˜∈U˜∩K∈PO(Ψ) and ax˜∈pClΨ(CY), then (U˜∩K)˜∩CY≠0A. Choose by˜∈(U˜∩K)˜∩CY. Since by˜∈CY, then fsv(by)=by. Since by˜∈K, then fsv(by)=by˜∈fsv(K)˜⊆ClΨ(V). Since U˜∩ClΨ(V)=0A, then by˜∉U. This is a contradiction.
Corollary 3.9. Let (G,Ψ,A) be soft Hausdorff, and let Y be a non-empty subset of G. If there is a soft weakly continuous function fsv:(G,Ψ,A)⟶(Y,ΨY,A) such that fsv(ax)=ax for all ax∈SP(G,A), then CY∈PC(Ψ).
Proof. Theorems 2.4 and 3.8 provide the proof.
Corollary 3.10. Let (G,Ψ,A) be soft Hausdorff, and let Y be a non-empty subset of G. If there is a soft pre-continuous function fsv:(G,Ψ,A)⟶(Y,ΨY,A) such that fsv(ax)=ax for all ax∈SP(G,A), then CY∈PC(Ψ).
Proof. Theorems 2.5 and 3.8 provide the proof.
Theorem 3.11. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c injection such that Graph(fsv) is soft strongly closed with respect to (G×H,pr(Ψ×Φ),A×B), then (G,Ψ,A) is soft Hausdorff.
Proof. Let ax,dz∈SP(G,A) such that ax≠dz. Since fsv is injective, then fsv(ax)≠fsv(dz). Thus, we have (a,v(d))(x,s(z))˜∈1A×B−Graph(fsv). Since Graph(fsv) is soft strongly closed with respect to (G×H,pr(Ψ×Φ),A×B), then there exist U∈Ψ and V∈Φ such that ax˜∈U, fsv(dz)˜∈V, and (U×ClΦ(V))˜∩Graph(fsv)=0A×B. Thus, we have U˜∩f−1sv(ClΦ(V))=0A and hence, U˜∩IntΨ(ClΨ(f−1sv(ClΦ(V))))=0A. Since fsv is soft a.w.c, then dz˜∈f−1sv(V)˜⊆IntΨ(ClΨ(f−1sv(ClΦ(V)))). It follows that (G,Ψ,A) is soft Hausdorff.
Theorem 3.12. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c injection and (H,Φ,B) is soft Urysohn, then (G,Ψ,A) is soft pre-T2.
Proof. ax,dz∈SP(G,A) such that ax≠dz. Since fsv is injective, then fsv(ax)≠fsv(dz). Since (H,Φ,B) is soft Urysohn, then there exist T,R∈Φ such that fsv(ax)˜∈T, fsv(dz)˜∈R, and ClΦ(T)˜∩ClΦ(R)=0B. Since fsv is soft a.w.c, then by Theorem 2.20 (d), there are K,M∈PO(Ψ) such that ax˜∈K, dz˜∈M, fsv(K)˜⊆ClΦ(T), and fsv(M)˜⊆ClΦ(R). Thus, fsv(K)˜∩fsv(M)˜⊆ClΦ(T)˜∩ClΦ(R)=0B and hence, fsv(K)˜∩fsv(M)=0B. Since fsv is injective, then fsv(K)˜∩fsv(M)=fsv(K˜∩M). Hence, K˜∩M=0A. It follows that (G,Ψ,A) is soft pre-T2.
Theorem 3.13. Let fsv,fpu:(G,Ψ,A)⟶(H,Φ,B) be soft a.w.c, (G,Ψ,A) is soft submaximal, and (H,Φ,B) is soft Urysohn. Then
˜∪{ax∈SP(G,A):fsv(ax)=fpu(ax)}∈Ψc.
Proof. Let N=˜∪{ax∈SP(G,A):fsv(ax)=fpu(ax)}. We will show that 1A−N˜⊆1A−ClΨ(N). Let ax˜∈1A−N. Then fsv(ax)≠fpu(ax). Since (H,Φ,B) is soft Urysohn, then there exist T,R∈Φ such that fsv(ax)˜∈T, fpu(ax)˜∈R, and ClΦ(T)˜∩ClΦ(R)=0B. Since fsv,fpu are soft a.w.c, by Theorem 2.20 (d), we find K,M∈PO(Ψ) such that ax˜∈K˜∩M, fsv(K)˜⊆ClΦ(T), and fpv(M)˜⊆ClΦ(R). Thus, fsv(K˜∩M)˜∩fpu(K˜∩M)˜⊆ClΦ(T)˜∩ClΦ(R)=0B. Since (G,Ψ,A) is soft submaximal, then by Theorem 4.2 of [60], we have K,M∈Ψ and so, K˜∩M∈Ψ. Since fsv(K˜∩M)˜∩fpu(K˜∩M)=0B, then (K˜∩M)˜∩N=0A. Since ax˜∈K˜∩M∈Ψ and (K˜∩M)˜∩N=0A, then ax˜∈1A−ClΨ(N).
Theorem 3.14. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c and (H,Φ,B) is soft Urysohn, then the soft set
E=˜∪{(a,d)(x,z)∈SP(G×G,A×A):fsv(ax)=fsv(dz)}∈PC(Ψ×Ψ).
Proof. We will show that 1A×A−E˜⊆1A×A−pClΨ×Ψ(E). Let (a,d)(x,z)˜∈1A×A−E. Then fsv(ax)≠fsv(dz). Since (H,Φ,B) is soft Urysohn, then there exist T,R∈Φ such that fsv(ax)˜∈T, fsv(dz)˜∈R, and ClΦ(T)˜∩ClΦ(R)=0B. Since fsv is soft a.w.c, then by Theorem 2.20 (d), there are K,M∈PO(Ψ) such that ax˜∈K, dz˜∈M, fsv(K)˜⊆ClΦ(T), and fsv(M)˜⊆ClΦ(R). Thus, fsv(K)˜∩fsv(M)˜⊆ClΦ(T)˜∩ClΦ(R)=0B and hence, fsv(K)˜∩fsv(M)=0B. Thus, we have (a,d)(x,z)˜∈K×M∈PO(Ψ×Ψ) and (K×M)˜∩E=0A×A. This implies that (a,d)(x,z)˜∈1A×A−pClΨ×Ψ(E).
Definition 3.15. Let fsv:(G,Ψ,A)⟶(H,Φ,B) be a soft function. Then Graph(fsv) is called soft strongly p -closed with respect to (G×H,pr(Ψ×Φ),A×B) if for each (a,d)(x,z)˜∈1A×B−Graph(fsv), there exist U∈PO(Ψ) and V∈Φ such that ax˜∈U, dz˜∈V, and (U×ClΦ(V))˜∩Graph(fsv)=0A×B.
Theorem 3.16. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c and (H,Φ,B) is soft Urysohn, then Graph(fsv) is soft strongly p-closed with respect to (G×H,pr(Ψ×Φ),A×B).
Proof. Let (a,d)(x,z)˜∈1A×B−Graph(fsv). Then fsv(ax)≠dz. Since (H,Φ,B) is soft Urysohn, then there exist T,R∈Φ such that fsv(ax)˜∈T, dz˜∈R, and ClΦ(T)˜∩ClΦ(R)=0B. Since fsv is soft a.w.c, then by Theorem 2.20 (d), there is K∈PO(Ψ) such that ax˜∈K and fsv(K)˜⊆ClΦ(T). This implies that fsv(K)˜∩ClΦ(R)=0B; hence (K×ClΦ(R))˜∩Graph(fsv)=0A×B. This shows that Graph(fsv) is soft strongly p-closed with respect to (G×H,pr(Ψ×Φ),A×B).
Theorem 3.17. If fsv:(G,Ψ,A)⟶(H,Φ,B) is a soft a.w.c injection with Graph(fsv) soft strongly p-closed with respect to (G×H,pr(Ψ×Φ),A×B), then (G,Ψ,A) is soft pre-T2.
Proof. Let ax,by∈SP(G,A) such that ax≠by. Since fsv is injective, then fsv(ax)≠fsv(by), and so (a,v(b))(x,s(y))˜∈1A×B−Graph(fsv). Since Graph(fsv) is soft strongly p-closed with respect to (G×H,pr(Ψ×Φ),A×B), then there exist U∈PO(Ψ) and V∈Φ such that ax˜∈U, fsv(by)˜∈V, and (U×ClΦ(V))˜∩Graph(fsv)=0A×B. Thus, we have fsv(U)˜∩ClΦ(V)=0B and hence, U˜∩f−1sv(ClΦ(V))=0A. It follows that U˜∩pIntΨ(f−1sv(ClΦ(V)))=0A. Since fsv is a soft a.w.c, then by Theorem 2.20 (c), by˜∈f−1sv(V)˜⊆pIntΨ(f−1sv(ClΦ(V)))∈PO(Ψ). This shows that (G,Ψ,A) is soft pre-T2.
This paper has successfully introduced and explored the novel concept of soft almost weakly continuous functions, a generalized family of soft continuous functions encompassing soft pre-continuous and weakly continuous functions. Through a comprehensive analysis, we have established various characterizations of soft almost weakly continuous functions and investigated their connections to their counterparts in general topology. We have also found the conditions that are needed for a soft almost weakly continuous function to change into a soft weakly continuous or soft pre-continuous function. This helps us understand how these function classes are related. The results on soft composition, limitation, preservation, product, and soft graph theorems in the setting of soft almost weakly continuous functions lay a solid foundation for future study in this field. As the subject of soft topology advances, this study sets the door for future investigation of soft continuous functions and their applications in a variety of disciplines.
Samer Al-Ghour and Jawaher Al-Mufarrij: Conceptualization, methodology, formal analysis, writing-original draft, writing-review and editing, and funding acquisition. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare that they have no conflicts of interest.
[1] | L. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X |
[2] | E. Zarei, M. Yazdi, R. Moradi, A. B. Toroody, Expert judgment and uncertainty in sociotechnical systems analysis, In: E. Zarei, Safety causation anal. sociotech. syst. adv. mod. tech., 2024. https://doi.org/10.1007/978-3-031-62470-4_18 |
[3] |
D. Molodtsov, Soft set theory-First results, Comput. Math. Appl., 37 (1999), 19–31. http://dx.doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
![]() |
[4] |
J. Yang, Y. Yao, Semantics of soft sets and three-way decision with soft sets, Knowl.-Based Syst., 194 (2020), 105538. http://dx.doi.org/10.1016/j.knosys.2020.105538 doi: 10.1016/j.knosys.2020.105538
![]() |
[5] |
J. C. R. Alcantud, The semantics of N-soft sets, their applications, and a coda about three-way decision, Inf. Sci., 606 (2022), 837–852. http://dx.doi.org/10.1016/j.ins.2022.05.084 doi: 10.1016/j.ins.2022.05.084
![]() |
[6] |
J. Gwak, H. Garg, N. Jan, Hybrid integrated decision-making algorithm for clustering analysis based on a bipolar complex fuzzy and soft sets, Alex. Eng. J., 67 (2023), 473–487. http://dx.doi.org/10.1016/j.aej.2022.12.003 doi: 10.1016/j.aej.2022.12.003
![]() |
[7] |
N. Malik, M. Shabir, T. M. Al-shami, R. Gul, A. Mhemdi, Medical decision-making techniques based on bipolar soft information, AIMS Math., 8 (2023), 18185–18205. http://dx.doi.org/10.3934/math.2023924 doi: 10.3934/math.2023924
![]() |
[8] |
O. Kguller, A soft set theoretic approach to network complexity and a case study for Turkish Twitter users, Appl. Soft Comput., 143 (2023), 110344. http://dx.doi.org/10.1016/j.asoc.2023.110344 doi: 10.1016/j.asoc.2023.110344
![]() |
[9] |
O. Dalkılıc, N. Demirtas, Algorithms for covid-19 outbreak using soft set theory: Estimation and application, Soft Comput., 27 (2022), 3203–3211. http://dx.doi.org/10.1007/s00500-022-07519-5 doi: 10.1007/s00500-022-07519-5
![]() |
[10] | M. A. Balci, L. M. Batrancea, O. Akguller, Network-induced soft sets and stock market applications, Mathematics, 10 ( 2022), 3964. http://dx.doi.org/10.3390/math10213964 |
[11] |
H. Qin, Q. Fei, X. Ma, W. Chen, A new parameter reduction algorithm for soft sets based on chi-square test, Appl. Intell., 51 (2021), 7960–7972. http://dx.doi.org/10.1007/s10489-021-02265-x doi: 10.1007/s10489-021-02265-x
![]() |
[12] |
X. Ma, H. Qin, Soft set based parameter value reduction for decision making application, IEEE Access, 7 (2019), 35499–35511. http://dx.doi.org/10.1109/ACCESS.2019.2905140 doi: 10.1109/ACCESS.2019.2905140
![]() |
[13] |
J. C. R. Alcantud, G. Varela, B. Santos-Buitrago, G. Santos-Garc ía, M. F. Jimenez, Analysis of survival for lung cancer resections cases with fuzzy and soft set theory in surgical decision making, Plos One, 14 (2019), e0218283. http://dx.doi.org/10.1371/journal.pone.0218283 doi: 10.1371/journal.pone.0218283
![]() |
[14] |
P. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X doi: 10.1016/S0898-1221(02)00216-X
![]() |
[15] |
Z. Xiao, K. Gong, Y. Zou, A combined forecasting approach based on fuzzy soft sets, J. Comput. Appl. Math., 228 (2009), 326–333. https://doi.org/10.1016/j.cam.2008.09.033 doi: 10.1016/j.cam.2008.09.033
![]() |
[16] | J. C. R. Alcantud, A. Z. Khameneh, G. Santos-Garcıa, M. Akram, A systematic literature review of soft set theory, Neural Comput. Applic., 36 (2024), 8951–8975. http://dx.doi.org/10.1007/s00521-024-09552-x |
[17] | J. C. R. Alcantud, Convex soft geometries, J. Comput. Cognitive Eng., 1 (2022), 2–12. https://doi.org/10.47852/bonviewJCCE597820 |
[18] |
F. Feng, C. X. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Comput., 14 (2010), 899–911. http://dx.doi.org/10.1007/s00500-009-0465-6 doi: 10.1007/s00500-009-0465-6
![]() |
[19] | M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786–1799. http://dx.doi.org/10.1016/j.camwa.2011.02.006 |
[20] |
T. M. Al-shami, M. E. El-Shafei, M. Abo-Elhamayel, Seven generalized types of soft semi-compact spaces, Korean J. Math., 27 (2019), 661–690. https://doi.org/10.11568/kjm.2019.27.3.661 doi: 10.11568/kjm.2019.27.3.661
![]() |
[21] | M. H. Alqahtani, Z. A. Ameen, Soft nodec spaces, AIMS Math., 9 (2024), 3289–3302. https://doi.org/10.3934/math.2024160 |
[22] | S. Al-Ghour, D. Abuzaid, M. Naghi, Soft weakly quasi-continuous functions between soft topological spaces, Mathematics, 12 ( 2024), 3280. https://doi.org/10.3390/math12203280 |
[23] |
S. Al Ghour, Between the Classes of soft open sets and soft omega open sets, Mathematics, 10 (2022), 719. https://doi.org/10.3390/math10050719 doi: 10.3390/math10050719
![]() |
[24] | T. M. Al-shami, A. Mhemdi, R. Abu-Gdairi, A Novel framework for generalizations of soft open sets and its applications via soft topologies, Mathematics, 11 (2023), 840. |
[25] | T. M. Al-shami, Soft somewhat open sets: Soft separation axioms and medical application to nutrition, Comput. Appl. Math., 41 (2022), 2016. https://doi.org/10.1007/s40314-022-01919-x |
[26] | S. Al Ghour, Boolean algebra of soft Q-Sets in soft topological spaces, Appl. Comput. Intell. Soft Comput., 2022 (2022), 5200590. https://doi.org/10.1155/2022/5200590 |
[27] |
A. Mhemdi, Novel types of soft compact and connected spaces inspired by soft Q-sets, Filomat, 37 (2023), 9617–9626. https://doi.org/10.2298/FIL2328617M doi: 10.2298/FIL2328617M
![]() |
[28] |
T. M. Al-shami, M. E. El-Shafei, Partial belong relation on soft separation axioms and decision-making problem, two birds with one stone, Soft Comput., 24 (2020), 5377–5387. https://doi.org/10.1007/s00500-019-04295-7 doi: 10.1007/s00500-019-04295-7
![]() |
[29] |
T. M. Al-shami, On soft separation axioms and their applications on decision-making problem, Math. Prob. Eng., 2021 (2021), 1–12. https://doi.org/10.1155/2021/8876978 doi: 10.1155/2021/8876978
![]() |
[30] |
X. Guan, Comparison of two types of separation axioms in soft topological spaces, J. Intell. Fuzzy Syst., 44 (2023), 2163–2171. https://doi.org/10.3233/JIFS-212432 doi: 10.3233/JIFS-212432
![]() |
[31] |
T. M. Al-shami, L. D. R. Kocinac, B. A. Asaad, Sum of soft topological spaces, Mathematics, 8 (2020), 990. https://doi.org/10.3390/math8060990 doi: 10.3390/math8060990
![]() |
[32] |
J. C. R. Alcantud, Soft open bases and a novel construction of soft topologies from bases for topologies, Mathematics, 8 (2020), 672. https://doi.org/10.3390/math8050672 doi: 10.3390/math8050672
![]() |
[33] | P. Majumdar, S. K. Samanta, On soft mappings, Comput. Math. Appl., 60 (2010), 2666–2672. http://dx.doi.org/10.1016/j.camwa.2010.09.004 |
[34] | A. Kharal, B. Ahmad, Mappings on soft classes, New Math. Nat. Comput., 7 (2011), 471–481. http://dx.doi.org/10.1142/S1793005711002025 |
[35] |
A. Aygunoglu, H. Aygun, Some notes on soft topological spaces, Neural Comput. Appl., 21 (2012), 113–119. http://dx.doi.org/10.1007/s00521-011-0722-3 doi: 10.1007/s00521-011-0722-3
![]() |
[36] |
M. Akdag, A. Ozkan, Soft α-open sets and soft α -continuous functions, Abstr. Appl. Anal., 2014 (2014), 891341. https://doi.org/10.1155/2014/891341 doi: 10.1155/2014/891341
![]() |
[37] |
M. Akdag, A. Ozkan, Soft β-open sets and soft β -continuous functions, Sci. World J., 2014 (2014), 843456. https://doi.org/10.1155/2014/843456 doi: 10.1155/2014/843456
![]() |
[38] |
I. Zorlutuna, H. Cakir, On continuity of soft mappings, Appl. Math. Inf. Sci., 9 (2015), 403–409. https://doi.org/10.12785/amis/090147 doi: 10.12785/amis/090147
![]() |
[39] |
T. Y. Ozturk, S. Bayramov, Topology on soft continuous function spaces, Math. Comput. Appl., 22 (2017), 32. https://doi.org/10.3390/mca22020032 doi: 10.3390/mca22020032
![]() |
[40] |
T. M. Al-shami, I. Alshammari, B. A. Asaad, Soft maps via soft somewhere dense sets, Filomat, 34 (2020), 3429–3440. https://doi.org/10.2298/FIL2010429A doi: 10.2298/FIL2010429A
![]() |
[41] |
S. Al Ghour, Soft ωp-open sets and soft ωp -continuity in soft topological spaces, Mathematics, 9 (2021), 2632. https://doi.org/10.3390/math9202632 doi: 10.3390/math9202632
![]() |
[42] |
S. Al Ghour, On some weaker forms of soft continuity and their decomposition theorems, J. Math. Comput. Sci., 29 (2023), 317–328. https://doi.org/10.22436/jmcs.029.04.02 doi: 10.22436/jmcs.029.04.02
![]() |
[43] |
T. M. Al-shami, Z. A. Ameen, B. A. Asaad, A. Mhemdi, Soft bi-continuity and related soft functions, J. Math. Comput. Sci., 30 (2023), 19–29. https://doi.org/10.22436/jmcs.030.01.03 doi: 10.22436/jmcs.030.01.03
![]() |
[44] |
S. Al Ghour, Soft functions via soft semi ω-open sets, J. Math. Comput. Sci., 30 (2023), 133–146. https://doi.org/10.22436/jmcs.030.02.05 doi: 10.22436/jmcs.030.02.05
![]() |
[45] | S. Al Ghour, A. Bin-Saadon, On some generated soft topological spaces and soft homogeneity, Heliyon, 5 (2019), e02061. http://dx.doi.org/10.1016/j.heliyon.2019.e02061 |
[46] |
S. Al Ghour, W. Hamed, On two classes of soft sets in soft topological spaces, Symmetry, 12 (2020), 265. https://doi.org/10.3390/sym12020265 doi: 10.3390/sym12020265
![]() |
[47] | D. S. Jankovic, θ-regular spaces, Internat. J. Math. Math. Sci., 8 (1985), 615–619. http://dx.doi.org/10.1155/S0161171285000667 |
[48] | B. Chen, Soft semi-open sets and related properties in soft topological spaces, Appl. Math. Inform. Sci., 7 (2013), 287–294. Available from: https://www.naturalspublishing.com/files/published/9n3942j17pww2p.pdf |
[49] | I. Arockiarani, A. Lancy, Generalized soft gβ-closed sets and soft gsβ-closed sets in soft topological spaces, Int. J. Math. Arch., 4 (2013), 1–7. Available from: https://api.semanticscholar.org/CorpusID: 124478886 |
[50] | S. Yuksel, N. Tozlu, Z. G. Ergul, Soft regular generalized closed sets in soft topological spaces, Int. J. Math. Anal., 8 (2014), 355–367. Available from: https://www.jatit.org/volumes/Vol37No1/2Vol37No1.pdf |
[51] | D. N. Georgiou, A. C. Megaritis, V. I. Petropoulos, On soft topological spaces, Appl. Math. Inf. Sci., 7 (2013), 1889–1901. http://dx.doi.org/10.12785/amis/070527 |
[52] |
J. Mahanta, P. K. Das, On soft topological space via semiopen and semiclosed soft sets, Kyungpook Math. J., 54 (2014), 221–236. http://dx.doi.org/10.5666/KMJ.2014.54.2.221 doi: 10.5666/KMJ.2014.54.2.221
![]() |
[53] |
O. R. Sayed, N. Hassan, A. M. Khalil, A decomposition of soft continuity in soft topological spaces, Afr. Mat., 28 (2017), 887–898. http://dx.doi.org/10.1007/s13370-017-0494-8 doi: 10.1007/s13370-017-0494-8
![]() |
[54] | S. S. Thakur, A. S. Rajput, Soft almost α-continuous mappings, J. Adv. Stud. Topol., 9 (2018), 94–99. |
[55] |
S. Ramkumar, V. Subbiah, Soft separation axioms and soft product of soft topological spaces, Maltepe J. Math., 2 (2020), 61–75. https://doi.org/10.47087/mjm.723886 doi: 10.47087/mjm.723886
![]() |
[56] | M. Akdag, A. Ozkan, On soft preopen sets and soft pre separation axioms, Gazi Univ. J. Sci., 27 (2014), 1077–1083. https://dergipark.org.tr/en/download/article-file/83676 |
[57] | G. Ilango, M. Ravindran, On soft preopen sets in soft topological spaces, Int. J. Math. Res., 4 (2013), 399–409. Available from: http://irphouse.com/ijmr/ijmrv5n4_05.pdf |
[58] | S. Al Ghour, Strong form of soft semi-open sets in soft topological spaces, Int. J. Fuzzy Logic Intell. Syst., 21 (2021), 159–168. http://dx.doi.org/10.5391/IJFIS.2021.21.2.159 |
[59] |
S. S. Thakur, A. S. Rajput, Connectedness between soft sets, New Math. Natural Comput., 14 (2018), 53–71. http://dx.doi.org/10.1142/S1793005718500059 doi: 10.1142/S1793005718500059
![]() |
[60] | S. Al Ghour, Z. A. Ameen, On soft submaximal spaces, Heliyon, 8 (2022), e10574. http://dx.doi.org/10.1016/j.heliyon.2022.e10574 |