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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 160-170

Published online June 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.2.160

© The Korean Institute of Intelligent Systems

Soft ω-Weak Continuity between Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to :
Samer Al Ghour (algore@just.edu.jo)

Received: June 17, 2023; Revised: December 3, 2023; Accepted: May 20, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

We define soft ω-weak continuity as a new soft continuity notion, which is strictly weaker than the soft ω-continuity and soft weak continuity. We present two characterizations and two composition theorems for the soft ω-weak continuity. Moreover, via soft ω-weak continuity, we give several preservation theorems related to soft connectedness and soft separation axioms. Additionally, we introduce w*-ω-continuous functions as a new class of soft functions strictly containing a class of soft ω-continuous functions. We show that soft ω-weak continuity and soft w*-ω-continuities are independent notions, and use them to obtain a decomposition theorem for the soft ω-continuity. Finally, we study the relationships between our new soft notions and their analogs in general topology.

Keywords: ω-weak continuity, Soft ω-continuity, Soft connectedness, Soft Urysohn, Generated soft topology

The soft sets concept was first proposed by the Russian mathematician Molodtsov [1] in 1999. Techniques for simulating mathematical issues, including uncertainty soft sets, are a more effective approach when there are no incomplete data because owing to nonrestrictive criteria for classifying objects in the present theory, thereby allowing researchers to select the type of characteristics required. Other theories, such as the fuzzy set theory [2], rough set theory [3], intuitionistic fuzzy sets [4], and vague set theory [5] can be as considered mathematical techniques for addressing uncertainty. However, each has its own set of challenges. Soft set theory, according to Molodtsov [1], can be applied in various areas. Maji et al. [6] investigated the (detailed) theoretical framework of soft set theory. They created various operators, specifically soft set theory. Since Maji’s contribution [6], other mathematical models, such as the soft group theory and soft ring theory have been examined in soft contexts. Shabir and Naz [7] began researching soft topological spaces as a generalization of topological space in 2011. Several soft topological notions, including soft separation axioms, soft-covering axioms, soft connectedness, and weak and strong soft continuities have been introduced and investigated in recent years [821]. Soft topologies and applications are currently an established field of study [2224]. A major goal of this study is to demonstrate how the definition of ω-weak continuity between topological spaces can be modified to define the soft ω-weak continuity between soft topological spaces. In this study, the soft ω-weak continuity is defined as a new concept of soft continuity, which is weaker than the soft ω-continuity and soft weak continuity. We present two characterizations of soft ω-weak continuity as two composition theorems. Furthermore, we provide various preservation theorems linked to the soft connectedness and several soft separation axioms through soft ω-weak continuity. Furthermore, we present w*-ω-continuous functions as a new class of soft functions containing a class of soft ω-continuous functions. We demonstrate that soft ω-weak continuity and soft w*-ω-continuity are separate concepts and use them to prove a soft ω-continuity decomposition theorem. Finally, we investigate the connections between our new soft notions and their topological analogs.

This study adheres to the notions and terminology provided in [11, 25, 26]. Soft topological space (STS) and topological space (TS) are utilized in this study.

Let M be a set of parameters and X be the initial universe. A function H : M → ℘ (X) is known as a soft set. The collection of all soft sets over X relative to M is denoted as SS(X,M). Let HSS (X,M). H is referred to as a null (resp. absolute) soft set over X relative to M and is denoted by 0M (resp. 1M) if H (m) = ∅︀ (resp. H (m) = X) for each mM. H is referred to as a soft point over X relative to M and is denoted by ax if there are xX and aM such that H (a) = {x} and H (m) = ∅︀ for each mM − {a}. The collection of all soft points over X relative to M is denoted as SP(X,M). If for some aM and YX, we have H (a) = Y and K (m) = ∅︀ for all mM − {a}, then H is denoted by aY . If for some H (m) = Y for each mM, then H is denoted as CY . A soft point axSP (X,M) is said to belong to H (notation ax∈̃H) if xH (a). Let A,BSS (X,M). Then A is a soft subset of B, denoted by A⊆̃B if A(m) ⊆ B(m) each mM. The soft union (resp. intersection, which is the difference) between A and B is denoted by A∪̃B (resp. A∩̃B, AB) and defined as (A∪̃B)(m) = A(m) ∪ B(m) (resp. (A∩̃B)(m) = A(m) ∩ B(m), (AB) (m) = A(m) − B (m)) for each mM.

Let (X, δ) and (X,M) be a TS and an STS, respectively. Let WX and KSS(X,M). In this paper, Intδ(W), Clδ(W), IntΩ(K), and ClΩ(K), respectively, will be used to refer to the interior of W in (X, δ), closure of W in (X, δ), soft interior of K in (X,M), and soft closure of K in (X,M). Additionally, δc , Ωc, CO (X, δ), and CO(X,M) represent the family of closed sets on (X, δ), family of soft closed sets on (X,M), family of clopen sets on (X, δ), and family of soft clopen set as (X,M).

The following definitions are used:

Definition 1.1

Let (A, δ), (B, β, ) be TSs. A soft function g : (A, δ) → (B, β) is said to be

(a) weak-continuous (w-c) at aA if for every Tβ such that g(a) ∈ T, there exists Sδ such that aS and g(S) ⊆ Clβ(T). If g is w-c for each aA, g is called w-c [27].

(b) ω-weak continuous (ω-w-c) at aA if, for every Tβ such that g(a) ∈ T, we find that Sδω such that aS and g(S) ⊆ Clβ(T). If g is ω-w-c at each aA, and g is called ω-w-c [28].

Definition 1.2

Let (A,M) and (B, ϒ,N) be STSs. A soft function fpu : (A,M) → (B, ϒ,N) is considered to be

(a) soft weak continuous (soft w-c) at maSP(A,M) if for every K ∈ ϒ such that fpu(ma)∈̃K, there exists H ∈ Ω such that ma∈̃H and fpu(H)⊆̃Clϒ (K) [29]. If fpu is soft w-c at each maSP(A,M), then fpu is said to be soft w-c.

(b) soft ω-continuous (soft ω-c) at maSP(A,M) if for every K ∈ ϒ such that fpu(ma)∈̃K, there exists H ∈ Ωω such that ma∈̃H and fpu(H)⊆̃K [15]. If fpu is soft ω-c at each maSP(A,M), then fpu is said to be soft ω-c.

Definition 1.3

An STS (A,M) is called

(a) soft connected if CO (A,M) = {0M, 1M} [30].

(b) soft disconnected if not soft-connected [30].

(c) soft Hausdorff space if, for every ma, nbSP(A,M) such manb, there exist S, T ∈ Ω such that ma∈̃S, nb∈̃T, and S∩̃T = 0M [31].

(d) soft regular if for each maSP(A,M) and each G ∈ Ω such that ma∈̃G, there exists L ∈ Ω such that

ma∈̃L⊆̃ClΩ (L) ⊆̃G [32].

(e) soft Urysohn space if, for every ma, nbSP(A,M) such that manb, there exist S, T ∈ Ω such that ma∈̃S, nb∈̃T, and ClΩ (S) ∩̃ClΩ (T) = 0M [33].

Definition 2.1

Let fpu : (X,M) → (Y, ϒ,N) be a soft function. Then

(a) fpu is called the soft ω-weak continuous (soft ω-w-c) at a soft point mxSP (X,M) if for each G ∈ ϒ such that fpu (mx) ∈̃G, there exists H ∈ Ωω such that mx∈̃H and fpu(H)⊆̃Clϒ (G);

(b) fpu is called soft ω-w-c if it is soft ω-w-c at each soft point, mxSP (X,M).

Theorem 2.2

For a soft function fpu : (X,M) → (Y, ϒ,N), the following are equivalent:

(a) fpu is soft ω-w-c;

(b) for each G ∈ ϒ, ClΩω(fpu-1(G))˜fpu-1(Clϒ(G));

(c) for every G ∈ ϒ, fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))).

Proof. (a) ⇒ (b) Let G ∈ ϒ. Suppose to the contrary that there exists mx˜ClΩω(fpu-1(G))-fpu-1(Clϒ(G)). Since fpu(mx)∉̃Clϒ(G), we find L ∈ ϒ such that fpu(mx) ∈ L and L∩̃G = 0N. From (a), there exists K ∈ Ωω such that mx∈̃K and fpu(K)⊆̃Clϒ (L). Since mx˜ClΩω(fpu-1(G)) and mx∈̃K ∈ Ωω, K˜fpu-1(G)0M. Choose az∈̃K such that fpu(az)∈̃G. Since az∈̃K and fpu(K)⊆̃Clϒ (L), fpu(az) ∈̃Clϒ (L). Since fpu(az)∈̃G ∈ ϒ and fpu(az)∈̃Clϒ (L), G∩̃L ≠ 0N, a contradiction.

(b) ⇒(c). Let G ∈ ϒ, then 1NClϒ(G) ∈ ϒ. So, by (b), ClΩω(fpu-1(1N-Clϒ(G)))˜fpu-1(Clϒ(1N-Clϒ(G))); thus,

1M-fpu-1(Clϒ(1N-Clϒ(G)))˜1M-ClΩω(fpu-1(1N-Clϒ(G))).

We have

1M-fpu-1(Clϒ(1N-Clϒ(G)))=fpu-1(1N-Clϒ(1N-Clϒ(G)))=fpu-1(Intϒ(Clϒ(G))).

Since G⊆̃Intϒ(Clϒ(G)), fpu-1(G)˜fpu-1(Intϒ(Clϒ(G)). Thus,

fpu-1(G)˜1M-fpu-1(Clϒ(1N-Clϒ(G)))˜1M-ClΩω(fpu-1(1N-Clϒ(G)))=1M-ClΩω((1M-fpu-1(Clϒ(G))))=IntΩω(fpu-1(Clϒ(G))).

(c) ⇒ (a). Let mxSP (X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. By (c), fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))). Let L=IntΩω(fpu-1(Clϒ(G))). Then fpu(mx)∈̃L∈Ωω and

fpu(L)=fpu(IntΩω(fpu-1(Clϒ(G))))˜fpu((fpu-1(Clϒ(G))))˜Clϒ(G).

This completes the proof.

Theorem 2.3

Let {(Xm) : mM} and {(Y, ϒn) : nN} be two families of TSs. Let p : XY be a function and u : MN be a bijective function. Then

fpu:(X,mMΩm,M)(Y,nNϒn,N)is soft ω-w-c iff p:(X,Ωm)(Y,ϒu(m))is ω-w-c for all mM.

Proof. Necessity. Suppose that fpu : (X,⊕mMΩm,M) → (Y,⊕nNϒn,N) is soft ω-w-c. Let sM. Let xX and V ∈ ϒu(m) such that p (x) ∈ V . Then fpu (sx) = (u(s))p(x) ∈̃ (u(s))V ∈ ⊕nNϒn. Therefore, there exists H ∈ (⊕mMΩm)ω such that mx∈̃H and fpu(H) ⊆̃Clϒ ((u(s))V ) = (u(s))Clϒu(s)(V). According to Theorem 8 in [26], we have (⊕mMΩm)ω = ⊕mMm)ω; thus, H(s) ∈ (Ωm)ω. Since

fpu(H)˜(u(s))Clϒu(s)(V),

(fpu(H)) (u(s)) ⊆ ((u(s))Clϒu(s)(V))(u(s)) = Clϒu(s) (V ). Since u is injective, (fpu(H)) (u (s)) = p (H(s)). This shows that p : (Xs) → (Y, ϒu(s)) is ω-w-c.

Sufficiency. Suppose that p : (Xm) → (Y, ϒu(m)) is ω-w-c for all mM. Let mxSP (X,M) and let G ∈ ϒ such that fpu (mx) = (u (m))p(x) ∈̃G. Then we have p (x) ∈ G(u (m)) ∈ ϒu(m). Since p : (Xm) → (Y, ϒu(m)) is ω-w-c, there exists U ∈ Ωω such that xU and p (U) ⊆ Clϒu(m) (G(u (m))). By Lemma 4.9 of [21], we have (ClnNϒn (G)) (u (m)) = Clϒu(m) (G(u (m))). Since xU, mx∈̃mU. Since U ∈ Ωω, mU ∈ ⊕mMm)ω and by Theorem 8 of [26], mU ∈ (⊕mMΩm)ω. Since p (U) ⊆ Clϒu(m) (G(u (m))),

(fpu(mU))(u(m))=((u(m))p(U))(u(m))=p(U)Clϒu(m)(G(u(m)))=(ClnNϒn(G))(u(m)).

If nNu (m), then (fpu (mU)) (n) = ((u (m))p(U))(n) = ∅︀ ⊆ (ClnNϒn (G)) (n). Therefore, fpu (mU) ⊆̃ClnNϒn (G). It follows that fpu : (X,⊕mMΩm,M) → (Y , ⊕nNϒn, N) is soft ω-w-c.

Corollary 2.4

Let p : (X, δ) → (Y, β) be a function between two TSs and let u : MN be a bijective function. Then p : (X, δ) → (Y, β) is ω-w-c if and only if fpu : (X, τ (δ) ,M) → (Y, τ (β) ,N) is soft ω-w-c.

Proof. For each mM and nN, let Ωm = δ and ϒn = β. Then τ (δ) = ⊕mMΩm and τ (β) = ⊕nNϒn. Thus, according to Theorem 2.3, we obtain the following result.

Theorem 2.5

Each soft ω-c function is a soft ω-w-c function.

Proof. The proof is straightforward.

Theorem 2.5’s converse is not always true, as demonstrated by the following illustration:

Example 2.6

Let δ be the typical topology for ℝ, β = {∅︀,ℝ,ℚ}, and M = ℤ. Define p : (ℝ, δ) → (ℝ, β) and u : MM by

p(x)={2,if x,π,if x-,

and u(m) = m for each mM. Then

p is ω-w-c: Let x ∈ ℝ and Vβ such that p (x) ∈ V . Then V = ℝ or V = ℚ and so Clβ (V ) = ℝ. Choose U = ℝ. Then xUδδω and p (U) = {2, π} ⊆ ℝ = Clβ (V ).

p is not ω-c: Since ℚ ∈ β while p−1 (ℚ) = ℚ ∉ δω.

Thus, from Corollaries 2.5 and 2.6 in [15], fpu : (X, τ (δ), M) → (Y, τ (β) ,M) is soft ω-w-c but not soft ω-c.

Theorem 2.7

If fpu : (X,M) → (Y, ϒ,N) is soft ω-w-c such that (Y, ϒ,N) is soft regular, then fpu is soft ω-c.

Proof. Let mxSP(X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. Since (Y, ϒ,N) is soft regular, there exists K ∈ ϒ such that fpu(mx)∈̃K⊆̃Clϒ (K) ⊆̃G. Since fpu is soft ω-w-c, we find H ∈ Ωω such that mx∈̃H and fpu(H)⊆̃Clϒ (K) ⊆̃G.

It follows that fpu is soft ω-c.

Theorem 2.8

Each soft w-c function is soft ω-w-c.

Proof. Let fpu : (X,M) → (Y, ϒ,N) be soft w-c. Let mxSP(X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. Then we find H ∈ Ω such that mx∈̃H and fpu(H)⊆̃Clϒ (G). Since Ω ⊆ Ωω, H ∈ Ωω. This shows that fpu is soft ω-w-c.

The following example shows that the converse of Theorem 2.8 need not be true in general:

Example 2.9

Let X = {1, 2, 3}, δ = {∅︀, X, {1}, {3}, {1, 3}}, β = {∅︀, X, {1}, {2}, {1, 2}}, and M = {a, b}. Consider the identity functions p : XX and u : MM. Then

p is ω-c, and from Theorem 2.5, it is soft ω-w-c, since δω is the discrete topology on X.

p is not w-c: Suppose that p is w-c, then since p(2) = 2 ∈ {2} ∈ β, there exists Wδ such that 2 ∈ W and p(W) = W = XClβ({2}) = {2, 3} which is impossible. Therefore, from Corollaries 2.4 and 3.4 in [11], fpu : (X, τ (δ) ,M) → (Y, τ (β) ,M) is soft ω-w-c but not soft w-c.

For any function g : ZW, the function w : ZZ × W defined by w (z) = (z, g (z)) is denoted g#. Hereinafter, the soft product topology of the two STSs (X,M) and (Y, ϒ,N) is denoted as pr (Ω × ϒ).

Theorem 2.10

The soft function fpu : (X,M) → (Y, ϒ,N) is soft w-c if and only if

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft w-c.

Proof. Necessity. Suppose fpu : (X,M) → (Y, ϒ,N) is soft w-c. Let mxSP(X,M) and let Gpr (Ω × ϒ) such that (fp#u# (mx)) = (u#)(m))(p#)(x) = (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃K × H⊆̃G. Since fpu is soft w-c and fpu (mx) = (u(m))p(x) ∈̃H ∈ ϒ, there exists L ∈ Ω such that mx∈̃L and fpu (L) ⊆̃Clϒ(H). Therefore,

fp#u#(L)˜K×Clϒ(H)˜Clpr(Ω×ϒ)(K×H)=Clpr(Ω×ϒ)(G).

This indicates that fp#u# is a soft w-c.

Sufficiency. Suppose that

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft w-c. Let mxSP(X,M) and H ∈ ϒ such that (fpu (mx)) = (u(m))p(x) ∈̃H. Then

(fp#u#(mx))=((u#)(m))(p#)(x)=(m,u(m))(x,p(x))=mx×(u(m))p(x)˜1M×Hpr(Ω×ϒ).

Since fp#u# is soft w-c, there exists T ∈ Ω such that mx∈̃T and fp#u# (T) ⊆̃Clpr(Ω× ϒ)(1M×H) = 1M×Clϒ(H); hence fpu (T) ⊆̃Clϒ(H). This shows that fpu is soft w-c.

Theorem 2.11

The soft function fpu : (X, Ω, M) → (Y, ϒ, N) is soft ω-w-c if and only if

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft ω-w-c.

Proof. Necessity. Suppose fpu : (X,M) → (Y, ϒ,N) is soft ω-w-c. Let mxSP(X,M) and Gpr (Ω × ϒ) such that (fp#u# (mx)) = (u#)(m))(p#)(x) = (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃K × H⊆̃G. Since fpu is soft ω-w-c and fpu (mx) = (u(m))p(x) ∈̃H ∈ ϒ, there exists L ∈ Ωω such that mx∈̃L and fpu (L) ⊆̃Clϒ(H). Therefore,

fp#u#(L)˜K×Clϒ(H)˜Clpr(Ω×ϒ)(K×H)=Clpr(Ω×ϒ)(G).

This indicates that fp#u# is soft ω-w-c.

Sufficiency. Suppose that

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft ω-w-c. Let mxSP(X,M) and H ∈ ϒ such that (fpu (mx)) = (u(m))p(x) ∈̃H. Then

(fp#u#(mx))=((u#)(m))(p#)(x)=(m,u(m))(x,p(x))=mx×(u(m))p(x)˜1M×Hpr(Ω×ϒ).

Since fp#u# is soft ω-w-c, there exists T ∈ Ωω such that mx∈̃T and fp#u# (T) ⊆̃Clpr(Ω×ϒ)(1M×H) = 1M×Clϒ(H); hence fpu (T) ⊆̃Clϒ(H). This shows that fpu is soft ω-w-c.

Definition 2.12

Let (X,M) be an STS. Then

(a) (X,M) is called soft ω-connected if CO (Xω,M) = {0M, 1M}.

(b) (X,M) is called soft ω-disconnected if it is not soft ω-connected.

Theorem 2.13

If (X,M) is a soft locally countable STS where SP (X,M) contains at least two elements, then (Xω,M) is soft ω-disconnected.

Proof. Let (X,M) be soft locally countable where SP(X, M) contains at least two elements. Then, from Corollary 5 in [26], (Xω,M) is a discrete STS. Since SP (X,M) contains at least two elements, SP (X,M) ∩ {0M, 1M} = ∅︀. Since SP (X,M) ⊆ Ωω, CO (Xω,M) ≠ {0M, 1M}. Hence, (Xω,M) is soft ω-disconnected.

Theorem 2.14

Each soft ω-connected STS is soft connected.

Proof. Let (X,M) be soft ω-connected. Then CO (Xω,M) = {0M, 1M}. Since Ω ⊆ Ωω, CO(X,M) ⊆ CO (Xω,M) = {0M, 1M}.

Therefore, CO(X,M) = {0M, 1M}; hence (X,M) is soft connected.

In general, soft connectedness does not imply soft ω-connectedness.

Example 2.15

Consider (ℕ,Ω,ℤ) where Ω = {0, 1}. Then CO (ℕ,Ω,ℤ) ⊆ Ω = {0, 1}; thus, CO (ℕ,Ω,ℤ) = {0, 1}. Thus, (ℕ,Ω,ℤ) is soft connected. Since (ℕ,Ω,ℤ) is soft locally countable where SP (X,M) contains more than one element, according to Theorem 2.13, (ℕ,Ω,ℤ) is soft ω-disconnected.

Theorem 2.16

Every soft anti-locally countable soft-connected STS is soft ω-connected.

Proof. Let (X,M) be soft anti-locally countable and soft connected. Suppose the contrary that (X,M) is soft ω-disconnected. Then there is GCO (Xω,M)−{0M, 1M}. Then we have ClΩω (G) = G and IntΩω (1MG) = 1MG. Since G ∈ Ωω and (1MG) ∈ (Ωω)c and (X,M) is soft anti-locally countable, according to Theorem 14 of [26], ClΩ (G) = ClΩω (G) = G and IntΩ(1MG) = IntΩω (1MG) = 1MG. Therefore, we have GCO(X, Ω, M) − {0M, 1M}; hence (X,M) is soft disconnected, which is contradictory.

Lemma 2.17

Let {(Xm) : mM} be a family of TSs. Then GCO (X,⊕mMΩm,M) if and only if G(s) ∈ CO (Xs) for all sM.

Proof. Necessity. Let GCO (X,⊕mMΩm,M) and let sM. Since GCO (X,⊕mMΩm,M), G, 1MG ∈ ⊕mMΩm; thus, G(s) ∈ Ωs and (1MG) (s) = XG(s) ∈ Ωs. Hence, G(s) ∈ CO (Xs).

Sufficiency. Let G(s) ∈ CO (Xs) for all sM. Then G(s) ∈ Ωs and XG(s) = (1MG) (s) ∈ Ωs for all sM. Hence, G, 1MG ∈ ⊕mMΩm. Therefore, GCO (X,⊕mMΩm,M).

Theorem 2.18

If {(Xm) : mM} is a family of TSs, where M contains at least two points, then (X,⊕mMΩm,M) is soft disconnected.

Proof. Choose sM. Consider the soft set GSS(X,M) defined by G(s) = ∅︀ and G(t) = X for all tM–{s}. Since for all mM, G(s) ∈ {∅︀,X} ⊆ CO(Xm), by Lemma 2.17, GCO(X,⊕mMΩm,M). Since M contains at least two points, G ∉ {0M, 1M}. Therefore, (X,⊕mMΩm, M) is soft disconnected.

Corollary 2.19

If (X, δ) is a TS and M is any set of parameters which contains at least two points, then (X, τ (δ),M) is soft ω-disconnected.

Proof. For each mM, put Ωm = δ. Then τ (δ) = ⊕mMΩm. Thus, according to Theorem 2.18 we obtain the result.

Corollary 2.20

If {(Xm) : mM} is a family of TSs, where M contains at least two points, then (X,⊕mMΩm, M) is soft ω-disconnected.

Proof. This follows from Theorems 2.14 and 2.18.

Corollary 2.21

If (X, δ) is a TS, then M is any set of parameters that contains at least two points, then (X, τ (δ),M) is soft ω-disconnected.

Proof. For each mM, put Ωm = δ. Then τ (δ) = ⊕mMΩm. Thus, by Corollary 2.20, we obtain the following results.

Theorem 2.22

If (X,M) is soft ω-connected and fpu : (X,M) → (Y,N) is a soft ω-w-c surjection, then (Y,N) is soft connected.

Proof. Suppose, in contrast to, (Y,N) is soft disconnected. Then there exists GCO(Y,N) – {0N, 1N}. Since G ∈ ϒc, Clϒ(G) = G. Since fpu : (X,M) → (Y,N) is soft ω-w-c and G ∈ ϒ, according to Theorem 2.2,

ClΩω(fpu-1(G))˜fpu-1(Clϒ(G))=fpu-1(G),andfpu-1(G)˜IntΩω(fpu-1(Clϒ(G)))=IntΩω(fpu-1(G)).

Thus, ClΩω(fpu-1(G))=fpu-1(G) and fpu-1(G)=IntΩω(fpu-1(G)).

Thus, fpu-1(G)CO(X,Ω,M). Since G ≠ 0N and fpu is surjective, fpu-1(G)0M. If fpu-1(G)=1M, then fpu(fpu-1(G))=fpu(1M)=1N˜G and so G = 1N. Thus, fpu-1(G)1M. This result indicates that (X,M) is soft ω-disconnected, which is a contradiction.

Theorem 2.23

If (Y,N) is a soft Urysohn STS and fpu : (X,M) → (Y,N) is a soft w-c injection, then (X,M) is soft Hausdorff.

Proof. Let ax, bySP (X,M) such that asbt. Since fpu is injective, fpu (ax) ≠ fpu (by). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃;G,

fpu(bt)˜H,and Clϒ(G)˜Clϒ(H)=0N.

Since fpu is soft w-c, according to Theorem 5.2 in [29],

fpu-1(G)˜IntΩ(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩ(fpu-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜IntΩ(fpu-1(Clϒ(G)))Ω,bt˜fpu-1(H)˜IntΩ(fpu-1(Clϒ(H)))Ω,andIntΩ(fpu-1(Clϒ(G)))˜IntΩ(fpu-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G))˜fpu-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G)˜Clϒ(H)))=IntΩ(fpu-1(0N))=0M.

It follows that (X,M) is soft Hausdorff.

The following result shows that the two soft w-c functions from an STS into a soft Urysohn TS agreed upon in a soft closed set:

Theorem 2.24

If fpu, fqv : (X,M) → (Y,N) are two soft w-c functions where (Y,N) is soft Urysohn, then ∪̃{mxSP(X,M) : fpu (mx) = fqv (mx)} ∈ Ωc.

Proof. Put K=∪̃{mxSP(X,M) : fpu(mx)=fqv(mx)}.

We demonstrate that 1MK ∈ Ω. Let as ∈̃ 1MK. Then fpu (as) ≠ fqv (as). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃ G, fqv (as) ∈̃ H, and Clϒ (G) ∩̃Clϒ (H) = 0N. Since fpu and fqv are soft w-c according to Theorem 5.2 in [29],

fpu-1(G)˜IntΩ(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩ(fqv-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜fqv-1(H)˜IntΩ(fpu-1(Clϒ(G)))˜IntΩ(fqv-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H)))Ω.

Claim

(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K=0M which ends the proof.

Proof of Claim. Suppose, by contrast, that there exists

bt˜(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K.

Then fpu (bt) ∈̃Clϒ (G), fqv (bt) ∈̃Clϒ (H), and fpu (bt) = fqv (bt). So, fpu (bt) ∈̃Clϒ (G) ∪̃Clϒ (H), a contradiction.

Corollary 2.25

Let fpu, fqv : (X,M) → (Y,N) be two soft w-c functions where (Y,N) denotes soft Urysohn. If K is a soft dense set in (Xω,M) such that fpu (mx) = fqv (mx) for each mx∈̃K, then fpu = fqv.

Theorem 2.26

If (Y,N) is a soft Urysohn STS and fpu : (X,M) → (Y,N) is a soft ω-w-c injection, then (Xω,M) is soft Hausdorff.

Proof. Let ax, bySP (X,M) such that asbt. Since fpu is injective, fpu (ax) ≠ fpu (by). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃G, fpu (bt) ∈̃H, and Clϒ (G) ∪̃Clϒ (H) = 0N.

Since fpu is soft ω-w-c, according to Theorem 2.2 (c),

fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩω(fpu-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜IntΩω(fpu-1(Clϒ(G)))Ωω,bt˜fpu-1(H)˜IntΩω(fpu-1(Clϒ(H)))Ωω,andIntΩω(fpu-1(Clϒ(G)))˜IntΩω(fpu-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G))˜fpu-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G)˜Clϒ(H)))=IntΩω(fpu-1(0N))=0M.

It follows that (Xω,M) is soft Hausdorff.

The following result shows that the two soft ω-w-c functions from an STS into a soft Urysohn TS agreed upon a soft ω-closed set:

Theorem 2.27

If fpu, fqv : (X,M) → (Y,N) are two soft ω-w-c functions where (Y,N) is soft Urysohn, then ∪̃{mxSP(X,M) : fpu (mx) = fqv (mx)} ∈ (Ωω)c.

Proof. We set K = ∪̃{mxSP(X,M) : fpu(mx) = fqv(mx)}. We show that 1MK ∈ Ωω. Let as∈̃1MK. Then fpu (as) ≠ fqv (as). Since (Y,N) is a soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃G, fqv (as) ∈̃H, and Clϒ (G) ∪̃Clϒ (H) = 0N.

Since fpu and fqv are soft ω-w-c, according to Theorem 2.2 (c),

fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))),andfqv-1(H)˜IntΩω(fqv-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜fqv-1(H)˜IntΩω(fpu-1(Clϒ(G)))˜IntΩω(fqv-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H)))Ωω.

Claim

(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K=0M which ends the proof.

Proof of Claim. Suppose, by contrast, that there exists bt˜(IntΩω(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K. Then fpu (bt) ∈̃Clϒ (G), fqv (bt) ∈̃Clϒ (H), and fpu (bt) = fqv (bt). So, fpu (bt) ∈̃Clϒ (G) ∪̃Clϒ (H), a contradiction.

Corollary 2.28

Let fpu, fqv : (X,M) → (Y,N) be two soft ω-w-c functions, where (Y,N) denotes a soft Urysohn. If K is a soft dense set in (Xω,M) such that fpu (mx) = fqv (mx) for each mx∈̃K, then fpu = fqv.

Theorem 2.29

If fp1u1 : (X,M) → (Y,N) is soft ω-w-c and fp2u2 : (Y,N) → (Z,R) is soft continuous, then f(p2p1)(u2u1) : (X,M) → (Z,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since fp2u2 : (Y,N) → (Z,R) is soft continuous, fp2u2-1(G)ϒ and Clϒ(fp2u2-1(G)))˜fp2u2-1(ClΠ(G)).

Since fp1u1 is soft ω-w-c, according to Theorem 2.2 (c),

f(p2p1)(u2u1)-1(G)=fp1u1-1(fp2u2-1(G))˜IntΩω(fp1u1-1(Clϒ(fp2u2-1(G))))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G))))=IntΩω(f(p2p1)(u2u1)-1(ClΠ(G)))).

Therefore, according to Theorem 2.2 (c), f(p2p1)(u2u1) is soft ω-w-c.

Theorem 2.30

Let fp1u1 : (X,M) → (Y,N) be a soft function such that fp1u1 : (Xω,M) → (Yω,N) is soft continuous and let fp2u2 : (Y,N) → (Z,R) be soft ω-w-c. Then f(p2p1)(u2u1) : (X,M) → (Z,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since fp2u2 : (Y,N) → (Z,R) is soft ω-w-c, from Theorem 2.2 (c), fp2u2-1(G)˜Intϒω(fp1u1-1(ClΠ(G)) and so fp1u1-1(fp2u2-1(G))˜fp1u1-1(Intϒω(fp2u2-1(ClΠ(G))). Since fp1u1 : (Xω,M) → (Yω,N) is soft continuous,

fp1u1-1(Intϒω(fp2u2-1(ClΠ(G)))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G)))).

Thus,

f(p2p1)(u2u1)-1(G)=fp1u1-1(fp2u2-1(G))˜fp1u1-1(Intϒω(fp2u2-1(ClΠ(G)))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G))))=IntΩω(f(p2p1)(u2u1)-1(ClΠ(G)))).

Therefore, from Theorem 2.2 (c), f(p2p1)(u2u1) is soft ω-w-c.

Definition 3.1

The soft function fpu : (X,M) → (Y,N) is called soft w*-ω-continuous (w*-ω-c) if fpu-1(Bdϒ(G))(Ωω)c for every G ∈ ϒ.

Lemma 3.2

Let {(Xm) : mM} be a family of TSs. Then for every HSP (X,M), (BdmMΩm(H)) (s) = BdΩm(H(s)) for all sM.

Proof. If sM, then from Lemma 4.9 in [21],

(BdmMΩm(H))(s)=((ClmMΩm(H))˜(ClmMΩm(1M-H)))(s)=(ClmMΩm(H))(s)(ClmMΩm(1M-H))(s)=ClΩs(H(s))ClΩs((1M-H)(s))=ClΩs(H(s))ClΩs((X-H(s)))=BdΩm(H(s)).

Theorem 3.3

Let {(Xm) : mM} and {(Yn) : nN} be two families of TSs. Let p : XY be a function and u : MN be a bijective function. Then fpu : (X,⊕mMΩm, M) → (Y,⊕nNϒn, N) is soft w*-ω-c if and only if p : (Xm) → (Yu(m)) is w*-ω-c for each mM.

Proof. Necessity. Suppose that fpu : (X, ⊕mMΩm, M) → (Y, ⊕nNϒn, N) is soft w*-ω-c. Let sM. Let V ∈ ϒu(s). Subsequently, (u(s))V ∈ ⊕nNϒn and so, fpu-1(BdnNϒn((u(s))V))((mMΩm)ω)c. Since by Theorem 8 in [26], (⊕mMΩm)ω = ⊕mMm)ω, we obtain

fpu-1(BdnNϒn((u(s))V))(mM(Ωm)ω)c.

Thus, (fpu-1(BdnNϒn((u(s))V)))(s)((Ωs)ω)c.

From Lemma 3.2, we can conclude that

BdnNϒn((u(s))V))=(u(s))Bdϒs(V).

Additionally, because u : MN is injective,

fpu-1((u(s))Bdϒs(V))=sp-1(.Bdϒs(V)).

Therefore,

(fpu-1(BdnNϒn((u(s))V)))(s)=(sp-1(Bdϒs(V)))   (s)=p-1(Bdϒs(V))((Ωs)ω)c.

This implies that p : (Xs) → (.Yu(s)) is w*-ω-c.

Sufficiency. Suppose that p : (Xm) → (Yu(m)) is w*-ω-c for each mM. Let G ∈ ϒ. Since u : MN is bijective, p : (Xu−1(n))→ (Yn) for each nN and so p−1(Bdϒn(G(n))) ∈ ((Ωu−1(n))ω)c. Thus, p−1(Bdϒm(G(u (m)))) ∈ ((Ωm)ω)c for all mM. Now, by Lemma 3.2, Bdϒm(G(u (m)) = (BdmMϒm(G)) (u (m)) for all mM.

Thus, for each mM,

(fpu-1(BdnNϒn(G)))   (m)=p-1((BdmMϒm(G))(u(m)))=p-1(Bdϒm(G(u(m)))((Ωm)ω)c.

Therefore, fpu-1(BdnNϒn(G))((mMΩm)ω)c.

This shows that

fpu:(X,mMΩm,M)(Y,nNϒn,N)

is soft w*-ω-c.

Corollary 3.4

Let p : (X, δ) → (Y, β) be a function between two TSs and let u : MN be a bijective function. Then p : (X, δ) → (Y, β) is w*-ω-c if and only if fpu : (X, τ (δ),M) →, (Y, τ (β),N) is soft w*-ω-c.

Proof. For each mM and nN, Ωm = δ and ϒn = β. Then τ (δ) = ⊕mMΩm and τ (β) = ⊕nNϒn. Thus, from Theorem 3.3, we obtain the following result:

The following two examples demonstrate the independence of the soft ω-weak continuity, and soft w*-ωcontinuity.

Example 3.5

Let δ and β be co-countable, and discrete topologies for ℝ, respectively. Let M = {a, b, c}. Consider the identity functions p : (ℝ, δ) → (ℝ, β) and u : MM. Then p is w*-ω-c: Let Wβ. Then Bdβ (W) = ∅︀ and so p−1 (Bdβ (W)) = ∅︀ ∈ (δω)c.

p is not ω-w-c: Suppose that p is ω-w-c. Since 2 ∈ p ({2}) = {2}, there exists Vδω such that 2 ∈ V and p(V) = VClβ({2}) = {2}. Thus, V = {2} ∈ δω = δ which is impossible. Therefore, according to Corollaries 2.4 and 3.4, fpu : (Y, τ (ℑ),E) → (Y, τ (ℵ),E) is soft w*-ω-c but not soft ω-w-c.

Example 3.6

LetX = ℝ, Y = {a, b}, δ be the co-countable topology on X, β = {∅︀, Y, {a}}, and E = [0, 1]. Define p : (ℝ, δ) → (ℝ, β) and u : MM, as follows:

p(x)={a,if xb,if x-,

and u (m) = m for all mM. Then

p is ω-w-c: Let xX and Wβ such that p (x) ∈ W. Then W = {a} or W = Y. In both cases, Clβ(W) = Y. Choose U = ℝ. Then Uδ = δω and p (U) = YY = Clβ(W).

p is not w*-ω-c, since {a} ∈ β whereas p−1 (Bdβ ({a})) = p−1 ({b}) = ℝ – ℤ ∉ δc = (δω)c.

Therefore, from Corollaries 2.4 and 3.4, fpu : (X, τ (δ),E) → (Y, τ (β),E) is soft ω-w-c but not soft w*-ω-c.

Theorem 3.7

Each soft ω-c function is soft w*-ω-c.

Proof. Let fpu : (X,M) → (Y,N) be soft ω-c and let G ∈ ϒ. Since fpu is soft ω-c and Bdϒ(G) ∈ ϒc, fpu-1(Bdϒ(K))(δω)c. This shows that fpu is soft w*-ω-c.

The opposite of Theorem 3.7 need not be true, as demonstrated in Example 3.5 and Theorem 2.5.

The following main result represents the decomposition of soft ω-continuity:

Theorem 3.8

The soft function fpu : (X, Ω, M) → (Y, ϒ, N) is soft ω-c if and only if both soft ω-w-c and soft w*-ω-c.

Proof. Necessity. The proof follows from Theorems 2.5 and 3.7.

Sufficiency. We assume that fpu : (X, Ω, M) → (Y, ϒ, N) is both soft ω-w-c and soft w*-ω-c. Let mxSP(X,M) and G ∈ ϒ such that fpu(mx)∈̃G. Since fpu is soft ω-w-c, there exists Lδω such that mx∈̃L and fpu(L)⊆̃Clϒ(G). Since fpu(mx)∈̃G and Bdϒ(G) = Clϒ(G) – G. fpu(mx)∉̃ Bdϒ(G). Thus, mx˜fpu-1(Bdϒ(G)). Since fpu is soft w*-ω-c,

fpu-1(Bdϒ(G))(Ωω)c.

Therefore, we have mx˜L-fpu-1(Bdϒ(G))Ωω.

Claim

fpu(L-fpu-1(Bdϒ(G)))˜G which ends the proof.

Proof of Claim. Suppose the contrary that there exists az˜fpu(L-fpu-1(Bdϒ(G)))-G. Choose br˜L-fpu-1(Bdϒ(G)) such that az = fpu(br). Since br∈̃L and fpu(L)⊆̃Clϒ(G), az = fpu(br)∈̃Clϒ(G) and so br˜fpu-1(Clϒ(G)). Since br˜fpu-1(Bdϒ(G))=fpu-1(Clϒ(G)-G)=fpu-1(Clϒ(G))-fpu-1(G), and br˜fpu-1(Clϒ(G)),br˜fpu-1(G).

However, since fpu(br)=az˜G,br˜fpu-1(G), which is contradictory.

We define soft ω-weak continuity (Definition 2.1) and soft w*-ω-continuity (Definition 3.1) as two new notions of soft functions. We proved that they are independent notions (Examples 3.5 and 3.6). Additionally, we prove that soft ω-weak continuity is strictly weaker than both the soft ω-continuity (Theorem 2.5 and Example 2.6) and soft weak continuity (Theorem 2.8 and Example 2.9), and soft w*-ω-continuity is strictly weaker than soft ω-continuity (Example 3.5 and Theorem 3.7). Moreover, we obtain a decomposition theorem for soft ω-continuity in terms of soft ω-weak continuity and soft w*-ω-continuity (Theorem 3.8). Regarding soft ω-weak continuity, we present two characterizations (Theorem 2.2); and examined soft graph (Theorems 2.10 and 2.11), soft preservation (Theorems 2.22, 2.23, and 2.26), and soft composition (Theorems 2.29 and 2.30) theorems. We prove that the two soft w-c (resp. ω-w-c) functions from an STS into a soft Urysohn TS agreed upon a soft closed set (Theorem 2.24) (resp. soft ω-closed set (Theorem 2.27). Finally, we examined the correspondence between each soft and weak continuity, soft ω–weak continuity, and soft w*ω-continuity and topological analogs (Theorems 2.3 and 3.3 and Corollaries 3.3 and 3.4).

In future work, we plan to (1) define and study weaker forms of ω-continuity; (2) define and study strong forms of soft continuity, such as super continuity; and (3) introduce new classes of soft open sets.

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Samer Al Ghour received the Ph.D. degree in mathematics from the University of Jordan, Amman, Jordan, in 1999. He is currently professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan. His research interests include general topology, fuzzy topology, and soft-set theory. He is included in the world’s top 2% of scientists in 2023 (Stanford University and Elsevier). E-mail: algore@just.edu.jo

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(2): 160-170

Published online June 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.2.160

Copyright © The Korean Institute of Intelligent Systems.

Soft ω-Weak Continuity between Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to:Samer Al Ghour (algore@just.edu.jo)

Received: June 17, 2023; Revised: December 3, 2023; Accepted: May 20, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define soft ω-weak continuity as a new soft continuity notion, which is strictly weaker than the soft ω-continuity and soft weak continuity. We present two characterizations and two composition theorems for the soft ω-weak continuity. Moreover, via soft ω-weak continuity, we give several preservation theorems related to soft connectedness and soft separation axioms. Additionally, we introduce w*-ω-continuous functions as a new class of soft functions strictly containing a class of soft ω-continuous functions. We show that soft ω-weak continuity and soft w*-ω-continuities are independent notions, and use them to obtain a decomposition theorem for the soft ω-continuity. Finally, we study the relationships between our new soft notions and their analogs in general topology.

Keywords: &omega,-weak continuity, Soft &omega,-continuity, Soft connectedness, Soft Urysohn, Generated soft topology

1. Introduction and Preliminaries

The soft sets concept was first proposed by the Russian mathematician Molodtsov [1] in 1999. Techniques for simulating mathematical issues, including uncertainty soft sets, are a more effective approach when there are no incomplete data because owing to nonrestrictive criteria for classifying objects in the present theory, thereby allowing researchers to select the type of characteristics required. Other theories, such as the fuzzy set theory [2], rough set theory [3], intuitionistic fuzzy sets [4], and vague set theory [5] can be as considered mathematical techniques for addressing uncertainty. However, each has its own set of challenges. Soft set theory, according to Molodtsov [1], can be applied in various areas. Maji et al. [6] investigated the (detailed) theoretical framework of soft set theory. They created various operators, specifically soft set theory. Since Maji’s contribution [6], other mathematical models, such as the soft group theory and soft ring theory have been examined in soft contexts. Shabir and Naz [7] began researching soft topological spaces as a generalization of topological space in 2011. Several soft topological notions, including soft separation axioms, soft-covering axioms, soft connectedness, and weak and strong soft continuities have been introduced and investigated in recent years [821]. Soft topologies and applications are currently an established field of study [2224]. A major goal of this study is to demonstrate how the definition of ω-weak continuity between topological spaces can be modified to define the soft ω-weak continuity between soft topological spaces. In this study, the soft ω-weak continuity is defined as a new concept of soft continuity, which is weaker than the soft ω-continuity and soft weak continuity. We present two characterizations of soft ω-weak continuity as two composition theorems. Furthermore, we provide various preservation theorems linked to the soft connectedness and several soft separation axioms through soft ω-weak continuity. Furthermore, we present w*-ω-continuous functions as a new class of soft functions containing a class of soft ω-continuous functions. We demonstrate that soft ω-weak continuity and soft w*-ω-continuity are separate concepts and use them to prove a soft ω-continuity decomposition theorem. Finally, we investigate the connections between our new soft notions and their topological analogs.

This study adheres to the notions and terminology provided in [11, 25, 26]. Soft topological space (STS) and topological space (TS) are utilized in this study.

Let M be a set of parameters and X be the initial universe. A function H : M → ℘ (X) is known as a soft set. The collection of all soft sets over X relative to M is denoted as SS(X,M). Let HSS (X,M). H is referred to as a null (resp. absolute) soft set over X relative to M and is denoted by 0M (resp. 1M) if H (m) = ∅︀ (resp. H (m) = X) for each mM. H is referred to as a soft point over X relative to M and is denoted by ax if there are xX and aM such that H (a) = {x} and H (m) = ∅︀ for each mM − {a}. The collection of all soft points over X relative to M is denoted as SP(X,M). If for some aM and YX, we have H (a) = Y and K (m) = ∅︀ for all mM − {a}, then H is denoted by aY . If for some H (m) = Y for each mM, then H is denoted as CY . A soft point axSP (X,M) is said to belong to H (notation ax∈̃H) if xH (a). Let A,BSS (X,M). Then A is a soft subset of B, denoted by A⊆̃B if A(m) ⊆ B(m) each mM. The soft union (resp. intersection, which is the difference) between A and B is denoted by A∪̃B (resp. A∩̃B, AB) and defined as (A∪̃B)(m) = A(m) ∪ B(m) (resp. (A∩̃B)(m) = A(m) ∩ B(m), (AB) (m) = A(m) − B (m)) for each mM.

Let (X, δ) and (X,M) be a TS and an STS, respectively. Let WX and KSS(X,M). In this paper, Intδ(W), Clδ(W), IntΩ(K), and ClΩ(K), respectively, will be used to refer to the interior of W in (X, δ), closure of W in (X, δ), soft interior of K in (X,M), and soft closure of K in (X,M). Additionally, δc , Ωc, CO (X, δ), and CO(X,M) represent the family of closed sets on (X, δ), family of soft closed sets on (X,M), family of clopen sets on (X, δ), and family of soft clopen set as (X,M).

The following definitions are used:

Definition 1.1

Let (A, δ), (B, β, ) be TSs. A soft function g : (A, δ) → (B, β) is said to be

(a) weak-continuous (w-c) at aA if for every Tβ such that g(a) ∈ T, there exists Sδ such that aS and g(S) ⊆ Clβ(T). If g is w-c for each aA, g is called w-c [27].

(b) ω-weak continuous (ω-w-c) at aA if, for every Tβ such that g(a) ∈ T, we find that Sδω such that aS and g(S) ⊆ Clβ(T). If g is ω-w-c at each aA, and g is called ω-w-c [28].

Definition 1.2

Let (A,M) and (B, ϒ,N) be STSs. A soft function fpu : (A,M) → (B, ϒ,N) is considered to be

(a) soft weak continuous (soft w-c) at maSP(A,M) if for every K ∈ ϒ such that fpu(ma)∈̃K, there exists H ∈ Ω such that ma∈̃H and fpu(H)⊆̃Clϒ (K) [29]. If fpu is soft w-c at each maSP(A,M), then fpu is said to be soft w-c.

(b) soft ω-continuous (soft ω-c) at maSP(A,M) if for every K ∈ ϒ such that fpu(ma)∈̃K, there exists H ∈ Ωω such that ma∈̃H and fpu(H)⊆̃K [15]. If fpu is soft ω-c at each maSP(A,M), then fpu is said to be soft ω-c.

Definition 1.3

An STS (A,M) is called

(a) soft connected if CO (A,M) = {0M, 1M} [30].

(b) soft disconnected if not soft-connected [30].

(c) soft Hausdorff space if, for every ma, nbSP(A,M) such manb, there exist S, T ∈ Ω such that ma∈̃S, nb∈̃T, and S∩̃T = 0M [31].

(d) soft regular if for each maSP(A,M) and each G ∈ Ω such that ma∈̃G, there exists L ∈ Ω such that

ma∈̃L⊆̃ClΩ (L) ⊆̃G [32].

(e) soft Urysohn space if, for every ma, nbSP(A,M) such that manb, there exist S, T ∈ Ω such that ma∈̃S, nb∈̃T, and ClΩ (S) ∩̃ClΩ (T) = 0M [33].

2. Soft ω-Weak Continuity

Definition 2.1

Let fpu : (X,M) → (Y, ϒ,N) be a soft function. Then

(a) fpu is called the soft ω-weak continuous (soft ω-w-c) at a soft point mxSP (X,M) if for each G ∈ ϒ such that fpu (mx) ∈̃G, there exists H ∈ Ωω such that mx∈̃H and fpu(H)⊆̃Clϒ (G);

(b) fpu is called soft ω-w-c if it is soft ω-w-c at each soft point, mxSP (X,M).

Theorem 2.2

For a soft function fpu : (X,M) → (Y, ϒ,N), the following are equivalent:

(a) fpu is soft ω-w-c;

(b) for each G ∈ ϒ, ClΩω(fpu-1(G))˜fpu-1(Clϒ(G));

(c) for every G ∈ ϒ, fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))).

Proof. (a) ⇒ (b) Let G ∈ ϒ. Suppose to the contrary that there exists mx˜ClΩω(fpu-1(G))-fpu-1(Clϒ(G)). Since fpu(mx)∉̃Clϒ(G), we find L ∈ ϒ such that fpu(mx) ∈ L and L∩̃G = 0N. From (a), there exists K ∈ Ωω such that mx∈̃K and fpu(K)⊆̃Clϒ (L). Since mx˜ClΩω(fpu-1(G)) and mx∈̃K ∈ Ωω, K˜fpu-1(G)0M. Choose az∈̃K such that fpu(az)∈̃G. Since az∈̃K and fpu(K)⊆̃Clϒ (L), fpu(az) ∈̃Clϒ (L). Since fpu(az)∈̃G ∈ ϒ and fpu(az)∈̃Clϒ (L), G∩̃L ≠ 0N, a contradiction.

(b) ⇒(c). Let G ∈ ϒ, then 1NClϒ(G) ∈ ϒ. So, by (b), ClΩω(fpu-1(1N-Clϒ(G)))˜fpu-1(Clϒ(1N-Clϒ(G))); thus,

1M-fpu-1(Clϒ(1N-Clϒ(G)))˜1M-ClΩω(fpu-1(1N-Clϒ(G))).

We have

1M-fpu-1(Clϒ(1N-Clϒ(G)))=fpu-1(1N-Clϒ(1N-Clϒ(G)))=fpu-1(Intϒ(Clϒ(G))).

Since G⊆̃Intϒ(Clϒ(G)), fpu-1(G)˜fpu-1(Intϒ(Clϒ(G)). Thus,

fpu-1(G)˜1M-fpu-1(Clϒ(1N-Clϒ(G)))˜1M-ClΩω(fpu-1(1N-Clϒ(G)))=1M-ClΩω((1M-fpu-1(Clϒ(G))))=IntΩω(fpu-1(Clϒ(G))).

(c) ⇒ (a). Let mxSP (X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. By (c), fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))). Let L=IntΩω(fpu-1(Clϒ(G))). Then fpu(mx)∈̃L∈Ωω and

fpu(L)=fpu(IntΩω(fpu-1(Clϒ(G))))˜fpu((fpu-1(Clϒ(G))))˜Clϒ(G).

This completes the proof.

Theorem 2.3

Let {(Xm) : mM} and {(Y, ϒn) : nN} be two families of TSs. Let p : XY be a function and u : MN be a bijective function. Then

fpu:(X,mMΩm,M)(Y,nNϒn,N)is soft ω-w-c iff p:(X,Ωm)(Y,ϒu(m))is ω-w-c for all mM.

Proof. Necessity. Suppose that fpu : (X,⊕mMΩm,M) → (Y,⊕nNϒn,N) is soft ω-w-c. Let sM. Let xX and V ∈ ϒu(m) such that p (x) ∈ V . Then fpu (sx) = (u(s))p(x) ∈̃ (u(s))V ∈ ⊕nNϒn. Therefore, there exists H ∈ (⊕mMΩm)ω such that mx∈̃H and fpu(H) ⊆̃Clϒ ((u(s))V ) = (u(s))Clϒu(s)(V). According to Theorem 8 in [26], we have (⊕mMΩm)ω = ⊕mMm)ω; thus, H(s) ∈ (Ωm)ω. Since

fpu(H)˜(u(s))Clϒu(s)(V),

(fpu(H)) (u(s)) ⊆ ((u(s))Clϒu(s)(V))(u(s)) = Clϒu(s) (V ). Since u is injective, (fpu(H)) (u (s)) = p (H(s)). This shows that p : (Xs) → (Y, ϒu(s)) is ω-w-c.

Sufficiency. Suppose that p : (Xm) → (Y, ϒu(m)) is ω-w-c for all mM. Let mxSP (X,M) and let G ∈ ϒ such that fpu (mx) = (u (m))p(x) ∈̃G. Then we have p (x) ∈ G(u (m)) ∈ ϒu(m). Since p : (Xm) → (Y, ϒu(m)) is ω-w-c, there exists U ∈ Ωω such that xU and p (U) ⊆ Clϒu(m) (G(u (m))). By Lemma 4.9 of [21], we have (ClnNϒn (G)) (u (m)) = Clϒu(m) (G(u (m))). Since xU, mx∈̃mU. Since U ∈ Ωω, mU ∈ ⊕mMm)ω and by Theorem 8 of [26], mU ∈ (⊕mMΩm)ω. Since p (U) ⊆ Clϒu(m) (G(u (m))),

(fpu(mU))(u(m))=((u(m))p(U))(u(m))=p(U)Clϒu(m)(G(u(m)))=(ClnNϒn(G))(u(m)).

If nNu (m), then (fpu (mU)) (n) = ((u (m))p(U))(n) = ∅︀ ⊆ (ClnNϒn (G)) (n). Therefore, fpu (mU) ⊆̃ClnNϒn (G). It follows that fpu : (X,⊕mMΩm,M) → (Y , ⊕nNϒn, N) is soft ω-w-c.

Corollary 2.4

Let p : (X, δ) → (Y, β) be a function between two TSs and let u : MN be a bijective function. Then p : (X, δ) → (Y, β) is ω-w-c if and only if fpu : (X, τ (δ) ,M) → (Y, τ (β) ,N) is soft ω-w-c.

Proof. For each mM and nN, let Ωm = δ and ϒn = β. Then τ (δ) = ⊕mMΩm and τ (β) = ⊕nNϒn. Thus, according to Theorem 2.3, we obtain the following result.

Theorem 2.5

Each soft ω-c function is a soft ω-w-c function.

Proof. The proof is straightforward.

Theorem 2.5’s converse is not always true, as demonstrated by the following illustration:

Example 2.6

Let δ be the typical topology for ℝ, β = {∅︀,ℝ,ℚ}, and M = ℤ. Define p : (ℝ, δ) → (ℝ, β) and u : MM by

p(x)={2,if x,π,if x-,

and u(m) = m for each mM. Then

p is ω-w-c: Let x ∈ ℝ and Vβ such that p (x) ∈ V . Then V = ℝ or V = ℚ and so Clβ (V ) = ℝ. Choose U = ℝ. Then xUδδω and p (U) = {2, π} ⊆ ℝ = Clβ (V ).

p is not ω-c: Since ℚ ∈ β while p−1 (ℚ) = ℚ ∉ δω.

Thus, from Corollaries 2.5 and 2.6 in [15], fpu : (X, τ (δ), M) → (Y, τ (β) ,M) is soft ω-w-c but not soft ω-c.

Theorem 2.7

If fpu : (X,M) → (Y, ϒ,N) is soft ω-w-c such that (Y, ϒ,N) is soft regular, then fpu is soft ω-c.

Proof. Let mxSP(X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. Since (Y, ϒ,N) is soft regular, there exists K ∈ ϒ such that fpu(mx)∈̃K⊆̃Clϒ (K) ⊆̃G. Since fpu is soft ω-w-c, we find H ∈ Ωω such that mx∈̃H and fpu(H)⊆̃Clϒ (K) ⊆̃G.

It follows that fpu is soft ω-c.

Theorem 2.8

Each soft w-c function is soft ω-w-c.

Proof. Let fpu : (X,M) → (Y, ϒ,N) be soft w-c. Let mxSP(X,M) and let G ∈ ϒ such that fpu(mx)∈̃G. Then we find H ∈ Ω such that mx∈̃H and fpu(H)⊆̃Clϒ (G). Since Ω ⊆ Ωω, H ∈ Ωω. This shows that fpu is soft ω-w-c.

The following example shows that the converse of Theorem 2.8 need not be true in general:

Example 2.9

Let X = {1, 2, 3}, δ = {∅︀, X, {1}, {3}, {1, 3}}, β = {∅︀, X, {1}, {2}, {1, 2}}, and M = {a, b}. Consider the identity functions p : XX and u : MM. Then

p is ω-c, and from Theorem 2.5, it is soft ω-w-c, since δω is the discrete topology on X.

p is not w-c: Suppose that p is w-c, then since p(2) = 2 ∈ {2} ∈ β, there exists Wδ such that 2 ∈ W and p(W) = W = XClβ({2}) = {2, 3} which is impossible. Therefore, from Corollaries 2.4 and 3.4 in [11], fpu : (X, τ (δ) ,M) → (Y, τ (β) ,M) is soft ω-w-c but not soft w-c.

For any function g : ZW, the function w : ZZ × W defined by w (z) = (z, g (z)) is denoted g#. Hereinafter, the soft product topology of the two STSs (X,M) and (Y, ϒ,N) is denoted as pr (Ω × ϒ).

Theorem 2.10

The soft function fpu : (X,M) → (Y, ϒ,N) is soft w-c if and only if

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft w-c.

Proof. Necessity. Suppose fpu : (X,M) → (Y, ϒ,N) is soft w-c. Let mxSP(X,M) and let Gpr (Ω × ϒ) such that (fp#u# (mx)) = (u#)(m))(p#)(x) = (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃K × H⊆̃G. Since fpu is soft w-c and fpu (mx) = (u(m))p(x) ∈̃H ∈ ϒ, there exists L ∈ Ω such that mx∈̃L and fpu (L) ⊆̃Clϒ(H). Therefore,

fp#u#(L)˜K×Clϒ(H)˜Clpr(Ω×ϒ)(K×H)=Clpr(Ω×ϒ)(G).

This indicates that fp#u# is a soft w-c.

Sufficiency. Suppose that

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft w-c. Let mxSP(X,M) and H ∈ ϒ such that (fpu (mx)) = (u(m))p(x) ∈̃H. Then

(fp#u#(mx))=((u#)(m))(p#)(x)=(m,u(m))(x,p(x))=mx×(u(m))p(x)˜1M×Hpr(Ω×ϒ).

Since fp#u# is soft w-c, there exists T ∈ Ω such that mx∈̃T and fp#u# (T) ⊆̃Clpr(Ω× ϒ)(1M×H) = 1M×Clϒ(H); hence fpu (T) ⊆̃Clϒ(H). This shows that fpu is soft w-c.

Theorem 2.11

The soft function fpu : (X, Ω, M) → (Y, ϒ, N) is soft ω-w-c if and only if

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft ω-w-c.

Proof. Necessity. Suppose fpu : (X,M) → (Y, ϒ,N) is soft ω-w-c. Let mxSP(X,M) and Gpr (Ω × ϒ) such that (fp#u# (mx)) = (u#)(m))(p#)(x) = (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))(x,p(x)) = mx × (u(m))p(x) ∈̃K × H⊆̃G. Since fpu is soft ω-w-c and fpu (mx) = (u(m))p(x) ∈̃H ∈ ϒ, there exists L ∈ Ωω such that mx∈̃L and fpu (L) ⊆̃Clϒ(H). Therefore,

fp#u#(L)˜K×Clϒ(H)˜Clpr(Ω×ϒ)(K×H)=Clpr(Ω×ϒ)(G).

This indicates that fp#u# is soft ω-w-c.

Sufficiency. Suppose that

fp#u#:(X,Ω,M)(X×Y,pr(Ω×ϒ),M×N)

is soft ω-w-c. Let mxSP(X,M) and H ∈ ϒ such that (fpu (mx)) = (u(m))p(x) ∈̃H. Then

(fp#u#(mx))=((u#)(m))(p#)(x)=(m,u(m))(x,p(x))=mx×(u(m))p(x)˜1M×Hpr(Ω×ϒ).

Since fp#u# is soft ω-w-c, there exists T ∈ Ωω such that mx∈̃T and fp#u# (T) ⊆̃Clpr(Ω×ϒ)(1M×H) = 1M×Clϒ(H); hence fpu (T) ⊆̃Clϒ(H). This shows that fpu is soft ω-w-c.

Definition 2.12

Let (X,M) be an STS. Then

(a) (X,M) is called soft ω-connected if CO (Xω,M) = {0M, 1M}.

(b) (X,M) is called soft ω-disconnected if it is not soft ω-connected.

Theorem 2.13

If (X,M) is a soft locally countable STS where SP (X,M) contains at least two elements, then (Xω,M) is soft ω-disconnected.

Proof. Let (X,M) be soft locally countable where SP(X, M) contains at least two elements. Then, from Corollary 5 in [26], (Xω,M) is a discrete STS. Since SP (X,M) contains at least two elements, SP (X,M) ∩ {0M, 1M} = ∅︀. Since SP (X,M) ⊆ Ωω, CO (Xω,M) ≠ {0M, 1M}. Hence, (Xω,M) is soft ω-disconnected.

Theorem 2.14

Each soft ω-connected STS is soft connected.

Proof. Let (X,M) be soft ω-connected. Then CO (Xω,M) = {0M, 1M}. Since Ω ⊆ Ωω, CO(X,M) ⊆ CO (Xω,M) = {0M, 1M}.

Therefore, CO(X,M) = {0M, 1M}; hence (X,M) is soft connected.

In general, soft connectedness does not imply soft ω-connectedness.

Example 2.15

Consider (ℕ,Ω,ℤ) where Ω = {0, 1}. Then CO (ℕ,Ω,ℤ) ⊆ Ω = {0, 1}; thus, CO (ℕ,Ω,ℤ) = {0, 1}. Thus, (ℕ,Ω,ℤ) is soft connected. Since (ℕ,Ω,ℤ) is soft locally countable where SP (X,M) contains more than one element, according to Theorem 2.13, (ℕ,Ω,ℤ) is soft ω-disconnected.

Theorem 2.16

Every soft anti-locally countable soft-connected STS is soft ω-connected.

Proof. Let (X,M) be soft anti-locally countable and soft connected. Suppose the contrary that (X,M) is soft ω-disconnected. Then there is GCO (Xω,M)−{0M, 1M}. Then we have ClΩω (G) = G and IntΩω (1MG) = 1MG. Since G ∈ Ωω and (1MG) ∈ (Ωω)c and (X,M) is soft anti-locally countable, according to Theorem 14 of [26], ClΩ (G) = ClΩω (G) = G and IntΩ(1MG) = IntΩω (1MG) = 1MG. Therefore, we have GCO(X, Ω, M) − {0M, 1M}; hence (X,M) is soft disconnected, which is contradictory.

Lemma 2.17

Let {(Xm) : mM} be a family of TSs. Then GCO (X,⊕mMΩm,M) if and only if G(s) ∈ CO (Xs) for all sM.

Proof. Necessity. Let GCO (X,⊕mMΩm,M) and let sM. Since GCO (X,⊕mMΩm,M), G, 1MG ∈ ⊕mMΩm; thus, G(s) ∈ Ωs and (1MG) (s) = XG(s) ∈ Ωs. Hence, G(s) ∈ CO (Xs).

Sufficiency. Let G(s) ∈ CO (Xs) for all sM. Then G(s) ∈ Ωs and XG(s) = (1MG) (s) ∈ Ωs for all sM. Hence, G, 1MG ∈ ⊕mMΩm. Therefore, GCO (X,⊕mMΩm,M).

Theorem 2.18

If {(Xm) : mM} is a family of TSs, where M contains at least two points, then (X,⊕mMΩm,M) is soft disconnected.

Proof. Choose sM. Consider the soft set GSS(X,M) defined by G(s) = ∅︀ and G(t) = X for all tM–{s}. Since for all mM, G(s) ∈ {∅︀,X} ⊆ CO(Xm), by Lemma 2.17, GCO(X,⊕mMΩm,M). Since M contains at least two points, G ∉ {0M, 1M}. Therefore, (X,⊕mMΩm, M) is soft disconnected.

Corollary 2.19

If (X, δ) is a TS and M is any set of parameters which contains at least two points, then (X, τ (δ),M) is soft ω-disconnected.

Proof. For each mM, put Ωm = δ. Then τ (δ) = ⊕mMΩm. Thus, according to Theorem 2.18 we obtain the result.

Corollary 2.20

If {(Xm) : mM} is a family of TSs, where M contains at least two points, then (X,⊕mMΩm, M) is soft ω-disconnected.

Proof. This follows from Theorems 2.14 and 2.18.

Corollary 2.21

If (X, δ) is a TS, then M is any set of parameters that contains at least two points, then (X, τ (δ),M) is soft ω-disconnected.

Proof. For each mM, put Ωm = δ. Then τ (δ) = ⊕mMΩm. Thus, by Corollary 2.20, we obtain the following results.

Theorem 2.22

If (X,M) is soft ω-connected and fpu : (X,M) → (Y,N) is a soft ω-w-c surjection, then (Y,N) is soft connected.

Proof. Suppose, in contrast to, (Y,N) is soft disconnected. Then there exists GCO(Y,N) – {0N, 1N}. Since G ∈ ϒc, Clϒ(G) = G. Since fpu : (X,M) → (Y,N) is soft ω-w-c and G ∈ ϒ, according to Theorem 2.2,

ClΩω(fpu-1(G))˜fpu-1(Clϒ(G))=fpu-1(G),andfpu-1(G)˜IntΩω(fpu-1(Clϒ(G)))=IntΩω(fpu-1(G)).

Thus, ClΩω(fpu-1(G))=fpu-1(G) and fpu-1(G)=IntΩω(fpu-1(G)).

Thus, fpu-1(G)CO(X,Ω,M). Since G ≠ 0N and fpu is surjective, fpu-1(G)0M. If fpu-1(G)=1M, then fpu(fpu-1(G))=fpu(1M)=1N˜G and so G = 1N. Thus, fpu-1(G)1M. This result indicates that (X,M) is soft ω-disconnected, which is a contradiction.

Theorem 2.23

If (Y,N) is a soft Urysohn STS and fpu : (X,M) → (Y,N) is a soft w-c injection, then (X,M) is soft Hausdorff.

Proof. Let ax, bySP (X,M) such that asbt. Since fpu is injective, fpu (ax) ≠ fpu (by). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃;G,

fpu(bt)˜H,and Clϒ(G)˜Clϒ(H)=0N.

Since fpu is soft w-c, according to Theorem 5.2 in [29],

fpu-1(G)˜IntΩ(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩ(fpu-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜IntΩ(fpu-1(Clϒ(G)))Ω,bt˜fpu-1(H)˜IntΩ(fpu-1(Clϒ(H)))Ω,andIntΩ(fpu-1(Clϒ(G)))˜IntΩ(fpu-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G))˜fpu-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G)˜Clϒ(H)))=IntΩ(fpu-1(0N))=0M.

It follows that (X,M) is soft Hausdorff.

The following result shows that the two soft w-c functions from an STS into a soft Urysohn TS agreed upon in a soft closed set:

Theorem 2.24

If fpu, fqv : (X,M) → (Y,N) are two soft w-c functions where (Y,N) is soft Urysohn, then ∪̃{mxSP(X,M) : fpu (mx) = fqv (mx)} ∈ Ωc.

Proof. Put K=∪̃{mxSP(X,M) : fpu(mx)=fqv(mx)}.

We demonstrate that 1MK ∈ Ω. Let as ∈̃ 1MK. Then fpu (as) ≠ fqv (as). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃ G, fqv (as) ∈̃ H, and Clϒ (G) ∩̃Clϒ (H) = 0N. Since fpu and fqv are soft w-c according to Theorem 5.2 in [29],

fpu-1(G)˜IntΩ(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩ(fqv-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜fqv-1(H)˜IntΩ(fpu-1(Clϒ(G)))˜IntΩ(fqv-1(Clϒ(H)))=IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H)))Ω.

Claim

(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K=0M which ends the proof.

Proof of Claim. Suppose, by contrast, that there exists

bt˜(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K.

Then fpu (bt) ∈̃Clϒ (G), fqv (bt) ∈̃Clϒ (H), and fpu (bt) = fqv (bt). So, fpu (bt) ∈̃Clϒ (G) ∪̃Clϒ (H), a contradiction.

Corollary 2.25

Let fpu, fqv : (X,M) → (Y,N) be two soft w-c functions where (Y,N) denotes soft Urysohn. If K is a soft dense set in (Xω,M) such that fpu (mx) = fqv (mx) for each mx∈̃K, then fpu = fqv.

Theorem 2.26

If (Y,N) is a soft Urysohn STS and fpu : (X,M) → (Y,N) is a soft ω-w-c injection, then (Xω,M) is soft Hausdorff.

Proof. Let ax, bySP (X,M) such that asbt. Since fpu is injective, fpu (ax) ≠ fpu (by). Since (Y,N) is soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃G, fpu (bt) ∈̃H, and Clϒ (G) ∪̃Clϒ (H) = 0N.

Since fpu is soft ω-w-c, according to Theorem 2.2 (c),

fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))),andfpu-1(H)˜IntΩω(fpu-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜IntΩω(fpu-1(Clϒ(G)))Ωω,bt˜fpu-1(H)˜IntΩω(fpu-1(Clϒ(H)))Ωω,andIntΩω(fpu-1(Clϒ(G)))˜IntΩω(fpu-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G))˜fpu-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G)˜Clϒ(H)))=IntΩω(fpu-1(0N))=0M.

It follows that (Xω,M) is soft Hausdorff.

The following result shows that the two soft ω-w-c functions from an STS into a soft Urysohn TS agreed upon a soft ω-closed set:

Theorem 2.27

If fpu, fqv : (X,M) → (Y,N) are two soft ω-w-c functions where (Y,N) is soft Urysohn, then ∪̃{mxSP(X,M) : fpu (mx) = fqv (mx)} ∈ (Ωω)c.

Proof. We set K = ∪̃{mxSP(X,M) : fpu(mx) = fqv(mx)}. We show that 1MK ∈ Ωω. Let as∈̃1MK. Then fpu (as) ≠ fqv (as). Since (Y,N) is a soft Urysohn, there exist G, H ∈ ϒ such that fpu (as) ∈̃G, fqv (as) ∈̃H, and Clϒ (G) ∪̃Clϒ (H) = 0N.

Since fpu and fqv are soft ω-w-c, according to Theorem 2.2 (c),

fpu-1(G)˜IntΩω(fpu-1(Clϒ(G))),andfqv-1(H)˜IntΩω(fqv-1(Clϒ(H))).

Thus, we have

as˜fpu-1(G)˜fqv-1(H)˜IntΩω(fpu-1(Clϒ(G)))˜IntΩω(fqv-1(Clϒ(H)))=IntΩω(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H)))Ωω.

Claim

(IntΩ(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K=0M which ends the proof.

Proof of Claim. Suppose, by contrast, that there exists bt˜(IntΩω(fpu-1(Clϒ(G))˜fqv-1(Clϒ(H))))˜K. Then fpu (bt) ∈̃Clϒ (G), fqv (bt) ∈̃Clϒ (H), and fpu (bt) = fqv (bt). So, fpu (bt) ∈̃Clϒ (G) ∪̃Clϒ (H), a contradiction.

Corollary 2.28

Let fpu, fqv : (X,M) → (Y,N) be two soft ω-w-c functions, where (Y,N) denotes a soft Urysohn. If K is a soft dense set in (Xω,M) such that fpu (mx) = fqv (mx) for each mx∈̃K, then fpu = fqv.

Theorem 2.29

If fp1u1 : (X,M) → (Y,N) is soft ω-w-c and fp2u2 : (Y,N) → (Z,R) is soft continuous, then f(p2p1)(u2u1) : (X,M) → (Z,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since fp2u2 : (Y,N) → (Z,R) is soft continuous, fp2u2-1(G)ϒ and Clϒ(fp2u2-1(G)))˜fp2u2-1(ClΠ(G)).

Since fp1u1 is soft ω-w-c, according to Theorem 2.2 (c),

f(p2p1)(u2u1)-1(G)=fp1u1-1(fp2u2-1(G))˜IntΩω(fp1u1-1(Clϒ(fp2u2-1(G))))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G))))=IntΩω(f(p2p1)(u2u1)-1(ClΠ(G)))).

Therefore, according to Theorem 2.2 (c), f(p2p1)(u2u1) is soft ω-w-c.

Theorem 2.30

Let fp1u1 : (X,M) → (Y,N) be a soft function such that fp1u1 : (Xω,M) → (Yω,N) is soft continuous and let fp2u2 : (Y,N) → (Z,R) be soft ω-w-c. Then f(p2p1)(u2u1) : (X,M) → (Z,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since fp2u2 : (Y,N) → (Z,R) is soft ω-w-c, from Theorem 2.2 (c), fp2u2-1(G)˜Intϒω(fp1u1-1(ClΠ(G)) and so fp1u1-1(fp2u2-1(G))˜fp1u1-1(Intϒω(fp2u2-1(ClΠ(G))). Since fp1u1 : (Xω,M) → (Yω,N) is soft continuous,

fp1u1-1(Intϒω(fp2u2-1(ClΠ(G)))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G)))).

Thus,

f(p2p1)(u2u1)-1(G)=fp1u1-1(fp2u2-1(G))˜fp1u1-1(Intϒω(fp2u2-1(ClΠ(G)))˜IntΩω(fp1u1-1(fp2u2-1(ClΠ(G))))=IntΩω(f(p2p1)(u2u1)-1(ClΠ(G)))).

Therefore, from Theorem 2.2 (c), f(p2p1)(u2u1) is soft ω-w-c.

3. Decomposition of ω-Continuity

Definition 3.1

The soft function fpu : (X,M) → (Y,N) is called soft w*-ω-continuous (w*-ω-c) if fpu-1(Bdϒ(G))(Ωω)c for every G ∈ ϒ.

Lemma 3.2

Let {(Xm) : mM} be a family of TSs. Then for every HSP (X,M), (BdmMΩm(H)) (s) = BdΩm(H(s)) for all sM.

Proof. If sM, then from Lemma 4.9 in [21],

(BdmMΩm(H))(s)=((ClmMΩm(H))˜(ClmMΩm(1M-H)))(s)=(ClmMΩm(H))(s)(ClmMΩm(1M-H))(s)=ClΩs(H(s))ClΩs((1M-H)(s))=ClΩs(H(s))ClΩs((X-H(s)))=BdΩm(H(s)).

Theorem 3.3

Let {(Xm) : mM} and {(Yn) : nN} be two families of TSs. Let p : XY be a function and u : MN be a bijective function. Then fpu : (X,⊕mMΩm, M) → (Y,⊕nNϒn, N) is soft w*-ω-c if and only if p : (Xm) → (Yu(m)) is w*-ω-c for each mM.

Proof. Necessity. Suppose that fpu : (X, ⊕mMΩm, M) → (Y, ⊕nNϒn, N) is soft w*-ω-c. Let sM. Let V ∈ ϒu(s). Subsequently, (u(s))V ∈ ⊕nNϒn and so, fpu-1(BdnNϒn((u(s))V))((mMΩm)ω)c. Since by Theorem 8 in [26], (⊕mMΩm)ω = ⊕mMm)ω, we obtain

fpu-1(BdnNϒn((u(s))V))(mM(Ωm)ω)c.

Thus, (fpu-1(BdnNϒn((u(s))V)))(s)((Ωs)ω)c.

From Lemma 3.2, we can conclude that

BdnNϒn((u(s))V))=(u(s))Bdϒs(V).

Additionally, because u : MN is injective,

fpu-1((u(s))Bdϒs(V))=sp-1(.Bdϒs(V)).

Therefore,

(fpu-1(BdnNϒn((u(s))V)))(s)=(sp-1(Bdϒs(V)))   (s)=p-1(Bdϒs(V))((Ωs)ω)c.

This implies that p : (Xs) → (.Yu(s)) is w*-ω-c.

Sufficiency. Suppose that p : (Xm) → (Yu(m)) is w*-ω-c for each mM. Let G ∈ ϒ. Since u : MN is bijective, p : (Xu−1(n))→ (Yn) for each nN and so p−1(Bdϒn(G(n))) ∈ ((Ωu−1(n))ω)c. Thus, p−1(Bdϒm(G(u (m)))) ∈ ((Ωm)ω)c for all mM. Now, by Lemma 3.2, Bdϒm(G(u (m)) = (BdmMϒm(G)) (u (m)) for all mM.

Thus, for each mM,

(fpu-1(BdnNϒn(G)))   (m)=p-1((BdmMϒm(G))(u(m)))=p-1(Bdϒm(G(u(m)))((Ωm)ω)c.

Therefore, fpu-1(BdnNϒn(G))((mMΩm)ω)c.

This shows that

fpu:(X,mMΩm,M)(Y,nNϒn,N)

is soft w*-ω-c.

Corollary 3.4

Let p : (X, δ) → (Y, β) be a function between two TSs and let u : MN be a bijective function. Then p : (X, δ) → (Y, β) is w*-ω-c if and only if fpu : (X, τ (δ),M) →, (Y, τ (β),N) is soft w*-ω-c.

Proof. For each mM and nN, Ωm = δ and ϒn = β. Then τ (δ) = ⊕mMΩm and τ (β) = ⊕nNϒn. Thus, from Theorem 3.3, we obtain the following result:

The following two examples demonstrate the independence of the soft ω-weak continuity, and soft w*-ωcontinuity.

Example 3.5

Let δ and β be co-countable, and discrete topologies for ℝ, respectively. Let M = {a, b, c}. Consider the identity functions p : (ℝ, δ) → (ℝ, β) and u : MM. Then p is w*-ω-c: Let Wβ. Then Bdβ (W) = ∅︀ and so p−1 (Bdβ (W)) = ∅︀ ∈ (δω)c.

p is not ω-w-c: Suppose that p is ω-w-c. Since 2 ∈ p ({2}) = {2}, there exists Vδω such that 2 ∈ V and p(V) = VClβ({2}) = {2}. Thus, V = {2} ∈ δω = δ which is impossible. Therefore, according to Corollaries 2.4 and 3.4, fpu : (Y, τ (ℑ),E) → (Y, τ (ℵ),E) is soft w*-ω-c but not soft ω-w-c.

Example 3.6

LetX = ℝ, Y = {a, b}, δ be the co-countable topology on X, β = {∅︀, Y, {a}}, and E = [0, 1]. Define p : (ℝ, δ) → (ℝ, β) and u : MM, as follows:

p(x)={a,if xb,if x-,

and u (m) = m for all mM. Then

p is ω-w-c: Let xX and Wβ such that p (x) ∈ W. Then W = {a} or W = Y. In both cases, Clβ(W) = Y. Choose U = ℝ. Then Uδ = δω and p (U) = YY = Clβ(W).

p is not w*-ω-c, since {a} ∈ β whereas p−1 (Bdβ ({a})) = p−1 ({b}) = ℝ – ℤ ∉ δc = (δω)c.

Therefore, from Corollaries 2.4 and 3.4, fpu : (X, τ (δ),E) → (Y, τ (β),E) is soft ω-w-c but not soft w*-ω-c.

Theorem 3.7

Each soft ω-c function is soft w*-ω-c.

Proof. Let fpu : (X,M) → (Y,N) be soft ω-c and let G ∈ ϒ. Since fpu is soft ω-c and Bdϒ(G) ∈ ϒc, fpu-1(Bdϒ(K))(δω)c. This shows that fpu is soft w*-ω-c.

The opposite of Theorem 3.7 need not be true, as demonstrated in Example 3.5 and Theorem 2.5.

The following main result represents the decomposition of soft ω-continuity:

Theorem 3.8

The soft function fpu : (X, Ω, M) → (Y, ϒ, N) is soft ω-c if and only if both soft ω-w-c and soft w*-ω-c.

Proof. Necessity. The proof follows from Theorems 2.5 and 3.7.

Sufficiency. We assume that fpu : (X, Ω, M) → (Y, ϒ, N) is both soft ω-w-c and soft w*-ω-c. Let mxSP(X,M) and G ∈ ϒ such that fpu(mx)∈̃G. Since fpu is soft ω-w-c, there exists Lδω such that mx∈̃L and fpu(L)⊆̃Clϒ(G). Since fpu(mx)∈̃G and Bdϒ(G) = Clϒ(G) – G. fpu(mx)∉̃ Bdϒ(G). Thus, mx˜fpu-1(Bdϒ(G)). Since fpu is soft w*-ω-c,

fpu-1(Bdϒ(G))(Ωω)c.

Therefore, we have mx˜L-fpu-1(Bdϒ(G))Ωω.

Claim

fpu(L-fpu-1(Bdϒ(G)))˜G which ends the proof.

Proof of Claim. Suppose the contrary that there exists az˜fpu(L-fpu-1(Bdϒ(G)))-G. Choose br˜L-fpu-1(Bdϒ(G)) such that az = fpu(br). Since br∈̃L and fpu(L)⊆̃Clϒ(G), az = fpu(br)∈̃Clϒ(G) and so br˜fpu-1(Clϒ(G)). Since br˜fpu-1(Bdϒ(G))=fpu-1(Clϒ(G)-G)=fpu-1(Clϒ(G))-fpu-1(G), and br˜fpu-1(Clϒ(G)),br˜fpu-1(G).

However, since fpu(br)=az˜G,br˜fpu-1(G), which is contradictory.

4. Conclusion

We define soft ω-weak continuity (Definition 2.1) and soft w*-ω-continuity (Definition 3.1) as two new notions of soft functions. We proved that they are independent notions (Examples 3.5 and 3.6). Additionally, we prove that soft ω-weak continuity is strictly weaker than both the soft ω-continuity (Theorem 2.5 and Example 2.6) and soft weak continuity (Theorem 2.8 and Example 2.9), and soft w*-ω-continuity is strictly weaker than soft ω-continuity (Example 3.5 and Theorem 3.7). Moreover, we obtain a decomposition theorem for soft ω-continuity in terms of soft ω-weak continuity and soft w*-ω-continuity (Theorem 3.8). Regarding soft ω-weak continuity, we present two characterizations (Theorem 2.2); and examined soft graph (Theorems 2.10 and 2.11), soft preservation (Theorems 2.22, 2.23, and 2.26), and soft composition (Theorems 2.29 and 2.30) theorems. We prove that the two soft w-c (resp. ω-w-c) functions from an STS into a soft Urysohn TS agreed upon a soft closed set (Theorem 2.24) (resp. soft ω-closed set (Theorem 2.27). Finally, we examined the correspondence between each soft and weak continuity, soft ω–weak continuity, and soft w*ω-continuity and topological analogs (Theorems 2.3 and 3.3 and Corollaries 3.3 and 3.4).

In future work, we plan to (1) define and study weaker forms of ω-continuity; (2) define and study strong forms of soft continuity, such as super continuity; and (3) introduce new classes of soft open sets.

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