In this paper, the Levenberg-Marquardt backpropagation scheme is used to develop a neural network model for the examination of the fluid flow on a magnetized flat surface with slip boundaries. The tangent hyperbolic fluid is considered along with heat generation, velocity, and thermal slip effects at the surface. The problem is modelled in terms of a non-linear differential system and Lie symmetry is used to get the scaling group of transformation. The order reduction of differential equations is done by using Lie transformation. The reduced system is solved by the shooting method. The surface quantity, namely skin friction, is evaluated at the surface for the absence and presence of an externally applied magnetic field. A total of 88 sample values are estimated for developing an artificial neural network model to predict skin friction coefficient (SFC). Weissenberg number, magnetic field parameter, and power law index are considered three inputs in the first layer, while 10 neurons are taken in the hidden layer. 62 (70%), 13 (15%), and 13 (15%) samples are used for training, validation, and testing, respectively. The Levenberg-Marquardt backpropagation is used to train the network by entertaining the random 62 sample values. Both mean square error and regression analysis are used to check the performance of the developed neural networking model. The SFC is noticed to be high at a magnetized surface for power law index and Weissenberg number.
Citation: Khalil Ur Rehman, Wasfi Shatanawi, Zead Mustafa. Artificial intelligence (AI) based neural networks for a magnetized surface subject to tangent hyperbolic fluid flow with multiple slip boundary conditions[J]. AIMS Mathematics, 2024, 9(2): 4707-4728. doi: 10.3934/math.2024227
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In this paper, the Levenberg-Marquardt backpropagation scheme is used to develop a neural network model for the examination of the fluid flow on a magnetized flat surface with slip boundaries. The tangent hyperbolic fluid is considered along with heat generation, velocity, and thermal slip effects at the surface. The problem is modelled in terms of a non-linear differential system and Lie symmetry is used to get the scaling group of transformation. The order reduction of differential equations is done by using Lie transformation. The reduced system is solved by the shooting method. The surface quantity, namely skin friction, is evaluated at the surface for the absence and presence of an externally applied magnetic field. A total of 88 sample values are estimated for developing an artificial neural network model to predict skin friction coefficient (SFC). Weissenberg number, magnetic field parameter, and power law index are considered three inputs in the first layer, while 10 neurons are taken in the hidden layer. 62 (70%), 13 (15%), and 13 (15%) samples are used for training, validation, and testing, respectively. The Levenberg-Marquardt backpropagation is used to train the network by entertaining the random 62 sample values. Both mean square error and regression analysis are used to check the performance of the developed neural networking model. The SFC is noticed to be high at a magnetized surface for power law index and Weissenberg number.
The need for theories that cope with uncertainty emerges from daily experiences with complicated challenges requiring ambiguous facts. Molodstov's [1] soft set is a contemporary mathematical approach to coping with these difficulties. Soft collection logic is founded on the parameterization principle, which argues that complex things must be seen from several perspectives, with each aspect providing only a partial and approximate representation of the full item. Molodstov [1] was a pioneer in the application of soft sets in a variety of domains, emphasizing their advantages over probability theory and fuzzy set theory, which deal with ambiguity or uncertainty.
Following that, Maji et al. [2] began researching soft set operations such as soft unions and soft intersections. To overcome the shortcomings of these operations, Ali et al. [3] created and showed new operations such as limited union, intersection, and complement of a soft set. Babitha and Sunil [4] investigated numerous aspects of linkages and functions in a soft setting. Qin and Hong [5] developed novel kinds of soft equal relations and showed some algebraic properties of them. Their pioneering work paved the way for subsequent papers (for more detail, see [6,7] and the references listed therein). Soft set theory has lately been a popular method among academics for dealing with uncertainty in a wide range of fields, including information theory [8], computer sciences [9], engineering [10], and medical sciences [11].
Soft topology was introduced by Shabir and Naz in [12]. Since then, many soft topological notions, including soft separation axioms [13,14,15,16], soft covering axioms [17,18,19,20,21,22], soft connectedness [23,24,25,26], and different weak and strong types of soft continuity, have been developed and investigated in recent years. The equivalence between the enriched and extended soft topologies was discussed in [27].
Separation axioms provide a way to study certain properties of compact and Lindelof spaces, as well as a way to categorize spaces and mappings into distinct families. As a result, topological scholars who presented various kinds of soft separation axioms became interested in soft separation axioms. Generally speaking, they can be separated into two classes: Soft points and ordinary points, based on the subjects being studied. While the authors in [14,15,16,28] examined soft separation axioms using ordinary points, the authors in [13,29,30,31,32,33] and others have applied the concept of soft points. In the present work, we introduce soft ω-almost-regularity, soft ω-semi-regularity, and soft ω-T212 as three novel soft separation axioms.
This article is organized as follows:
In Section 1, after the introduction, we provide a few definitions that are relevant to this paper.
In Section 2, we define soft ω-almost-regularity as a new soft separation axiom that lies between soft regularity and soft almost-regularity. We introduce many characterizations of this type of soft separation axiom. Also, we provide several sufficient conditions establishing the equivalence between this newly introduced axiom and its relevant counterparts. Moreover, we establish that soft ω -almost-regularity is heritable for specific types of soft subspaces. Furthermore, we show that soft ω-almost-regularity is a productive soft property. In addition, we investigated the links between this class of soft topological spaces and its analogs in general topology.
In Section 3, we define soft ω-semi-regularity and soft ω-T212 as two new soft separation axioms. We show that soft ω-semi-regularity is a weaker form of both soft semi-regularity and soft ω-regularity, and soft ω-T212 lies strictly between soft T212 and soft T2. Also, we provide several sufficient conditions establishing the equivalence between these newly introduced axioms and their relevant counterparts. Moreover, a decomposition theorem for soft regularity through the interplay of soft ω-semi-regularity and soft ω-almost-regularity is obtained. In addition, we investigated the links between these classes of soft topological spaces and their analogs in general topology.
This paper follows the notions and terminologies as appear in [34,35,36]. Topological spaces and soft topological spaces, respectively, shall be abbreviated as TS and STS.
The following definitions will be used in the remainder of the paper:
Definition 1.1. A TS (H,β) is called
(a) [37] almost-regular (A-R, for simplicity) if for every z∈H and every N∈SC(H,β) such that z∈H−N, we find U,V∈β such that z∈U, N⊆V, and U∩V=∅;
(b) [38] semi-regular (S-R, for simplicity) if RO(H,β) forms a base for β;
(c) [39] ω-almost-regular (ω-A-R, for simplicity) if for every z∈H and every N∈SωC(H,β) such that z∈H−N, we find U,V∈β such that z∈U, N⊆V, and U∩V=∅;
(d) [39] ω-semi-regular (ω-S-R, for simplicity) if RωO(H,β) forms a base for β.
Definition 1.2. A STS (H,φ,Σ) is called
(a) [13] soft T2 if for every two soft points as,bt∈SP(H,Σ), we find K,W∈φ such that as˜∈K, by˜∈W, and K˜∩W=0Σ;
(b) [13] soft regular if for every az∈SP(H,Σ) and every K∈φ such that az˜∈K, we find G∈φ such that az˜∈G˜⊆Clφ(G)˜⊆K;
(c) [32] soft T212 if for every two soft points as,bt∈SP(H,Σ), we find K,W∈φ such that as˜∈K, by˜∈W, and Clφ(K)˜∩Clφ(W)=0Σ;
(d) [31] soft almost-regular (soft A-R, for simplicity) if for every rz∈SP(H,Σ) and every G∈SC(H,φ,Σ) such that rz˜∈1Σ−G, we find S,T∈φ such that rz˜∈S, G˜⊆T, and S˜∩T=0Σ.
(d) [33] soft ω-regular for every az∈SP(H,Σ) and every K∈φ such that az˜∈K, we find G∈φ such that az˜∈G˜⊆Clφω(G)˜⊆K.
(e) [22] fully if G(r)≠∅ for every G∈φ−{0Σ} and r∈Σ.
In this section, we define soft ω-almost-regularity as a new soft separation axiom that lies between soft regularity and soft almost-regularity. We introduce many characterizations of this type of soft separation axiom. Also, we provide several sufficient conditions establishing the equivalence between this newly introduced axiom and its relevant counterparts. Moreover, we establish that soft ω -almost-regularity is heritable for specific types of soft subspaces. Furthermore, we show that soft ω-almost-regularity is a productive soft property. In addition, we investigated the links between this class of soft topological spaces and its analogs in general topology.
Definition 2.1. An STS (H,φ,Σ) is called soft ω-almost-regular (soft ω-A-R, for simplicity) if for every rz∈SP(H,Σ) and every G∈SωC(H,φ,Σ) such that rz˜∈1Σ−G, we find S,T∈φ such that rz˜∈S, G˜⊆T, and S˜∩T=0Σ.
Several characterizations of soft ω-almost-regularity are listed in the following theorem.
Theorem 2.2. The following are equivalent for any STS (H,φ,Σ):
(1) (H,φ,Σ) is soft ω-A-R.
(2) For every rz∈SP(H,Σ) and every K∈SωO(H,φ,Σ) such that rz˜∈K, we find L∈φ such that rz˜∈L˜⊆Clφ(L)˜⊆K.
(3) For every rz∈SP(H,Σ) and every K∈SωO(H,φ,Σ) such that rz˜∈K, we find L∈SO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆K.
(4) For every rz∈SP(H,Σ) and every K∈SωO(H,φ,Σ) such that rz˜∈K, we find L∈SωO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆K.
(5) For every rz∈SP(H,Σ) and every K∈φ such that rz˜∈K, there is L∈SωO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφω(K)).
(6) For every rz∈SP(H,Σ) and every K∈φ such that rz˜∈K, there is L∈φ such that rz˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφω(K)).
(7) For every rz∈SP(H,Σ) and every G∈SωC(H,φ,Σ) such that rz˜∈1Σ−G, there are S,T∈φ such that rz˜∈S, G˜⊆T, and Clφ(S)˜∩Clφ(T)=0Σ.
(8) For every G∈SωC(H,φ,Σ), G=˜∩{Clφ(K):K∈φ and G˜⊆K}.
(9) For every G∈SωC(H,φ,Σ), G=˜∩{Y:Y∈φc and G˜⊆Intφ(Y)}.
(10) For every L∈SS(H,Σ) and every M∈SωO(H,φ,Σ) such that L˜∩M≠0Σ, there is K∈φ such that L˜∩K≠0Σ and Clφ(K)˜⊆M.
(11) For every L∈SS(H,Σ)−{0Σ} and every M∈SωC(H,φ,Σ) such that L˜∩M=0Σ, there are S,T∈φ such that L˜∩S≠0Σ and M˜⊆T.
Proof. (1) ⟶ (2): Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. Then, rz˜∉1Σ−K∈SωC(H,φ,Σ) and by (a) there exist L,T∈φ such that rz˜∈L, 1Σ−K˜⊆T, and L˜∩T=0Σ. Thus, rz˜∈L˜⊆1Σ−T˜⊆K with 1Σ−T∈φc, and so rz˜∈L˜⊆Clφ(L)˜⊆1Σ−T˜⊆K. This ends the proof.
(2) ⟶ (3): Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. By (2) we find M∈φ such that rz˜∈M˜⊆Clφ(M)˜⊆K. Set L=Intφ(Clφ(M)). Then, L∈SO(H,φ,Σ). Since L˜⊆Clφ(M)˜⊆K, Clφ(L)˜⊆Clφ(M)˜⊆K. This completes the proof.
(3) ⟶ (4): Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. By (3) we find L∈SO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆K. Since L∈SO(H,φ,Σ), and by Theorem 3 of [36], we have RO(H,φ,Σ)˜⊆RωO(H,φ,Σ), L∈SωO(H,φ,Σ). This completes the proof.
(4) ⟶ (5): Let rz∈SP(H,Σ) and K∈φ such that rz˜∈K. Since by Theorem 9 of [36] Intφ(Clφω(K))∈SωO(H,φ,Σ), by (4) there is L∈SωO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφω(K)). This completes the proof.
(5) ⟶ (6): Let rz∈SP(H,Σ) and K∈φ such that rz˜∈K. Then, by (5) we find L∈SωO(H,φ,Σ) such that rz˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφω(K)). Since by Theorem 3 of [36] we have RωO(H,φ,Σ)˜⊆φ, then L∈φ. This completes the proof.
(6) ⟶ (7): Let rz∈SP(H,Σ) and G∈SωC(H,φ,Σ) such that rz˜∈1Σ−G. Since by Theorem 3 of [36] RωO(H,φ,Σ)˜⊆φ, we have rz˜∈1Σ−G∈φ. So, by (6) we find N∈φ such that rz˜∈N˜⊆Clφ(N)˜⊆Intφ(Clφω(1Σ−G))=1Σ−G. Again, by (6) we find S∈φ such that rz˜∈S˜⊆Clφ(S)˜⊆Intφ(Clφω(N))˜⊆Clφ(N)˜⊆1Σ−G. Let T=1Σ−Clφ(N). Then, S,T∈φ and rz˜∈S. Since Clφ(N)˜⊆1Σ−G, then G˜⊆1Σ−Clφ(N)=T.
Claim. Clφ(S)˜∩Clφ(T)=0Σ.
Proof of Claim. Suppose to the contrary that there is ax˜∈Clφ(S)˜∩Clφ(T). Since ax˜∈Clφ(T) and ax˜∈Clφ(S)˜⊆Intφ(Clφω(N))∈φ, then Intφ(Clφω(N))˜∩T≠0Σ. Since Intφ(Clφω(N))˜⊆Clφ(N), then Clφ(N)˜∩T=Clφ(N)˜∩(1Σ−Clφ(N))≠0Σ, a contradiction.
This completes the proof.
(7) ⟶ (8): Let G∈SωC(H,φ,Σ). Then, for each rz˜∈1Σ−G, there exist Srz,Trz∈φ such that rz˜∈Srz, G˜⊆Trz, and Clφ(Srz)˜∩Clφ(Trz)=0Σ. Thus, G˜⊆Trz and rz˜∉Clφ(Trz).
Claim. G=˜∩{Clφ(Trz):rz˜∈1Σ−G}.
Proof of Claim. For every rz˜∈1Σ−G, we have G˜⊆Trz˜⊆Clφ(Trz), and so G˜⊆˜∩{Clφ(Trz):rz˜∈1Σ−G}. To show that ˜∩{Clφ(Trz):rz˜∈1Σ−G}˜⊆G, let rz˜∈1Σ−G. Then, rz˜∉Clφ(Trz), and thus rz˜∉˜∩{Clφ(Trz):rz˜∈1Σ−G}.
By the above claim, we conclude that G˜⊆˜∩{Clφ(T):T∈φ with G˜⊆T}˜⊆˜∩{Clφ(Trz):rz˜∈1Σ−G}=G. This completes the proof.
(8) ⟶ (9): Obvious.
(9) ⟶ (10): Let L∈SS(H,Σ) and M∈SωO(H,φ,Σ) such that L˜∩M≠0Σ. Pick ax˜∈L˜∩M. Since M∈SωO(H,φ,Σ), 1Σ−M∈SωC(H,φ,Σ), and by (9) 1Σ−M=˜∩{Y:Y∈φc with 1Σ−M˜⊆Intφ(Y)}. Since ax˜∈M, then ax˜∉˜∩{Y:Y∈φc with 1Σ−M˜⊆Intφ(Y)}, and thus we find Y∈φc such that 1Σ−M˜⊆Intφ(Y) and ax˜∉Y. Let S=1Σ−Y. Then, S∈φ, S˜⊆1Σ−Intφ(Y)˜⊆M, and ax˜∈S˜∩L. Since 1Σ−Intφ(Y)∈φc and S˜⊆1Σ−Intφ(Y)˜⊆M, then Clφ(S)˜⊆M. This completes the proof.
(10) ⟶ (11): Let L∈SS(H,Σ)−{0Σ} and M∈SωC(H,φ,Σ) such that L˜∩M=0Σ. Then, 1Σ−M∈SωO(H,φ,Σ) such that L˜∩(1Σ−M)=L≠0Σ. Thus, by (10) we find S∈φ such that L˜∩S≠0Σ and Clφ(S)˜⊆1Σ−M. Let T=1Σ−Clφ(S). Then, T∈φ, M˜⊆T, and S˜∩T=S˜∩(1Σ−Clφ(S))=0Σ.
(11) ⟶ (1): rz∈SP(H,Σ) and every G∈SωC(H,φ,Σ) such that rz˜∈1Σ−G. Then, rz˜∩G=0Σ, and by (11) there exist S,T∈φ such that rz˜∩S≠0Σ, G˜⊆T, and S˜∩T=0Σ. Since rz˜∩S≠0Σ, then rz˜∈S. This ends the proof.
In Theorems 2.3, 2.4, 2.7, and Corollary 2.8, we discuss the connections between soft almost-regularity and its analog in traditional topological spaces. Also, in Theorems 2.5, 2.6, 2.9, and Corollary 2.10, we discuss the connections between soft ω-almost-regularity and its analog in traditional topological spaces.
Theorem 2.3. If (H,φ,Σ) is full and soft A-R, then (H,φr) is A-R for all r∈Σ.
Proof. Let (H,φ,Σ) be full and soft A-R. Let r∈Σ. Let z∈H and let W∈φr such that z∈W. Choose K∈φ such that K(r)=W. Since rz˜∈K∈φ, by Theorem 3.4 (ⅳ) of [31], we find L∈φ such that rz˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφ(K)). By Proposition 7 of [12], Clφr(L(r))⊆(Clφ(L))(r). Also, by Theorem 12 (c) of [36], (Intφ(Clφ(K)))(r) =Intφ(Clφ(K(r))). Therefore, we have
z∈L(r)⊆Clφr(L(r))⊆(Clφ(L))(r)⊆(Intφ(Clφ(K)))(r)=Intφ(Clφ(K(r)))=Intφr(Clφr(W)). |
Hence, by Theorem 2.2 (d) of [37], it follows that (H,φr) is A-R.
Theorem 2.4. Let (D,L) be a TS. Then, for any set Σ, (D,C(L),Σ) is soft A-R iff (D,L) is A-R.
Proof. Necessity. Let (D,C(L),Σ) be soft A-R. Pick r∈Σ. Since it is clear that (D,C(L),Σ) is full, then by Theorem 2.3, (D,(C(L))r)=(D,L) is A-R.
Sufficiency. Let (D,L) be A-R. Let rz∈SP(D,Σ) and let CU∈C(L) such that rz˜∈CU. Then, we have z∈U∈L. So, by Theorem 2.2 (d) of [37], we find V∈L such that z∈V⊆ClL(V)⊆IntL(ClL(U)). Thus, we have CV∈C(L) and rz˜∈CV˜⊆ClC(L)(CV)= CClL(V)˜⊆CIntL(ClL(U))=IntC(L)(ClC(L)(CU)). Therefore, by Theorem 3.4 (ⅳ) of [39], (D,C(L),Σ) is soft A-R.
Theorem 2.5. Let (D,L) be a TS. Then, for any set Σ, (D,C(L),Σ) is soft ω-A-R iff (D,L) is ω-A-R.
Proof. Necessity. Let (D,C(L),Σ) be soft ω-A-R. Let z∈D and U∈L such that z∈U. Pick r∈Σ. Then, we have rz˜∈CU∈C(L). Since (D,C(L),Σ) is soft ω-A-R, by Theorem 2.2 (5) we find V∈L such that rz˜∈CV˜⊆ClC(L)(CV)=CClL(V)˜⊆IntC(L)(Cl(C(L))ω(CV))=CIntL(ClLω(U)). Therefore, z∈V⊆ClL(V)⊆IntL(ClLω(U)). This shows that (D,L) is ω-A-R.
Sufficiency. Let (D,L) be ω-A-R. Let rz∈SP(D,Σ) and let CU∈C(L) such that rz˜∈CU. Then, we have z∈U∈L. So, by Theorem 2.1 (e) of [39], we find V∈L such that z∈V⊆ClL(V)⊆IntL(ClLω(U)). Thus, we have CV∈C(L) and rz˜∈CV˜⊆ClC(L)(CV)=CClL(V)˜⊆CIntL(ClLω(U))=IntC(L)(Cl(C(L))ω(CV)). Therefore, (D,C(L),Σ) is soft ω-A-R.
Theorem 2.6. Let (D,L) be a TS. Then, for any set Σ, (D,C(L),Σ) is soft regular iff (D,L) is regular.
Proof. Necessity. Let (D,C(L),Σ) be soft regular. Let z∈D and U∈L such that z∈U. Pick r∈Σ. Then, we have rz˜∈CU∈C(L). Since (D,C(L),Σ) is soft regular, we find V∈L such that rz˜∈CV˜⊆ClC(L)(CV)=CClL(V)˜⊆CU. Therefore, z∈V⊆ClL(V)⊆U. This shows that (D,L) is regular.
Sufficiency. Let (D,L) be regular. Let rz∈SP(D,Σ) and let CU∈C(L) such that rz˜∈CU. Then, we have z∈U∈L. So, we find V∈L such that z∈V⊆ClL(V)⊆U. Thus, we have CV∈C(L) and rz˜∈CV˜⊆CClL(V)=ClC(L)(CV)˜⊆CU. Therefore, (D,C(L),Σ) is soft regular.
Theorem 2.7. Let {(H,Lr):r∈Σ} be a collection of TSs. Then, (H,⊕r∈ΣLr,Σ) is soft A-R iff (H,Lr) is A-R for every r∈Σ.
Proof. Necessity. Let (H,⊕r∈ΣLr,Σ) be soft A-R and let r∈Σ. Let z∈H and let U∈Lr such that z∈U. Then, rz˜∈rU∈⊕r∈ΣLr. So, by Theorem 3.4 (ⅳ) of [31], we find L∈⊕r∈ΣLr such that rz˜∈L˜⊆Cl⊕r∈ΣLr(L)˜⊆Int⊕r∈ΣLr(Cl⊕r∈ΣLr(rU)). Thus, we have z∈L(r)∈Lr and z∈L(r)⊆(Cl⊕r∈ΣLr(L))(r)⊆(Int⊕r∈ΣLr(Cl⊕r∈ΣLr(rU)))(r).
In contrast, by Lemma 4.9 of [40], (Cl⊕r∈ΣLr(L))(r)=ClLr(L(r)) and (Int⊕r∈ΣLr(Cl⊕r∈ΣLr(rU)))(r) =IntLr((ClLr(rU))(r))=IntLr(ClLr(U)). Thus, by Theorem 3.4 (ⅳ) of [31], (H,Lr) is A-R.
Sufficiency. Let (H,Lr) be A-R for every r∈Σ. Let rz∈SP(H,Σ) and let K∈⊕r∈ΣLr such that rz˜∈K. By Theorem 3.5 of [34], we find U∈Lr such that rz˜∈rU˜⊆K. Then, we have z∈U∈Lr. So, by Theorem 2.1 (d) of [39], we find V∈Lr such that z∈V⊆ClLr(V)⊆IntLr(ClLr(U)). Thus, we have rV∈⊕r∈ΣLr and
rz˜∈rV˜⊆rClLr(V)=Cl⊕r∈ΣLr(rV)˜⊆rIntLr(ClLr(U))=Int⊕r∈ΣLr(Cl⊕r∈ΣLr(rU))˜⊆Int⊕r∈ΣLr(Cl⊕r∈ΣLr(K)). |
Corollary 2.8. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft A-R iff (D,L) is A-R.
Proof. For each r∈Σ, set Lr=L. Then, τ(L)=⊕r∈ΣLr, and by Theorem 2.7 we get the result.
Theorem 2.9. Let {(H,Lr):r∈Σ} be a collection of TSs. Then, (H,⊕r∈ΣLr,Σ) is soft ω-A-R iff (H,Lr) is ω-A-R for every r∈Σ.
Proof. Necessity. Let (H,⊕r∈ΣLr,Σ) be soft ω-A-R and let r∈Σ. Let z∈H and let U∈Lr such that z∈U. Then, rz˜∈rU∈⊕r∈ΣLr. So, by Theorem 2.2 (e), we find L∈⊕r∈ΣLr such that rz˜∈L˜⊆Cl⊕r∈ΣLr(L)˜⊆Int⊕r∈ΣLr(Cl(⊕r∈ΣLr)ω(rU)). Thus, we have z∈L(r)∈Lr and z∈L(r)⊆(Cl⊕r∈ΣLr(L))(r)⊆(Int⊕r∈ΣLr(Cl⊕r∈ΣLr(rU)))(r).
In contrast, by Lemma 4.9 of [40] and Theorem 8 of [35], (Cl⊕r∈ΣLr(L))(r)=ClLr(L(r)) and Int⊕r∈ΣLr(Cl(⊕r∈ΣLr)ω(rU))(r) =IntLr((Cl(Lr)ω(rU))(r)) =IntLr(Cl(Lr)ω(U)). Thus, by Theorem 2.1 (e) of [39] (H,Lr) is ω-A-R.
Sufficiency. Let (H,Lr) be ω-A-R for every r∈Σ. Let rz∈SP(H,Σ) and let K∈⊕r∈ΣLr such that rz˜∈K. By Theorem 3.5 of [34] we find U∈Lr such that rz˜∈rU˜⊆K. Then, we have z∈U∈Lr. So, by Theorem 2.1 (e) of [39] we find V∈Lr such that z∈V⊆ClLr(V)⊆IntLr(Cl(Lr)ω(U)). Thus, we have rV∈⊕r∈ΣLr and
rz˜∈rV˜⊆rClLr(V)=Cl⊕r∈ΣLr(rV)˜⊆rIntLr(Cl(Lr)ω(U))=Int⊕r∈ΣLr(Cl(⊕r∈Σ(Lr)ω)(rU))˜⊆Int⊕r∈ΣLr(Cl⊕r∈Σ(Lr)ω(K))=Int⊕r∈ΣLr(Cl(⊕r∈ΣLr)ω(K)). |
Corollary 2.10. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft ω-A-R iff (D,L) is ω-A-R.
Proof. For each r∈Σ, set Lr=L. Then, τ(L)=⊕r∈ΣLr and by Theorem 2.9 we get the result.
Theorem 2.11. Soft regular STSs are soft ω-A-R.
Proof. Let (H,φ,Σ) be soft regular. Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. Since by Theorem 3 of [36] we have RωO(H,φ,Σ)⊆φ, then K∈φ. Since (H,φ,Σ) is soft regular, then we find G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Thus, by Theorem 2.2 (2) (H,φ,Σ) is soft ω-A-R.
Theorem 2.12. Soft ω-A-R STSs are soft A-R.
Proof. Let (H,φ,Σ) be soft ω-A-R. Let rz∈SP(H,Σ) and K∈SO(H,φ,Σ) such that rz˜∈K. Since by Theorem 3 of [36] we have RO(H,φ,Σ)⊆RωO(H,φ,Σ), then K∈SωO(H,φ,Σ). Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (b) there is G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Thus, by Theorem 2.2 (b) of [31], (H,φ,Σ) is soft A-R.
Theorem 2.13. Soft L-C soft ω-A-R STSs are soft regular.
Proof. Let (H,φ,Σ) be soft L-C and soft ω -A-R. Let rz∈SP(H,Σ) and K∈φ such that rz˜∈K. Since (H,φ,Σ) is soft L-C, then by Theorem 5 of [36] K∈SωO(H,φ,Σ). Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (2) there is G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Therefore, (H,φ,Σ) is soft regular.
Theorem 2.14. Soft anti-L-C soft A-R STSs are soft ω-A-R.
Proof. Let (H,φ,Σ) be soft anti-L-C and soft A-R. Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. Since (H,φ,Σ) is anti-L-C, then by Theorem 6 of [36] K∈SO(H,φ,Σ). Since (H,φ,Σ) is soft A-R, then by Theorem 3.4 (ⅱ) of [31] there is G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Therefore, by Theorem 2.2 (b), (H,φ,Σ) is soft ω-A-R.
Theorem 2.15. For any STS (H,φ,Σ), (H,φω,Σ) is soft A-R iff (H,φ,Σ) is soft ω -A-R.
Proof. Necessity. Let (H,φω,Σ) be soft A-R. Let rz∈SP(H,Σ) and K∈SωO(H,φ,Σ) such that rz˜∈K. By Theorem 7 of [36] K∈SO(H,φω,Σ). Since (H,φω,Σ) is soft A-R, then by Theorem 3.4 (ⅱ) of [31] there is G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Therefore, by Theorem 2.2 (b) (H,φ,Σ) is soft ω-A-R.
Sufficiency. Let (H,φ,Σ) be soft ω-A-R. Let rz∈SP(H,Σ) and K∈SO(H,φω,Σ) such that rz˜∈K. By Theorem 7 of [36] K∈SωO(H,φ,Σ). Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (2) there is G∈φ such that rz˜∈G˜⊆Clφ(G)˜⊆K. Therefore, by Theorem 3.4 (ⅱ) of [31] (H,φω,Σ) is soft A-R.
The previously mentioned theorems lead to the following implications, yet Examples 2.16 and 2.17 that follow demonstrate that the opposite of these implications is false.
Soft regular ⟶Soft ω-A-R ⟶Soft A-R. |
The following two examples show that any of the conditions soft L-C and soft anti-L-C in Theorems 2.13 and 2.14 cannot be dropped:
Example 2.16. Consider (R,C(Θ),Z), where Θ is the cofinite topology on R. Since (R,Θ) is not regular, by Theorem 2.6 (R,C(Θ),Z) is not soft regular. In contrast, since (R,C(Θ),Z) is anti-L-C, then by Theorem 6 of [36] RωO(R,C(Θ),Z)=RO(R,C(Θ),Z)={0Z,1Z}, and thus (R,C(Θ),Z) is soft ω-A-R.
Example 2.17. Consider (N,C(Θ),{a,b}), where Θ is the cofinite topology on N. Since (N,Θ) is not regular, by Theorem 2.6 (N,C(Θ),{a,b}) is not soft regular. Since (N,C(Θ),{a,b}) is soft L-C, then by Theorem 2.13 (N,C(Θ),{a,b})is not soft ω-A-R. In contrast, since RO(N,C(Θ),{a,b})={0{a,b},1{a,b}}, then (N,C(Θ),{a,b}) is soft A-R.
The following lemma will be used in the next main result:
Lemma 2.18. Let (H,φ,Σ) be an STS. If CY is a soft dense subset of (H,φω,Σ), then for any soft subset H∈SS(Y,Σ) IntφY(Cl(φω)Y(H))=Intφ(Clφω(H))˜∩CY.
Proof. Suppose that CY is a soft dense subset of (H,φω,Σ) and let H∈SS(Y,Σ). To see that IntφY(Cl(φω)Y(H))˜⊆Intφ(Clφω(H))˜∩CY, let ax˜∈IntφY(Cl(φω)Y(H)). Since IntφY(Cl(φω)Y(H))∈φY, then there is M∈φ such that IntφY(Cl(φω)Y(H))=M˜∩CY. Thus, we have ax˜∈M˜∩CY˜⊆Cl(φω)Y(H)=(Clφω(H))˜∩CY.
Claim. M˜⊆Clφω(H).
Proof of Claim. Suppose to the contrary that M˜∩(1Σ−Clφω(H))≠0Σ. Since 1Σ−Clφω(H)∈φω and M∈φ⊆φω, then M˜∩(1Σ−Clφω(H))∈φω. Since CY is soft dense in (H,φω,Σ), then M˜∩(1Σ−Clφω(H))˜∩CY≠0Σ. Choose by˜∈M˜∩(1Σ−Clφω(H))˜∩CY. Thus, we have by˜∈1Σ−Clφω(H) and by˜∈M˜∩CY˜⊆(Clφω(H))˜∩CY˜⊆Clφω(H), a contradiction.
Therefore, by the above Claim, we must have ax˜∈M˜⊆Clφω(H), and hence ax˜∈Intφ(Clφω(H). Hence, ax˜∈Intφ(Clφω(H))˜∩CY.
To see that Intφ(Clφω(H))˜∩CY˜⊆IntφY(Cl(φω)Y(H)), let ax˜∈Intφ(Clφω(H))˜∩CY. Since ax˜∈Intφ(Clφω(H))∈φ, then there is M∈φ such that ax˜∈M˜⊆Clφω(H) and so ax˜∈M˜∩CY˜⊆Clφω(H)˜∩CY=Cl(φω)Y(H). Since M˜∩CY∈φω, then ax˜∈IntφY(Cl(φω)Y(H)).
Theorems 2.19 and 2.21 establish that soft ω-almost-regularity is heritable for specific types of soft subspaces.
Theorem 2.19. If (H,φ,Σ) is a soft ω-A-R STS and CY is a soft dense subspace of (H,φω,Σ), then (Y,φY,Σ) is soft ω-A-R.
Proof. Let ax∈SP(Y,Σ) and let H∈SωO(Y,φY,Σ) such that ax˜∈H. Since H∈SωO(Y,φY,Σ), then IntφY(Cl(φY)ω(H))=H. Since by Theorem 15 of [35] (φω)Y=(φY)ω, then IntφY(Cl(φω)Y(H))=H. So, by Lemma 2.18 H=Intφ(Clφω(H))˜∩CY. Thus, we have ax˜∈Intφ(Clφω(H))∈SωO(H,φ,Σ). Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (2) there is L∈φ such that ax˜∈L˜⊆Clφ(L)˜⊆Intφ(Clφω(H)). Therefore, we have ax˜∈L˜∩CY∈φY and ClφY(L˜∩CY)=Clφ(L˜∩CY)˜∩CY˜⊆Intφ(Clφω(H))˜∩CY=H. This shows that (Y,φY,Σ) is soft ω -A-R.
The following lemma will be used in the next main result:
Lemma 2.20. Let (H,φ,Σ) be an STS and let CY∈SωO(H,φ,Σ)−{0Σ}, then RωO(Y,φY,Σ)⊆RωO(H,φ,Σ).
Proof. Let CY∈SωO(H,φ,Σ)−{0Σ} and let H∈SωO(Y,φY,Σ). Then, H=IntφY(Cl(φY)ω(H)). Since by Theorem 15 of [35], (φω)Y=(φY)ω, then Cl(φY)ω(H)=Cl(φω)Y(H)=Clφω(H)˜∩CY. Since by Theorem 3 of [36] RωO(H,φ,Σ)⊆φ, then CY∈φ and so IntφY(Cl(φY)ω(H))=Intφ((Cl(φY)ω(H))). Thus, H=Intφ(Clφω(H)˜∩CY) =Intφ(Clφω(H))˜∩Intφ(CY)=Intφ(Clφω(H))˜∩CY. Since H˜⊆CY, then Intφ(Clφω(H))˜⊆Intφ(Clφω(CY))=CY and thus, Intφ(Clφω(H))˜∩CY=Intφ(Clφω(H)). Therefore, H=Intφ(Clφω(H)). Hence, H∈SωO(H,φ,Σ).
Theorem 2.21. If (H,φ,Σ) is a soft ω-A-R STS and CY∈SωO(H,φ,Σ)−{0Σ}, then (Y,φY,Σ) is soft ω-A-R.
Proof. Let ax∈SP(Y,R) and let H∈SωO(Y,φY,Σ) such that ax˜∈H. By Lemma 2.20, H∈SωO(H,φ,Σ). Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (2) there is L∈φ such that ax˜∈L˜⊆Clφ(L)˜⊆H. Therefore, we have ax˜∈L˜∩CY∈φY and ClφY(L)=Clφ(L)˜∩CY˜⊆H. Hence, (Y,φY,Σ) is soft ω-A-R.
The following lemma will be used in Theorems 2.23 and 3.33:
Lemma 2.22. For any two STSs (Z,δ,Σ) and (W,ρ,Ψ), (δ×ρ)δω⊆δδω×ρδω.
Proof. Let T∈(δ×ρ)δω and (e,f)(z,w)˜∈T. Then, by Theorem 20 of [36] we find S∈SωO(Z×W,δ×ρ,Σ×Ψ) such that (e,f)(z,w)˜∈S=Intδ×ρ(Cl(δ×ρ)ω(S))˜⊆T. Choose L∈δ and M∈ρ such that (e,f)(z,w)˜∈L×M˜⊆S˜⊆T. By Proposition 3 (b) of [33] we have Clδω(L)×Clρω(M)˜⊆Cl(δ×ρ)ω(L×M), and so
L×M˜⊆Intδ(Clδω(L))×Intρ(Clρω(M))˜⊆Intδ×ρ(Clδω(L)×Clρω(M))˜⊆Intδ×ρ(Cl(δ×ρ)ω(L×M))˜⊆T. |
By Theorem 9 and Corollary 7 of [36], Intδ(Clδω(L))∈δδω and Intρ(Clρω(M))∈ρδω. It follows that T∈δδω×ρδω.
The following result shows that soft ω-almost-regularity is a productive soft property:
Theorem 2.23. The soft product of two soft ω-A-R STSs is soft ω-A-R.
Proof. Let (Z,δ,Σ) and (W,ρ,Ψ) be two soft ω-A-R STSs. Let (e,f)(z,w)˜∈SP(Z×W,Σ×Ψ) and let K∈SωO(Z×W,δ×ρ,Σ×Ψ) such that (e,f)(z,w)˜∈K. Then, by Corollary 7 of [36] G∈(δ×ρ)δω. So, by Lemma 2.22 K∈δδω×ρδω. Thus, there are L∈δδω and M∈ρδω such that (e,f)(z,w)˜∈L×M˜⊆K. By Corollary 7 of [36] we find S∈SωO(Z,δ,Σ) and T∈SωO(W,ρ,Ψ) such that (e,f)(z,w)˜∈S×T˜⊆L×M˜⊆G. So, by Theorem 2.2 (2) there are M∈δ and N∈ρ such that ez˜∈M˜⊆Clδ(M)˜⊆S and fw˜∈N˜⊆Clρ(N)˜⊆T. Therefore, we have M×N∈δ×ρ and (e,f)(z,w)˜∈M×N˜⊆Clδ×ρ(M×N)=Clδ(M) ×Clρ(N)˜⊆S×T˜⊆L×M˜⊆K. Again, by Theorem 2.2 (2) (Z×W,δ×ρ,Σ×Ψ) is soft ω-A-R.
The following result shows that soft almost-regularity is a productive soft property:
Theorem 2.24. Let (Z,δ,Σ) and (W,ρ,Ψ) be two STSs. Then (Z×W,δ×ρ,Σ×Ψ) is soft A-R iff (Z,δ,Σ) and (W,ρ,Ψ) are both soft A-R.
Proof. Necessity. Let (Z×W,δ×ρ,Σ×Ψ) be soft A-R. To see that (Z,δ,Σ) is soft A-R, let ez∈SP(Z,Σ) and G∈SO(Z,δ,Σ) such that ez˜∈G. Choose fw˜∈SP(W,F). Then, (e,f)(z,w)˜∈∈G×1Ψ∈SO(Z×W,δ×ρ,Σ×Ψ). Thus, by Theorem 3.4 (ⅱ) of [31] we find H∈δ×ρ such that (e,f)(z,w)˜∈H˜⊆Clδ×ρ(H)˜⊆G×1Ψ. Choose M∈δ and N∈ρ such that (e,f)(z,w)˜∈M×N˜⊆H. Thus,
(e,f)(z,w)˜∈M×N˜⊆Clδ(M)×Clρ(N)=Clδ×ρ(M×N)˜⊆Clδ×ρ(H)˜⊆G×1Ψ. |
Therefore, we have ez˜∈M˜⊆Clδ(M)˜⊆G. Hence, by Theorem 3.4 (ⅱ) of [31] (Z,δ,Σ) is soft A-R. Similarly, we can show that (W,ρ,Ψ) is soft A-R.
Sufficiency. Let (Z,δ,Σ) and (W,ρ,Ψ) be soft A-R. Let (e,f)(z,w)˜∈SP(Z×W,Σ×Ψ) and let K∈SO(Z×W,δ×ρ,Σ×Ψ) such that (e,f)(z,w)˜∈K. Choose M∈δ and N∈ρ such that (e,f)(z,w)˜∈M×N˜⊆K. Since ez˜∈M and fw˜∈N, by Theorem 3.4 (ⅳ) of [31] there are S∈SO(Z,δ,Σ) and T∈SO(W,ρ,Ψ) such that ez˜∈S˜⊆Clδ(S)˜⊆Intδ(Clδ(M)) and fw˜∈T˜⊆Clρ(T)˜⊆Intρ(Clρ(N)). Thus, we have S×T∈SO(Z×W,δ×ρ,Σ×Ψ) and
(e,f)(z,w)˜∈S×T˜⊆Clδ(S)×Clρ(T)=Clδ×ρ(S×T)˜⊆Intδ(Clδ(M))×Intρ(Clρ(N))=Intδ×ρ(Clδ×ρ(M×N))˜⊆Intφ×ρ(Clδ×ρ(M×N))=K. |
Therefore, by Theorem 3.4 (ⅳ) of [31] (Z×W,δ×ρ,Σ×Ψ) is soft A-R.
In this section, we define soft ω-semi-regularity and soft ω- T212 as two new soft separation axioms. We show that soft ω-semi-regularity is a weaker form of both soft semi-regularity and soft ω-regularity, and soft ω-T212 lies strictly between soft T212 and soft T2. Also, we provide several sufficient conditions establishing the equivalence between these newly introduced axioms and their relevant counterparts. Moreover, a decomposition theorem for soft regularity through the interplay of soft ω-semi-regularity and soft ω-almost-regularity is obtained. In addition, we investigated the links between these classes of soft topological spaces and their analogs in general topology.
Definition 3.1. An STS (H,φ,Σ) is called soft ω-semi-regular (soft ω-S-R, for simplicity) if RωO(H,φ,Σ) forms a soft base for φ.
Two characterizations of soft ω-semi-regularity are listed in the following theorem.
Theorem 3.2. For any STS (H,φ,Σ), T.F.A.E:
(a) (H,φ,Σ) is soft ω-S-R.
(b) For every H∈φ−{0Σ} and every rz˜∈H, we find K∈φ such that rz˜∈K˜⊆Intφ(Clφω(K))˜⊆H.
(c) φδω=φ.
Proof. (a) ⟶ (b): Let H∈φ−{0Σ} and let rz˜∈H. By (a) we find K∈SωO(H,φ,Σ) such that rz˜∈K=Intφ(Clφω(K))˜⊆H.
(b) ⟶ (c): By Theorem 21 of [36] we have φδω˜⊆φ. To show that φ⊆φδω, let H∈φ−{0Σ}, then for every rz˜∈H we find Krz∈φ such that rz˜∈Krz˜⊆Intφ(Clφω(Krz))˜⊆H. Let K=˜∪rz˜∈HIntφ(Clφω(Krz)). Since for every rz˜∈H Intφ(Clφω(Krz))∈SωO(H,φ,Σ)⊆φδω, then K∈φδω.
(c) ⟶ (a): Since RωO(H,φ,Σ) is a soft base for φδω, and by (c) φδω=φ, then RωO(H,φ,Σ) is a soft base for φ. Therefore, (H,φ,Σ) is soft ω-S-R.
Corollary 3.3. Every soft ω-regular STS is soft ω -S-R.
Proof. The proof follows from Theorem 25 of [36] and Theorem 3.2.
Theorem 3.4. Every soft S-R STS is soft ω-S-R.
Proof. Let (H,φ,Σ) be soft S-R. Then φδ=φ. So, by Theorem 21 of [36] φ=φδ⊆φδω⊆φ, and thus φδω=φ. Therefore, by Theorem 3.2, (H,φ,Σ) is soft ω-S-R.
Theorem 3.5. Every soft ω-S-R soft anti-L-C STS is soft S-R.
Proof. Let (H,φ,Σ) be soft ω-S-R soft anti-L-C. Since (H,φ,Σ) is soft ω-S-R, then RωO(H,φ,Σ) is a soft base for φ. Since (H,φ,Σ) is soft anti-L-C, then by Theorem 6 of [36], RO(H,φ,Σ)=RωO(H,φ,Σ). So, RO(H,φ,Σ) is a soft base for φ. Hence, (H,φ,Σ) is soft S-R.
The following implications come from the previous theorems; nevertheless, Examples 3.15 and 3.16 show that the converses of these implications are not true.
soft S-R⟶soft ω-S-R↑soft ω-regular. |
Theorem 3.6. Soft L-C STSs are soft ω-S-R.
Proof. Let (D,φ,Σ) be soft L-C. Then, by Theorem 5 of [36] RωO(D,φ,Σ)=φ. So, RωO(D,φ,Σ) is a soft base for φ. Hence, (D,φ,Σ) is soft ω-S-R.
Theorem 3.7. Let (D,φ,Σ) be an STS. If (D,φω,Σ) is soft ω-S-R, then (D,φω,Σ) is soft S-R.
Proof. Let (D,φω,Σ) be soft ω-S-R. Then, RωO(D,φω,Σ) is a soft base for φω. Since by Theorem 7 of [36] RωO(D,φω,Σ)=RO(D,φω,Σ), then RO(D,φω,Σ) is a soft base for φω. Hence, (D,φω,Σ) is soft S-R.
In Theorems 3.8, 3.9, 3.11, and Corollary 3.12, we discuss the connections between soft semi-regularity and its analog in traditional topological spaces. Also, in Theorems 3.10, 3.13, and Corollary 3.14, we discuss the connections between soft ω-semi-regularity and its analog in traditional topological spaces.
Theorem 3.8. If (H,φ,Σ) is full and soft S-R, then (H,φr) is S-R for all r∈Σ.
Proof. Let (H,φ,Σ) be full and soft S-R. Let r∈Σ. Let z∈H and let W∈φr such that z∈W. Choose K∈φ such that K(r)=W. Since (H,φ,Σ) is soft S-R and rz˜∈K∈φ, we find L∈SO(H,φ,Σ) such that rz˜∈L˜⊆K and so z∈L(r)⊆K(r)=W. In contrast, by Theorem 13 of [36], L(r)∈SO(H,φr). This shows that (H,φr) is S-R for all r∈Σ.
Theorem 3.9. Let (D,L) be a TS. Then, for any set Σ, (D,C(L),Σ) is soft S-R iff (D,L) is S-R.
Proof. Necessity. Let (D,C(L),Σ) be soft S-R. Pick r∈Σ. Since it is clear that (D,C(L),Σ) is full, then by Theorem 3.8 (D,(C(L))r)=(D,L) is S-R.
Sufficiency. Let (D,L) be S-R. Let rz∈SP(D,Σ) and let CU∈C(L) such that rz˜∈CU. Then, we have z∈U∈L. So, we find V∈SO(D,L) such that z∈IntL(ClL(V))⊆U. Thus, we have CV∈C(L) and rz˜∈CIntL(ClL(V))=IntC(L)(ClC(L)(CV))˜⊆CU. This shows that (D,C(L),Σ) is soft S-R.
Theorem 3.10. Let (D,L) be a TS. Then, for any set Σ, (D,C(L),Σ) is soft ω-S-R iff (D,L) is ω-S-R.
Proof. Necessity. Let (D,C(L),Σ) be soft ω-S-R. Let z∈D and U∈L such that z∈U. Pick r∈Σ. Then, we have rz˜∈CU∈C(L). Since (D,C(L),Σ) is soft ω-S-R, we find CV∈SωO(D,C(L),Σ) such that rz˜∈CV˜⊆CU. Therefore, we have z∈V∈SωO(D,L) and V⊆U. This shows that (D,L) is ω-S-R.
Sufficiency. Let (D,L) be ω-S-R. Let rz∈SP(H,Σ) and let CU∈C(L) such that rz˜∈CU. Then, we have z∈U∈L. So, we find V∈SωO(D,L) such that z∈V⊆U. Thus, we have RωO(D,C(L),Σ) and rz˜∈CV˜⊆CU. This shows that (D,C(L),Σ) is soft ω-S-R.
Theorem 3.11. Let {(H,Lr):r∈Σ} be a collection of TSs. Then (H,⊕r∈ΣLr,Σ) is soft S-R iff (H,Lr) is S-R for every r∈Σ.
Proof. Necessity. Let (H,⊕r∈ΣLr,Σ) be soft S-R and let r∈Σ. Let z∈H and let U∈Lr such that z∈U. Then, rz˜∈rU∈⊕r∈ΣLr. So, we find L∈SO(H,⊕r∈ΣLr,Σ) such that rz˜∈L˜⊆rU and thus, z∈L(r)⊆(rU)(r)=U. In contrast, by Theorem 14 of [36] L(r)∈SO(H,Lr). This shows that (H,Lr) is S-R.
Sufficiency. Let (H,Lr) be S-R for every r∈Σ. Let rz∈SP(H,Σ) and let K∈⊕r∈ΣLr such that rz˜∈K. Then, we have z∈K(r)∈Lr and so we find V∈SO(H,Lr) such that z∈V⊆U. Now, we have rz˜∈rV˜⊆rU˜⊆K, and by Theorem 14 of [36] rV∈SO(H,⊕r∈ΣLr,Σ). This shows that (H,⊕r∈ΣLr,Σ) is soft S-R.
Corollary 3.12. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft S-R iff (D,L) is S-R.
Proof. For each r∈Σ, set Lr=L. Then, τ(L)=⊕r∈ΣLr and by Theorem 3.11 we get the result.
Theorem 3.13. Let {(H,Lr):r∈Σ} be a collection of TSs. Then, (H,⊕r∈ΣLr,Σ) is soft ω-S-R iff (H,Lr) is ω-S-R for every r∈Σ.
Proof. Necessity. Let (H,⊕r∈ΣLr,Σ) be soft ω-S-R and let r∈Σ. Let z∈H and let U∈Lr such that z∈U. Then, rz˜∈rU∈⊕r∈ΣLr. So, we find L∈SωO(H,⊕r∈ΣLr,Σ) such that rz˜∈L˜⊆rU, and thus z∈L(r)⊆(rU)(r)=U. In contrast, by Theorem 15 of [36] L(r)∈SωO(H,Lr). This shows that (H,Lr) is ω-S-R.
Sufficiency. Let (H,Lr) be ω-S-R for every r∈Σ. Let rz∈SP(H,Σ) and let K∈⊕r∈ΣLr such that rz˜∈K. Then, we have z∈K(r)∈Lr and so we find V∈SωO(H,Lr) such that z∈V⊆U. Now, we have rz˜∈rV˜⊆rU˜⊆K, and by Theorem 15 of [36] rV∈SωO(H,⊕r∈ΣLr,Σ). This shows that (H,⊕r∈ΣLr,Σ) is soft ω-S-R.
Corollary 3.14. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft ω-S-R iff (D,L) is ω-S-R.
Proof. For each r∈Σ, set Lr=L. Then, τ(L)=⊕r∈ΣLr and by Theorem 3.13 we get the result.
The following two examples show, respectively, that each of Theorem 3.4 and Corollary 3.3 does not have to be true in all cases:
Example 3.15. Consider (H,φ,Σ) in Example 2.17. RO(H,φ,Σ)={0Σ,1Σ} is not a soft base for φ and thus (H,φ,Σ) is not soft S-R. In contrast, by Theorem 3.6 (H,φ,Σ) is soft ω-S-R.
Example 3.16. Let (D,L) be as in Example 3.9 of [39]. It is proved in [39] that (D,L) is ω-S-R but not ω-regular. Therefore, by Corollaries 3.14 and 19 of [33], (D,τ(L),Σ) is soft ω-S-R but not soft ω-regular.
The following main result introduces a decomposition of soft regularity in terms of soft ω-semi-regularity and soft ω-almost-regularity:
Theorem 3.17. An STS (H,φ,Σ) is soft regular iff it is soft ω-S-R and soft ω-A-R.
Proof. Necessity. Let (H,φ,Σ) be soft regular. Then, by Theorem 15 of [33] and Corollary 3.3 (H,φ,Σ) is soft ω-S-R. In contrast, by Theorem 2.11 (H,φ,Σ) is soft ω-A-R.
Sufficiency. Let (H,φ,Σ) be soft ω -S-R and soft ω-A-R. Let H∈φ−{0Σ} and let rz˜∈H. Since (H,φ,Σ) is soft ω-S-R, then there is G∈SωO(H,φ,Σ) such that rz˜∈G˜⊆H. Since (H,φ,Σ) is soft ω-A-R, then by Theorem 2.2 (2) there is T∈φ such that rz˜∈T˜⊆Clφ(T)˜⊆G˜⊆H. Hence, (H,φ,Σ) is soft regular.
Definition 3.18. An STS (H,φ,Σ) is called soft ω-T212 if for every rx,sy∈SP(H,Σ) such that rx≠sy, we find K,G∈φ such that rx˜∈K, sy˜∈G, and Clφω(K)˜∩Clφω(G)=0Σ.
In Theorems 3.19, 3.21, and Corollary 3.12, we discuss the connections between soft T212 spaces and their analogs in traditional topological spaces. Also, in Theorems 3.20, 3.23, and Corollary 3.24, we discuss the connections between soft ω-T212 spaces and their analogs in traditional topological spaces.
Theorem 3.19. If (H,φ,Σ) is soft T212, then (H,φr) is T212 for every r∈Σ.
Proof. Suppose that (H,φ,Σ) is soft T212 and let r∈Σ. Let x,y∈Z such that x≠y. Then, rx,ry∈SP(S,D) such that rx≠ry. Since (H,φ,Σ) is soft T212, we find K,G∈φ such that rx˜∈K, ry˜∈G, and Clφ(K)˜∩Clφ(G)=0Σ. Thus, we have x∈K(r)∈φr, y∈G(r)∈φr, and by Proposition 7 of [12] Clφr(K(r))∩Clφr(G(r))⊆(Clφ(K))(r)∩(Clφ(G))(r) =(Clφ(K)˜∩Clφ(G))(r)=∅. This shows that (H,φr) is T212.
Theorem 3.20. If (H,φ,Σ) is soft ω-T212, then (H,φr) is ω-T212 for every r∈Σ.
Proof. Suppose that (H,φ,Σ) is soft ω-T212 and let r∈Σ. Let x,y∈Z such that x≠y. Then, rx,ry∈SP(S,D) such that rx≠ry. Since (H,φ,Σ) is soft ω-T212, we find K,G∈φ such that rx˜∈K, ry˜∈G, and Clφω(K)˜∩Clφω(G)=0Σ. Thus, we have x∈K(r)∈φr, y∈G(r)∈φr, and by Proposition 7 of [12] Cl(φω)r(K(r))∩Cl(φω)r(G(r))⊆(Clφω(K))(r) ∩(Clφω(G))(r) =(Clφω(K)˜∩Clφω(G))(r)=∅. But by Theorem 7 of [35], (φω)r=(φr)ω. This shows that (H,φr) is ω-T212.
Theorem 3.21. Let {(H,LR):r∈Σ} be a collection of TSs. Then, (H,⊕r∈ΣLr,Σ) is soft T212 iff (H,Lr) is T212 for every r∈Σ.
Proof. Necessity. Suppose that (H,⊕r∈ΣLr,Σ) is soft T212 and let r∈Σ. Then, by Theorem 3.19 (H,(⊕r∈ΣLr)r) is T212. On the other hand, by Theorem 3.7 of [34] (⊕r∈ΣLr)r=Lr.
Sufficiency. Suppose that (H,Lr) is T212 for every r∈Σ. Let rx,sy∈SP(H,Σ) such that rx≠sy.
Case 1. r≠s. Then, rx˜∈rZ∈⊕r∈ΣLr, sy˜∈sZ∈⊕r∈ΣLr, and Cl⊕r∈ΣLr(rZ)˜∩Cl⊕r∈ΣLr(sZ)=0Σ.
Case 2. r=s. Then, x≠y. Since (H,Lr) is T212, we find U,V∈Lr such that x∈U, y∈V, and ClLr(U)∩ClLr(V)=∅. Then, we have rx˜∈rU∈⊕r∈ΣLr, sy˜∈sV∈⊕r∈ΣLr and Cl⊕r∈ΣLr(rU)∩Cl⊕r∈ΣLr(sV)=0Σ.
Corollary 3.22. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft T212 iff (D,L) is T212.
Proof. For each r∈Σ, put Lr=L. Then, τ(L)=⊕r∈ΣLr. We get the result as a consequence of Theorem 3.21.
Theorem 3.23. Let {(H,Lr):r∈Σ} be a collection of TSs. Then, (H,⊕r∈ΣLr,Σ) is soft ω-T212 iff (H,Lr) is ω-T212 for every r∈Σ.
Proof. Necessity. Suppose that (H,⊕r∈ΣLr,Σ) is soft ω-T212 and let r∈Σ. Then, by Theorem 3.20 (H,(⊕r∈ΣLr)r) is ω-T212. In contrast, by Theorem 3.7 of [35], (⊕r∈ΣLr)r=Lr.
Sufficiency. Suppose that (H,Lr) is ω-T212 for every r∈Σ. Let rx,sy∈SP(H,Σ) such that rx≠sy.
Case 1. r≠s. Then, rx˜∈rZ∈⊕r∈ΣLr, sy˜∈sZ∈⊕r∈ΣLr, and Cl(⊕r∈ΣLr)ω(rZ)˜∩Cl(⊕r∈ΣLr)ω(sZ)=0Σ.
Case 2. r=s. Then, x≠y. Since (D,L) is ω-T212, we find A,B∈Lr such that x∈A, y∈B, and A∩B=∅. Then, we have rx˜∈rA∈⊕r∈ΣLr, sy˜∈sB∈⊕r∈ΣLr and Cl(⊕r∈ΣLr)ω(rA)˜∩Cl(⊕r∈ΣLr)ω(sB)=0Σ.
Corollary 3.24. Let (D,L) be a TS. Then, for any set Σ, (D,τ(L),Σ) is soft ω-T212 iff (D,L) is ω-T212.
Proof. For each r∈Σ, put Lr=L. Then, τ(L)=⊕r∈ΣLr. The result follows from Theorem 3.23.
Theorem 3.25. If (H,φ,Σ) is soft T212 , then (H,φ,Σ) is soft ω-T212.
Proof. Let (H,φ,Σ) be soft T212 and let rx,sy∈SP(H,Σ) such that rx≠sy. Then, we find K,G∈φ such that rx˜∈K, sy˜∈G, and Clφ(K)˜∩Clφ(G)=0Σ. Since Clφω(K)˜∩Clφω(G)˜⊆Clφ(K)˜∩Clφ(G)=0Σ, then Clφω(K)˜∩Clφω(G)=0Σ. This shows that (H,φ,Σ) is soft ω-T212.
Theorem 3.26. If (H,φ,Σ) is soft anti-L-C and soft ω-T212, then (H,φ,Σ) is soft T212.
Proof. Let (H,φ,Σ) be soft anti-L-C and soft ω-T212. Let rx,sy∈SP(H,Σ) such that rx≠sy. Since (H,φ,Σ) is soft ω-T212, then we find K,G∈φ such that rx˜∈K, sy˜∈G, and Clφω(K)˜∩Clφω(G)=0Σ. Since (H,φ,Σ) is anti-L-C, then by Theorem 14 of [35] Clφ(K)˜∩Clφ(G)=Clφω(K)˜∩Clφω(G)=0Σ. Hence, (H,φ,Σ) is soft T212.
Theorem 3.27. Every soft ω-T212 STS is soft T2.
Proof. Let (H,φ,Σ) be ω-T212 and let rx,sy∈SP(H,Σ) such that rx≠sy. Then, we find K,G∈φ such that rx˜∈K, sy˜∈G, and Clφω(K)˜∩Clφω(G)=0Σ. Since K˜∩G˜⊆Clφω(K)˜∩Clφω(G)=0Σ, then K˜∩G=0Σ. Hence, (H,φ,Σ) is soft T2.
Theorem 3.28. If (H,φ,Σ) is soft L-C and soft T2, then (H,φ,Σ) is soft ω-T212.
Proof. Let (H,φ,Σ) be soft L-C and soft T2. Let rx,sy∈SP(H,Σ) such that rx≠sy. Since (H,φ,Σ) is soft T2, then we find K,G∈φ such that rx˜∈K, sy˜∈G, and K˜∩G=0Σ. Since (H,φ,Σ) is soft L-C, then by Corollary 5 of [35] Clφω(K)˜∩Clφω(G)=K˜∩G=0Σ. Hence, (H,φ,Σ) is soft ω-T212.
The following example demonstrates that Theorem 3.25's converse does not have to be true in general:
Example 3.29. Let (D,L) be the TS in Example 75 of [41]. Then (D,L) is T2 but not T212. Since (D,τ(L),N) is soft L-C, by Corollary 5 of [35] it is soft T2. Thus, by Theorem 3.28 (D,τ(L),N) is soft ω-T212. On the other hand, by Corollary 2.22 (D,τ(L),N) is not soft T212.
The following example demonstrates why Theorem 3.27 does not have to be true in general:
Example 3.30. Let (D,L) be the TS in Example 81 of [41]. It is known that (D,L) is T2 but not T212. Then, by Corollary 7 of [33] and Corollary 2.22 (D,τ(L),[0,1]) is soft T2 but not soft T212. Since (D,τ(L),[0,1]) is soft anti-L-C, then by Theorem 3.26 (D,τ(L),[0,1]) is not ω-T212.
Theorem 3.31. Every soft ω-regular T2 STS is soft ω-T212.
Proof. Let (H,φ,Σ) be soft ω-regular and soft T2. Let rx,sy∈SP(H,Σ) such that rx≠sy. Since (H,φ,Σ) is soft T2, then we find K,G∈φ such that rx˜∈K, sy˜∈G, and K˜∩G=0Σ. Since (H,φ,Σ) is soft ω-regular, then we find L,M∈φ such that rx˜∈L˜⊆Clφω(L)˜⊆K and sy˜∈M˜⊆Clφω(M)˜⊆G. Therefore, we have rx˜∈L, sy˜∈M, and Clφω(L)˜∩Clφω(M)˜⊆K˜∩G=0Σ. This proves that (H,φ,Σ) is soft ω-T212.
Question 3.32. Is it true that every soft ω-T212 STS is soft ω-regular?
Theorem 3.33. If (Z,β,Σ) and (W,ρ,Ψ) are two soft ω-S-R STSs such that the soft product (Z×W,β×ρ,Σ×Ψ) is soft ω-S-R, then both of (Z,β,Σ) and (W,ρ,Ψ) are soft ω-S-R.
Proof. Since (Z×W,β×ρ,Σ×Ψ) is soft ω-S-R, then by Theorem 3.2 (β×ρ)δω=β×ρ. So, by Lemma 2.22, β×ρ˜⊆ βδω×ρδω. Hence, β = βδω and ρ=ρδω. Therefore, again by Theorem 3.2 (Z,β,Σ) and (W,ρ,Ψ) are soft ω-S-R.
Soft separation axioms are a collection of requirements for categorizing a system of STSs based on certain soft topological features. These axioms are often expressed in terms of classes of soft sets.
In this work, "soft ω-almost-regular", "soft ω -semi-regular", and "soft ω-T212" are defined as three new notions of soft separation axioms (Definitions 2.1, 3.1, 3.18). Several characterizations of soft ω-almost-regularity (Theorems 2.2) and soft ω-semi-regularity (Theorem 3.2) are given. It is proved that soft ω-almost-regularity lies strictly between regularity and almost-regularity (Theorems 2.11, 2.12 and Examples 2.16, 2.17); soft ω-semi-regularity is a weaker form of both soft semi-regularity and soft ω-regularity (Corollary 3.3, Theorem 3.4 and Examples 3.15, 3.16); soft ω-T212 lies strictly between soft T212 and soft T2 (Theorems 3.25, 3.27 and Examples 3.29, 3.30). Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given (Theorems 2.13, 2.14, 3.5, 3.6, 3.26, 3.28). A decomposition theorem of soft regularity by means of soft ω-semi-regularity and soft ω -almost-regularity is given (Theorem 3.17). It is shown that soft ω -almost-regularity is heritable for specific kinds of soft subspaces (Theorems 2.19, 2.21). Soft product theorems regarding soft almost regular spaces (Theorem 2.23), soft ω-almost regular spaces (Theorem 2.24), and soft ω-semi-regular spaces (Theorem 3.33). Finally, the article delves into the connections between the newly proposed as well as some known soft axioms and their counterparts in traditional topological spaces, facilitating a bridging of concepts between the soft and classical realms (Theorems 2.3–2.7, 3.8–3.11, 3.13, 3.19–3.21, 3.23, and Corollaries 2.8, 2.10, 3.12, 3.14, 3.22, 3.24).
In the next work, we intend to: 1) Define and investigate soft ω -almost-normality; 2) investigate the behavior of these new soft separation ideas under various kinds of soft mappings; and 3) find an application for our new two conceptions in the "decision-making problem", "information systems", or "expert systems".
The authors declare that they have not used Artificial Intelligence tools in the creation of this article.
The authors declare that they have no conflicts of interest.
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