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 [8–21]. Soft topologies and applications are currently an established field of study [22–24]. 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 H ∈ SS (X,M). H is referred to as a null (resp. absolute) soft set over X relative to M and is denoted by 0_M (resp. 1_M) if H (m) = ∅︀ (resp. H (m) = X) for each m ∈ M. H is referred to as a soft point over X relative to M and is denoted by a_x if there are x ∈ X and a ∈ M such that H (a) = {x} and H (m) = ∅︀ for each m ∈ M − {a}. The collection of all soft points over X relative to M is denoted as SP(X,M). If for some a ∈ M and Y ⊆ X, we have H (a) = Y and K (m) = ∅︀ for all m ∈ M − {a}, then H is denoted by a_Y . If for some H (m) = Y for each m ∈ M, then H is denoted as C_Y . A soft point a_x ∈ SP (X,M) is said to belong to H (notation a_x∈̃H) if x ∈ H (a). Let A,B ∈ SS (X,M). Then A is a soft subset of B, denoted by A⊆̃B if A(m) ⊆ B(m) each m ∈ M. The soft union (resp. intersection, which is the difference) between A and B is denoted by A∪̃B (resp. A∩̃B, A − B) and defined as (A∪̃B)(m) = A(m) ∪ B(m) (resp. (A∩̃B)(m) = A(m) ∩ B(m), (A − B) (m) = A(m) − B (m)) for each m ∈ M.

Let (X, δ) and (X,Ω,M) be a TS and an STS, respectively. Let W ⊆ X and K ∈ SS(X,M). In this paper, Int_δ(W), Cl_δ(W), Int_Ω(K), and Cl_Ω(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,Ω,M), and soft closure of K in (X,Ω,M). Additionally, δ^c , Ω^c, CO (X, δ), and CO(X,Ω,M) represent the family of closed sets on (X, δ), family of soft closed sets on (X,Ω,M), family of clopen sets on (X, δ), and family of soft clopen set as (X,Ω,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 a ∈ A if for every T ∈ β such that g(a) ∈ T, there exists S ∈ δ such that a ∈ S and g(S) ⊆ Cl_β(T). If g is w-c for each a ∈ A, g is called w-c [27].

(b) ω-weak continuous (ω-w-c) at a ∈ A if, for every T ∈ β such that g(a) ∈ T, we find that S ∈ δ_ω such that a ∈ S and g(S) ⊆ Cl_β(T). If g is ω-w-c at each a ∈ A, and g is called ω-w-c [28].

Definition 1.2

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

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

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

Definition 1.3

An STS (A,Ω,M) is called

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

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

(c) soft Hausdorff space if, for every m_a, n_b ∈ SP(A,M) such m_a ≠ n_b, there exist S, T ∈ Ω such that m_a∈̃S, n_b∈̃T, and S∩̃T = 0_M [31].

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

m_a∈̃L⊆̃Cl_Ω (L) ⊆̃G [32].

(e) soft Urysohn space if, for every m_a, n_b ∈ SP(A,M) such that m_a ≠ n_b, there exist S, T ∈ Ω such that m_a∈̃S, n_b∈̃T, and Cl_Ω (S) ∩̃Cl_Ω (T) = 0_M [33].

Definition 2.1

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

(a) f_pu is called the soft ω-weak continuous (soft ω-w-c) at a soft point m_x ∈ SP (X,M) if for each G ∈ ϒ such that f_pu (m_x) ∈̃G, there exists H ∈ Ω_ω such that m_x∈̃H and f_pu(H)⊆̃Cl_ϒ (G);

(b) f_pu is called soft ω-w-c if it is soft ω-w-c at each soft point, m_x ∈ SP (X,M).

Theorem 2.2

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

(a) f_pu 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 f_pu(m_x)∉̃Cl_ϒ(G), we find L ∈ ϒ such that f_pu(m_x) ∈ L and L∩̃G = 0_N. From (a), there exists K ∈ Ω_ω such that m_x∈̃K and f_pu(K)⊆̃Cl_ϒ (L). Since mx∈˜ClΩω (fpu-1 (G)) and m_x∈̃K ∈ Ω_ω, K∩˜fpu-1 (G)≠0M. Choose a_z∈̃K such that f_pu(a_z)∈̃G. Since a_z∈̃K and f_pu(K)⊆̃Cl_ϒ (L), f_pu(a_z) ∈̃Cl_ϒ (L). Since f_pu(a_z)∈̃G ∈ ϒ and f_pu(a_z)∈̃Cl_ϒ (L), G∩̃L ≠ 0_N, a contradiction.

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

We have

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

(c) ⇒ (a). Let m_x ∈ SP (X,M) and let G ∈ ϒ such that f_pu(m_x)∈̃G. By (c), fpu-1 (G)⊆˜IntΩω (fpu-1 (Clϒ(G))). Let L=IntΩω (fpu-1 (Clϒ(G))). Then f_pu(m_x)∈̃L∈Ω_ω and

This completes the proof.

Theorem 2.3

Let {(X,Ω_m) : m ∈ M} and {(Y, ϒ_n) : n ∈ N} be two families of TSs. Let p : X → Y be a function and u : M → N be a bijective function. Then

Proof. Necessity. Suppose that f_pu : (X,⊕_m∈MΩ_m,M) → (Y,⊕_n∈Nϒ_n,N) is soft ω-w-c. Let s ∈ M. Let x ∈ X and V ∈ ϒ_u(m) such that p (x) ∈ V . Then f_pu (s_x) = (u(s))_p(x) ∈̃ (u(s))_V ∈ ⊕_n∈Nϒ_n. Therefore, there exists H ∈ (⊕_m∈MΩ_m)_ω such that m_x∈̃H and f_pu(H) ⊆̃Cl_ϒ ((u(s))_V ) = (u(s))_Clϒu(s)(V). According to Theorem 8 in [26], we have (⊕_m∈MΩ_m)_ω = ⊕_m∈M (Ω_m)_ω; thus, H(s) ∈ (Ω_m)_ω. Since

(f_pu(H)) (u(s)) ⊆ ((u(s))_Clϒu(s)(V))(u(s)) = Cl_ϒu(s) (V ). Since u is injective, (f_pu(H)) (u (s)) = p (H(s)). This shows that p : (X,Ω_s) → (Y, ϒ_u(s)) is ω-w-c.

Sufficiency. Suppose that p : (X,Ω_m) → (Y, ϒ_u(m)) is ω-w-c for all m ∈ M. Let m_x ∈ SP (X,M) and let G ∈ ϒ such that f_pu (m_x) = (u (m))_p(x) ∈̃G. Then we have p (x) ∈ G(u (m)) ∈ ϒ_u(m). Since p : (X,Ω_m) → (Y, ϒ_u(m)) is ω-w-c, there exists U ∈ Ω_ω such that x ∈ U and p (U) ⊆ Cl_ϒu(m) (G(u (m))). By Lemma 4.9 of [21], we have (Cl_⊕n∈Nϒn (G)) (u (m)) = Cl_ϒu(m) (G(u (m))). Since x ∈ U, m_x∈̃m_U. Since U ∈ Ω_ω, m_U ∈ ⊕_m∈M (Ω_m)_ω and by Theorem 8 of [26], m_U ∈ (⊕_m∈MΩ_m)_ω. Since p (U) ⊆ Cl_ϒu(m) (G(u (m))),

If n ∈ N−u (m), then (f_pu (m_U)) (n) = ((u (m))_p(U))(n) = ∅︀ ⊆ (Cl_⊕n∈Nϒn (G)) (n). Therefore, f_pu (m_U) ⊆̃Cl_⊕n∈Nϒn (G). It follows that f_pu : (X,⊕_m∈MΩ_m,M) → (Y , ⊕_n∈Nϒ_n, N) is soft ω-w-c.

Corollary 2.4

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

Proof. For each m ∈ M and n ∈ N, let Ω_m = δ and ϒ_n = β. Then τ (δ) = ⊕_m∈MΩ_m and τ (β) = ⊕_n∈Nϒ_n. 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 : M → M by

and u(m) = m for each m ∈ M. Then

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

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

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

Theorem 2.7

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

Proof. Let m_x ∈ SP(X,M) and let G ∈ ϒ such that f_pu(m_x)∈̃G. Since (Y, ϒ,N) is soft regular, there exists K ∈ ϒ such that f_pu(m_x)∈̃K⊆̃Cl_ϒ (K) ⊆̃G. Since f_pu is soft ω-w-c, we find H ∈ Ω_ω such that m_x∈̃H and f_pu(H)⊆̃Cl_ϒ (K) ⊆̃G.

It follows that f_pu is soft ω-c.

Theorem 2.8

Each soft w-c function is soft ω-w-c.

Proof. Let f_pu : (X,Ω,M) → (Y, ϒ,N) be soft w-c. Let m_x ∈ SP(X,M) and let G ∈ ϒ such that f_pu(m_x)∈̃G. Then we find H ∈ Ω such that m_x∈̃H and f_pu(H)⊆̃Cl_ϒ (G). Since Ω ⊆ Ω_ω, H ∈ Ω_ω. This shows that f_pu 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 : X → X and u : M → M. Then

p is ω-c, and from Theorem 2.5, it is soft ω-w-c, since δ_ω 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 W ∈ δ such that 2 ∈ W and p(W) = W = X ⊆ Cl_β({2}) = {2, 3} which is impossible. Therefore, from Corollaries 2.4 and 3.4 in [11], f_pu : (X, τ (δ) ,M) → (Y, τ (β) ,M) is soft ω-w-c but not soft w-c.

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

Theorem 2.10

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

is soft w-c.

Proof. Necessity. Suppose f_pu : (X,Ω,M) → (Y, ϒ,N) is soft w-c. Let m_x ∈ SP(X,M) and let G ∈ pr (Ω × ϒ) such that (f_p^#u^# (m_x)) = (u^#)(m))_(p^#)(x) = (m, u(m))_(x,p(x)) = m_x × (u(m))_p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))_(x,p(x)) = m_x × (u(m))_p(x) ∈̃K × H⊆̃G. Since f_pu is soft w-c and f_pu (m_x) = (u(m))_p(x) ∈̃H ∈ ϒ, there exists L ∈ Ω such that m_x∈̃L and f_pu (L) ⊆̃Cl_ϒ(H). Therefore,

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

Sufficiency. Suppose that

is soft w-c. Let m_x ∈ SP(X,M) and H ∈ ϒ such that (f_pu (m_x)) = (u(m))_p(x) ∈̃H. Then

Since f_p^#u^# is soft w-c, there exists T ∈ Ω such that m_x∈̃T and f_p^#u^# (T) ⊆̃Cl_{pr(Ω× ϒ)}(1_M×H) = 1_M×Cl_ϒ(H); hence f_pu (T) ⊆̃Cl_ϒ(H). This shows that f_pu is soft w-c.

Theorem 2.11

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

is soft ω-w-c.

Proof. Necessity. Suppose f_pu : (X,Ω,M) → (Y, ϒ,N) is soft ω-w-c. Let m_x ∈ SP(X,M) and G ∈ pr (Ω × ϒ) such that (f_p^#u^# (m_x)) = (u^#)(m))_(p^#)(x) = (m, u(m))_(x,p(x)) = m_x × (u(m))_p(x) ∈̃G. Then there exist K ∈ Ω and H ∈ ϒ, such that (m, u(m))_(x,p(x)) = m_x × (u(m))_p(x) ∈̃K × H⊆̃G. Since f_pu is soft ω-w-c and f_pu (m_x) = (u(m))_p(x) ∈̃H ∈ ϒ, there exists L ∈ Ω_ω such that m_x∈̃L and f_pu (L) ⊆̃Cl_ϒ(H). Therefore,

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

Sufficiency. Suppose that

is soft ω-w-c. Let m_x ∈ SP(X,M) and H ∈ ϒ such that (f_pu (m_x)) = (u(m))_p(x) ∈̃H. Then

Since f_p^#u^# is soft ω-w-c, there exists T ∈ Ω_ω such that m_x∈̃T and f_p^#u^# (T) ⊆̃Cl_pr(Ω×ϒ)(1_M×H) = 1_M×Cl_ϒ(H); hence f_pu (T) ⊆̃Cl_ϒ(H). This shows that f_pu is soft ω-w-c.

Definition 2.12

Let (X,Ω,M) be an STS. Then

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

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

Theorem 2.13

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

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

Theorem 2.14

Each soft ω-connected STS is soft connected.

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

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

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

Example 2.15

Consider (ℕ,Ω,ℤ) where Ω = {0_ℤ, 1_ℤ}. Then CO (ℕ,Ω,ℤ) ⊆ Ω = {0_ℤ, 1_ℤ}; thus, CO (ℕ,Ω,ℤ) = {0_ℤ, 1_ℤ}. 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,Ω,M) be soft anti-locally countable and soft connected. Suppose the contrary that (X,Ω,M) is soft ω-disconnected. Then there is G ∈ CO (X,Ω_ω,M)−{0_M, 1_M}. Then we have Cl_Ωω (G) = G and Int_Ωω (1_M − G) = 1_M − G. Since G ∈ Ω_ω and (1_M − G) ∈ (Ω_ω)^c and (X,Ω,M) is soft anti-locally countable, according to Theorem 14 of [26], Cl_Ω (G) = Cl_Ωω (G) = G and Int_Ω(1_M − G) = Int_Ωω (1_M − G) = 1_M − G. Therefore, we have G ∈ CO(X, Ω, M) − {0_M, 1_M}; hence (X,Ω,M) is soft disconnected, which is contradictory.

Lemma 2.17

Let {(X,Ω_m) : m ∈ M} be a family of TSs. Then G ∈ CO (X,⊕_m∈MΩ_m,M) if and only if G(s) ∈ CO (X,Ω_s) for all s ∈ M.

Proof. Necessity. Let G ∈ CO (X,⊕_m∈MΩ_m,M) and let s ∈ M. Since G ∈ CO (X,⊕_m∈MΩ_m,M), G, 1_M − G ∈ ⊕_m∈MΩ_m; thus, G(s) ∈ Ω_s and (1_M − G) (s) = X−G(s) ∈ Ω_s. Hence, G(s) ∈ CO (X,Ω_s).

Sufficiency. Let G(s) ∈ CO (X,Ω_s) for all s ∈ M. Then G(s) ∈ Ω_s and X − G(s) = (1_M − G) (s) ∈ Ω_s for all s ∈ M. Hence, G, 1_M − G ∈ ⊕_m∈MΩ_m. Therefore, G ∈ CO (X,⊕_m∈MΩ_m,M).

Theorem 2.18

If {(X,Ω_m) : m ∈ M} is a family of TSs, where M contains at least two points, then (X,⊕_m∈MΩ_m,M) is soft disconnected.

Proof. Choose s ∈ M. Consider the soft set G ∈ SS(X,M) defined by G(s) = ∅︀ and G(t) = X for all t ∈ M–{s}. Since for all m ∈ M, G(s) ∈ {∅︀,X} ⊆ CO(X,Ω_m), by Lemma 2.17, G ∈ CO(X,⊕_m∈MΩ_m,M). Since M contains at least two points, G ∉ {0_M, 1_M}. Therefore, (X,⊕_m∈MΩ_m, 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 m ∈ M, put Ω_m = δ. Then τ (δ) = ⊕_m∈MΩ_m. Thus, according to Theorem 2.18 we obtain the result.

Corollary 2.20

If {(X,Ω_m) : m ∈ M} is a family of TSs, where M contains at least two points, then (X,⊕_m∈MΩ_m, 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 m ∈ M, put Ω_m = δ. Then τ (δ) = ⊕_m∈MΩ_m. Thus, by Corollary 2.20, we obtain the following results.

Theorem 2.22

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

Proof. Suppose, in contrast to, (Y,ϒ,N) is soft disconnected. Then there exists G ∈ CO(Y,ϒ,N) – {0_N, 1_N}. Since G ∈ ϒ^c, Cl_ϒ(G) = G. Since f_pu : (X,Ω,M) → (Y,ϒ,N) is soft ω-w-c and G ∈ ϒ, according to Theorem 2.2,

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

Theorem 2.23

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

Proof. Let a_x, b_y ∈ SP (X,M) such that a_s ≠ b_t. Since f_pu is injective, f_pu (a_x) ≠ f_pu (b_y). Since (Y,ϒ,N) is soft Urysohn, there exist G, H ∈ ϒ such that f_pu (a_s) ∈̃;G,

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

Thus, we have

It follows that (X,Ω,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 f_pu, f_qv : (X,Ω,M) → (Y,ϒ,N) are two soft w-c functions where (Y,ϒ,N) is soft Urysohn, then ∪̃{m_x ∈ SP(X,M) : f_pu (m_x) = f_qv (m_x)} ∈ Ω^c.

Proof. Put K=∪̃{m_x∈SP(X,M) : f_pu(m_x)=f_qv(m_x)}.

We demonstrate that 1_M – K ∈ Ω. Let a_s ∈̃ 1_M – K. Then f_pu (a_s) ≠ f_qv (a_s). Since (Y,ϒ,N) is soft Urysohn, there exist G, H ∈ ϒ such that f_pu (a_s) ∈̃ G, f_qv (a_s) ∈̃ H, and Cl_ϒ (G) ∩̃Cl_ϒ (H) = 0_N. Since f_pu and f_qv are soft w-c according to Theorem 5.2 in [29],

Thus, we have

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

Then f_pu (b_t) ∈̃Cl_ϒ (G), f_qv (b_t) ∈̃Cl_ϒ (H), and f_pu (b_t) = f_qv (b_t). So, f_pu (b_t) ∈̃Cl_ϒ (G) ∪̃Cl_ϒ (H), a contradiction.

Corollary 2.25

Let f_pu, f_qv : (X,Ω,M) → (Y,ϒ,N) be two soft w-c functions where (Y,ϒ,N) denotes soft Urysohn. If K is a soft dense set in (X,Ω_ω,M) such that f_pu (m_x) = f_qv (m_x) for each m_x∈̃K, then f_pu = f_qv.

Theorem 2.26

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

Proof. Let a_x, b_y ∈ SP (X,M) such that a_s ≠ b_t. Since f_pu is injective, f_pu (a_x) ≠ f_pu (b_y). Since (Y,ϒ,N) is soft Urysohn, there exist G, H ∈ ϒ such that f_pu (a_s) ∈̃G, f_pu (b_t) ∈̃H, and Cl_ϒ (G) ∪̃Cl_ϒ (H) = 0_N.

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

Thus, we have

It follows that (X,Ω_ω,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 f_pu, f_qv : (X,Ω,M) → (Y,ϒ,N) are two soft ω-w-c functions where (Y,ϒ,N) is soft Urysohn, then ∪̃{m_x ∈ SP(X,M) : f_pu (m_x) = f_qv (m_x)} ∈ (Ω_ω)^c.

Proof. We set K = ∪̃{m_x ∈ SP(X,M) : f_pu(m_x) = f_qv(m_x)}. We show that 1_M – K ∈ Ω_ω. Let a_s∈̃1_M – K. Then f_pu (a_s) ≠ f_qv (a_s). Since (Y,ϒ,N) is a soft Urysohn, there exist G, H ∈ ϒ such that f_pu (a_s) ∈̃G, f_qv (a_s) ∈̃H, and Cl_ϒ (G) ∪̃Cl_ϒ (H) = 0_N.

Since f_pu and f_qv are soft ω-w-c, according to Theorem 2.2 (c),

Thus, we have

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 f_pu (b_t) ∈̃Cl_ϒ (G), f_qv (b_t) ∈̃Cl_ϒ (H), and f_pu (b_t) = f_qv (b_t). So, f_pu (b_t) ∈̃Cl_ϒ (G) ∪̃Cl_ϒ (H), a contradiction.

Corollary 2.28

Let f_pu, f_qv : (X,Ω,M) → (Y,ϒ,N) be two soft ω-w-c functions, where (Y,ϒ,N) denotes a soft Urysohn. If K is a soft dense set in (X,Ω_ω,M) such that f_pu (m_x) = f_qv (m_x) for each m_x∈̃K, then f_pu = f_qv.

Theorem 2.29

If f_p1u1 : (X,Ω,M) → (Y,ϒ,N) is soft ω-w-c and f_p2u2 : (Y,ϒ,N) → (Z,Π,R) is soft continuous, then f_{(p2∘p1)(u2∘u1)} : (X,Ω,M) → (Z,Π,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since f_p2u2 : (Y,ϒ,N) → (Z,Π,R) is soft continuous, fp2u2-1 (G)∈ϒ and Clϒ(fp2u2-1 (G)))⊆˜fp2u2-1 (ClΠ(G)).

Since f_p1u1 is soft ω-w-c, according to Theorem 2.2 (c),

Therefore, according to Theorem 2.2 (c), f_{(p2∘p1)(u2∘u1)} is soft ω-w-c.

Theorem 2.30

Let f_p1u1 : (X,Ω,M) → (Y,ϒ,N) be a soft function such that f_p1u1 : (X,Ω_ω,M) → (Y,ϒ_ω,N) is soft continuous and let f_p2u2 : (Y,ϒ,N) → (Z,Π,R) be soft ω-w-c. Then f_{(p2∘p1)(u2∘u1)} : (X,Ω,M) → (Z,Π,R) is soft ω-w-c.

Proof. Let G ∈ Π. Since f_p2u2 : (Y,ϒ,N) → (Z,Π,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 f_p1u1 : (X,Ω_ω,M) → (Y,ϒ_ω,N) is soft continuous,

Thus,