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, ${Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)\tilde{\subseteq}{f}_{pu}^{-1}\left({Cl}_{\mathrm{\Upsilon}}(G)\right)$;
(c) for every G ∈ ϒ, ${f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)$.
Proof. (a) ⇒ (b) Let G ∈ ϒ. Suppose to the contrary that there exists ${m}_{x}\tilde{\in}{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)-{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))$. 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 ${m}_{x}\tilde{\in}{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)$ and m_{x}∈̃K ∈ Ω_{ω}, $K\tilde{\cap}{f}_{pu}^{-1}\hspace{0.17em}(G)\ne {0}_{M}$. 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), ${Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G))\right)\tilde{\subseteq}{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G)))$; thus,
$$\begin{array}{l}{1}_{M}-{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G)))\\ \tilde{\subseteq}{1}_{M}-{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G))\right).\end{array}$$We have
$$\begin{array}{l}{1}_{M}-{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G)))\\ ={f}_{pu}^{-1}\hspace{0.17em}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G)))\\ ={f}_{pu}^{-1}\hspace{0.17em}(In{t}_{\mathrm{\Upsilon}}({Cl}_{\mathrm{\Upsilon}}(G))).\end{array}$$Since G⊆̃Int_{ϒ}(Cl_{ϒ}(G)), ${f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}{f}_{pu}^{-1}\hspace{0.17em}(In{t}_{\mathrm{\Upsilon}}({Cl}_{\mathrm{\Upsilon}}(G))$. Thus,
$$\begin{array}{l}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}{1}_{M}-{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G)))\\ \tilde{\subseteq}{1}_{M}-{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({1}_{N}-{Cl}_{\mathrm{\Upsilon}}(G))\right)\\ ={1}_{M}-{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left(({1}_{M}-{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G)))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right).\end{array}$$(c) ⇒ (a). Let m_{x} ∈ SP (X,M) and let G ∈ ϒ such that f_{pu}(m_{x})∈̃G. By (c), ${f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)$. Let $L=In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)$. Then f_{pu}(m_{x})∈̃L∈Ω_{ω} and
$$\begin{array}{l}{f}_{pu}(L)={f}_{pu}(In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right))\\ \tilde{\subseteq}{f}_{pu}(\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right))\\ \tilde{\subseteq}{Cl}_{\mathrm{\Upsilon}}(G).\end{array}$$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
$$\begin{array}{l}{f}_{pu}:(X,{\oplus}_{m\in M}{\mathrm{\Omega}}_{m},M)\\ \to (Y,{\oplus}_{n\in N}{\mathrm{\Upsilon}}_{n},N)\hspace{0.17em}\text{is\hspace{0.17em}soft\hspace{0.17em}}\omega \text{-w-c\hspace{0.17em}iff\hspace{0.17em}}p:(X,{\mathrm{\Omega}}_{m})\\ \to \left(Y,{\mathrm{\Upsilon}}_{u(m)}\right)\hspace{0.17em}\text{is\hspace{0.17em}}\omega \text{-w-c\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}m\in M.\end{array}$$Proof. Necessity. Suppose that f_{pu} : (X,⊕_{m∈M}Ω_{m},M) → (Y,⊕_{n∈N}ϒ_{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∈N}ϒ_{n}. Therefore, there exists H ∈ (⊕_{m∈M}Ω_{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∈M}Ω_{m})_{ω} = ⊕_{m∈M} (Ω_{m})_{ω}; thus, H(s) ∈ (Ω_{m})_{ω}. Since
$${f}_{pu}(H)\tilde{\subseteq}{(u(s))}_{{Cl}_{{\mathrm{\Upsilon}}_{u(s)}}(V)},$$(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∈M}Ω_{m})_{ω}. Since p (U) ⊆ Cl_{ϒu(m)} (G(u (m))),
$$\begin{array}{l}({f}_{pu}\hspace{0.17em}({m}_{U}))\hspace{0.17em}(u(m))=\left({(u\hspace{0.17em}(m))}_{p(U)}\right)\hspace{0.17em}(u(m))\\ =p\hspace{0.17em}(U)\\ \subseteq {Cl}_{{\mathrm{\Upsilon}}_{u(m)}}\hspace{0.17em}(G(u\hspace{0.17em}(m)))\\ =({Cl}_{{\oplus}_{n\in N}{\mathrm{\Upsilon}}_{n}}\hspace{0.17em}(G))\hspace{0.17em}(u\hspace{0.17em}(m)).\end{array}$$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∈M}Ω_{m},M) → (Y , ⊕_{n∈N}ϒ_{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∈M}Ω_{m} and τ (β) = ⊕_{n∈N}ϒ_{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
$$p(x)=\{\begin{array}{ll}2,\hfill & \text{if\hspace{0.17em}}x\in \mathbb{Q},\hfill \\ \pi ,\hfill & \text{if\hspace{0.17em}}x\in \mathbb{R}-\mathbb{Q},\hfill \end{array}$$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
$${f}_{{p}^{\#}{u}^{\#}}:(X,\mathrm{\Omega},M)\to (X\times Y,pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}),M\times N)$$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,
$$\begin{array}{l}{f}_{{p}^{\#}{u}^{\#}}(L)\tilde{\subseteq}K\times {Cl}_{\mathrm{\Upsilon}}(H)\\ \tilde{\subseteq}{Cl}_{pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon})}(K\times H)\\ ={Cl}_{pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon})}(G).\end{array}$$This indicates that f_{p#u#} is a soft w-c.
Sufficiency. Suppose that
$${f}_{{p}^{\#}{u}^{\#}}:(X,\mathrm{\Omega},M)\to (X\times Y,pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}),M\times N)$$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
$$\begin{array}{l}\left({f}_{{p}^{\#}{u}^{\#}}({m}_{x})\right)={\left(\left({u}^{\#}\right)\hspace{0.17em}(m)\right)}_{({p}^{\#})\hspace{0.17em}(x)}\\ ={(m,u(m))}_{(x,p(x))}\\ ={m}_{x}\times {(u(m))}_{p(x)}\\ \tilde{\in}{1}_{M}\times H\\ \in pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}).\end{array}$$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
$${f}_{{p}^{\#}{u}^{\#}}:(X,\mathrm{\Omega},M)\to (X\times Y,pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}),M\times N)$$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,
$$\begin{array}{l}{f}_{{p}^{\#}{u}^{\#}}(L)\tilde{\subseteq}K\times {Cl}_{\mathrm{\Upsilon}}(H)\\ \tilde{\subseteq}{Cl}_{pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon})}(K\times H)\\ ={Cl}_{pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon})}(G).\end{array}$$This indicates that f_{p#u#} is soft ω-w-c.
Sufficiency. Suppose that
$${f}_{{p}^{\#}{u}^{\#}}:(X,\mathrm{\Omega},M)\to (X\times Y,pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}),M\times N)$$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
$$\begin{array}{l}\left({f}_{{p}^{\#}{u}^{\#}}({m}_{x})\right)={\left(\left({u}^{\#}\right)\hspace{0.17em}(m)\right)}_{({p}^{\#})\hspace{0.17em}(x)}\\ ={(m,u(m))}_{(x,p(x))}\\ ={m}_{x}\times {(u(m))}_{p(x)}\\ \tilde{\in}{1}_{M}\times H\\ \in pr\hspace{0.17em}(\mathrm{\Omega}\times \mathrm{\Upsilon}).\end{array}$$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∈M}Ω_{m},M) if and only if G(s) ∈ CO (X,Ω_{s}) for all s ∈ M.
Proof. Necessity. Let G ∈ CO (X,⊕_{m∈M}Ω_{m},M) and let s ∈ M. Since G ∈ CO (X,⊕_{m∈M}Ω_{m},M), G, 1_{M} − G ∈ ⊕_{m∈M}Ω_{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∈M}Ω_{m}. Therefore, G ∈ CO (X,⊕_{m∈M}Ω_{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∈M}Ω_{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∈M}Ω_{m},M). Since M contains at least two points, G ∉ {0_{M}, 1_{M}}. Therefore, (X,⊕_{m∈M}Ω_{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∈M}Ω_{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∈M}Ω_{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∈M}Ω_{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,
$$\begin{array}{l}{Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)\tilde{\subseteq}{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))={f}_{pu}^{-1}\hspace{0.17em}(G),\hspace{0.17em}\text{and}\\ {f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)=In{t}_{{\mathrm{\Omega}}_{\omega}}({f}_{pu}^{-1}\hspace{0.17em}(G)).\end{array}$$Thus, ${Cl}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)={f}_{pu}^{-1}\hspace{0.17em}(G)$ and ${f}_{pu}^{-1}\hspace{0.17em}(G)=In{t}_{{\mathrm{\Omega}}_{\omega}}({f}_{pu}^{-1}\hspace{0.17em}(G))$.
Thus, ${f}_{pu}^{-1}\hspace{0.17em}(G)\in CO(X,\mathrm{\Omega},M)$. Since G ≠ 0_{N} and f_{pu} is surjective, ${f}_{pu}^{-1}\hspace{0.17em}(G)\ne {0}_{M}$. If ${f}_{pu}^{-1}\hspace{0.17em}(G)={1}_{M}$, then ${f}_{pu}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}(G)\right)={f}_{pu}({1}_{M})={1}_{N}\tilde{\subseteq}G$ and so G = 1_{N}. Thus, ${f}_{pu}^{-1}\hspace{0.17em}(G)\ne {1}_{M}$. 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,
$${f}_{pu}({b}_{t})\tilde{\in}H,\hspace{0.17em}\text{and\hspace{0.17em}}{Cl}_{\mathrm{\Upsilon}}(G)\tilde{\cap}{Cl}_{\mathrm{\Upsilon}}(H)={0}_{N}.$$Since f_{pu} is soft w-c, according to Theorem 5.2 in [29],
$$\begin{array}{l}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))),\hspace{0.17em}\text{and}\\ {f}_{pu}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right).\end{array}$$Thus, we have
$$\begin{array}{l}{a}_{s}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\in \mathrm{\Omega},\\ {b}_{t}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\in \mathrm{\Omega},\hspace{0.17em}\text{and}\\ In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\tilde{\cap}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G)\tilde{\cap}{Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({0}_{N})\right)\\ ={0}_{M}.\end{array}$$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],
$$\begin{array}{l}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right),\hspace{0.17em}\text{and}\\ {f}_{pu}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right).\end{array}$$Thus, we have
$$\begin{array}{l}{a}_{s}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}(H)\\ \tilde{\subseteq}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\tilde{\cap}In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ \in \mathrm{\Omega}.\end{array}$$Claim
$\left(In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\right)\tilde{\cap}K={0}_{M}$ which ends the proof.
Proof of Claim. Suppose, by contrast, that there exists
$${b}_{t}\tilde{\in}\left(In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\right)\tilde{\cap}K.$$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 2.25
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),
$$\begin{array}{l}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right),\hspace{0.17em}\text{and}\\ {f}_{pu}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right).\end{array}$$Thus, we have
$$\begin{array}{l}{a}_{s}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\in {\mathrm{\Omega}}_{\omega},\\ {b}_{t}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\in {\mathrm{\Omega}}_{\omega},\hspace{0.17em}\text{and}\\ In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\tilde{\cap}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G)\tilde{\cap}{Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({0}_{N})\right)\\ ={0}_{M}.\end{array}$$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),
$$\begin{array}{l}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right),\hspace{0.17em}\text{and}\\ {f}_{qv}^{-1}\hspace{0.17em}(H)\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right).\end{array}$$Thus, we have
$$\begin{array}{l}{a}_{s}\tilde{\in}{f}_{pu}^{-1}\hspace{0.17em}(G)\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}(H)\\ \tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\right)\tilde{\cap}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\in {\mathrm{\Omega}}_{\omega}.\end{array}$$Claim
$\left(In{t}_{\mathrm{\Omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\right)\tilde{\cap}K={0}_{M}$ which ends the proof.
Proof of Claim. Suppose, by contrast, that there exists ${b}_{t}\tilde{\in}\left(In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{pu}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(G))\tilde{\cap}{f}_{qv}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}(H))\right)\right)\tilde{\cap}K$. 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, ${f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)\in \mathrm{\Upsilon}$ and ${Cl}_{\mathrm{\Upsilon}}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)))\tilde{\subseteq}{f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}(G))$.
Since f_{p1u1} is soft ω-w-c, according to Theorem 2.2 (c),
$$\begin{array}{l}{f}_{({p}_{2}\circ {p}_{1})\hspace{0.17em}({u}_{2}\circ {u}_{1})}^{-1}\hspace{0.17em}(G)\\ ={f}_{{p}_{1}{u}_{1}}^{-1}\left({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)\right)\\ \tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Upsilon}}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)))\right)\\ \tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}(G)))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{({p}_{2}\circ {p}_{1})\hspace{0.17em}({u}_{2}\circ {u}_{1})}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\right).\end{array}$$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), ${f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)\tilde{\subseteq}In{t}_{{\mathrm{\Upsilon}}_{\omega}}({f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G))$ and so ${f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G))\tilde{\subseteq}{f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}(In{t}_{{\mathrm{\Upsilon}}_{\omega}}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))$. Since f_{p1u1} : (X,Ω_{ω},M) → (Y,ϒ_{ω},N) is soft continuous,
$${f}_{{p}_{1}{u}_{1}}^{-1}\left(In{t}_{{\mathrm{\Upsilon}}_{\omega}}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}({f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\right).$$Thus,
$$\begin{array}{l}{f}_{({p}_{2}\circ {p}_{1})\hspace{0.17em}({u}_{2}\circ {u}_{1})}^{-1}\hspace{0.17em}(G)\\ ={f}_{{p}_{1}{u}_{1}}^{-1}\left({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}(G)\right)\\ \tilde{\subseteq}{f}_{{p}_{1}{u}_{1}}^{-1}\left(In{t}_{{\mathrm{\Upsilon}}_{\omega}}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\tilde{\subseteq}In{t}_{{\mathrm{\Omega}}_{\omega}}({f}_{{p}_{1}{u}_{1}}^{-1}\hspace{0.17em}({f}_{{p}_{2}{u}_{2}}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\right)\\ =In{t}_{{\mathrm{\Omega}}_{\omega}}\hspace{0.17em}\left({f}_{({p}_{2}\circ {p}_{1})\hspace{0.17em}({u}_{2}\circ {u}_{1})}^{-1}\hspace{0.17em}({Cl}_{\mathrm{\Pi}}\hspace{0.17em}(G)))\right).\end{array}$$Therefore, from Theorem 2.2 (c), f_{(p2∘p1)(u2∘u1)} is soft ω-w-c.