Somewhat omega continuity and somewhat omega openness in soft topological spaces

1 Introduction and Preliminaries

In this paper, we follow the notions and terminologies presented in [1, 2]. Soft topological space and topological space will be referred to as STS and TS, respectively, throughout this paper. Molodtsov [3] has initiated soft sets as a new tool for dealing with uncertainty. Let Y be a universal set and E be a set of parameters. A soft set over Y relative to E is a mapping H : E ⟶ P ( Y ) . In this paper, SS(Y, E) will denote the collection of all soft sets over Y relative to E, 0 _E and 1 _E will denote the null soft set and the absolute soft set, respectively. STSs were defined in [4] as follows: A STS is a triplet (Y, δ, E), where δ ⊆ SS(Y, E), δ is closed under arbitrary soft union, δ is closed under finite soft intersection, and δ contains 0 _E and 1 _E . Let K be a soft set in (Y, δ, E). Then K is a soft open set in (Y, δ, E) if K ∈ δ and K is called a soft closed set in (Y, δ, E) if 1 _E  - K ∈ δ. Soft topology concepts and their applications are still a hot research topic ([1, 5–18]).

Somewhat continuous mappings and somewhat open mappings were introduced in [19] as weaker forms of semi-continuous mappings and semi-open mappings in TSs, respectively. These notions have proved to be very useful in topology [20–23]. The notions of somewhat continuous mappings and somewhat open mappings have been extended to include STSs in [24]. In this paper, we introduce soft somewhat ω-continuous soft mappings and soft somewhat ω-open soft mappings as two new classes of soft mappings. We characterize these two concepts. Also, we prove that the class of soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings contains the class of soft somewhat continuous (resp. soft somewhat open) soft mappings. Moreover, we obtain some sufficient conditions for the composition of two soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mappings to be a soft somewhat ω-continuous (resp. a soft somewhat ω-open) soft mapping. Furthermore, we introduce some sufficient conditions for restricting a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping to being a soft somewhat ω-continuous (resp. soft somewhat ω-open) soft mapping. In addition to these, we introduce extension theorems regarding soft somewhat ω-continuity and soft somewhat ω-openness. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs. In the next work, we hope to find an application for our new soft topological notions in a decision-making problem.

Let (Y, δ, E) be a STS, (Y, ℑ) be a TS, K ∈ SS(Y, E), and V ⊆ Y. Throughout this paper, Cl _δ (K), Int _δ (K), Cl_ℑ(V), and Int_ℑ(V) will denote the soft closure of K in (Y, δ, E), the soft interior of K in (Y, δ, E), the closure of V in (Y, ℑ), and the interior of V in (Y, ℑ), respectively.

The following definitions and results will be used in the sequel:

Definition 1.1. [25] Let (Y, ℑ) be a TS, D ⊆ Y, and y ∈ Y. Then y is a condensation point of D if for each V∈ ℑ with y ∈ V, the set V ∩ D is uncountable. D is called an ω-closed set in (Y, ℑ) if it contains all its condensation points. D is called an ω-open set in (Y, ℑ) if Y - D is an ω-closed set in (Y, ℑ). The family of all ω-open sets in (Y, ℑ) will be denoted by ℑ _ω.

Definition 1.2. A mapping g :(Y, ℑ) ⟶(Z, ℵ) between the TSs is said to be

(a) [21] somewhat continuous if for any V∈ ℵ such that g^-1(V)≠ ∅, there exists U∈ ℑ - { ∅ } such that U⊆ g^-1(V).

(b) [21] somewhat open if for any U∈ ℑ - { ∅ }, there is V∈ ℵ - { ∅ } such that V ⊆ g(U).

(c) [26] ω-continuous if g^-1(V) ∈ ℑ  _ω for all V∈ ℵ.

(d) [26] ω-open if g(U) ∈ ℵ  _ω for all U∈ ℑ.

Definition 1.3. [27] Let SS(Y, E) and SS(Z, D) be families of soft sets. Let p : Y ⟶ Z and u : E ⟶ D be functions. Then a mapping f _pu  : SS(Y, E) ⟶ SS(Z, D) is defined as:

(a) Let K ∈ SS(Y, E). The image of K under f _pu , written as f _pu (K) ∈ SS(Z, D) is defined by (f _pu (K))(d) =

{ ∪ { p ( K ( e ) ) : u ( e ) = d } if u - 1 ( d ) ≠ ∅ ∅ if u - 1 ( d ) = ∅ .

(b) Let G ∈ SS(Z, D). The inverse of G under f _pu , written as f pu - 1 ( G ) ∈ SS ( Y , E ) is defined by

( f pu - 1 ( G ) ) ( e ) = p - 1 ( G ( u ( e ) ) ) .

Definition 1.4. A soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D) is said to be

(a) [24] soft somewhat continuous if for any G ∈ β such that f pu - 1 ( G ) ≠ 0 E , there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) .

(b) [24] soft somewhat open if for every H∈ δ - { 0 _E  }, there exists G∈ β - { 0 _D  } such that G ⊆ ˜ f pu ( H ) .

(b) [5] soft ω-continuous if g^-1(G) ∈ δ _ω for all G ∈ β.

Definition 1.5. A STS (Y, σ, E) is said to be

(a) [28] soft separable if there exists a countable soft set F ∈ SS(Y, E) such that Cl _σ (F) =1 _E .

(b) [29] a soft D-space if every G∈ σ - { 0 _E  } is soft dense in (Y, σ, E).

Definition 1.6. A soft set K ∈ SS(Y, E) defined by

(a) [1] K ( e ) = { Z if e = a ∅ if e ≠ a is denoted by a _Z .

(b) [1] K(e) = Z for all e ∈ E is denoted by C _Z .

(c) [30] K ( e ) = { { y } if e = a ∅ if e ≠ a is denoted by a _y and is called a soft point. The set of all soft points in SS(Y, E) is denoted by SP(Y, E).

Definition 1.7. [30] Let K ∈ SS(Y, E) and e _y  ∈ SP(Y, E). Then e _y is said to belong to K (notation: e y ∈ ˜ K ) if y ∈ K(e).

Theorem 1.8. [31] For any TS (Y, ℑ), the family

{K∈ SS(Y, E) : K(e) ∈ ℑ forall e ∈ E }

is a soft topology on Y relative to E. This soft topology will be denoted by τ(ℑ).

2 Soft Somewhat ω-Continuity

Definition 2.1. A soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D) is said to be soft somewhat ω-continuous if for any G ∈ β such that f pu - 1 ( G ) ≠ 0 E , there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) .

Theorem 2.2. Every soft ω-continuous soft mapping is soft somewhat ω-continuous.

Proof. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be a soft ω-continuous soft mapping. Let G ∈ β and f pu - 1 ( G ) ≠ 0 E . Since f _pu is soft ω-continuous, then f pu - 1 ( G ) ∈ δ ω . Put H = f pu - 1 ( G ) . Then H∈ δ _ω - { 0 _E  } and H = f pu - 1 ( G ) ⊆ ˜ f pu - 1 ( G ) . Therefore, f _pu is soft somewhat ω-continuous.

The converse of Theorem 2.2 need not be true in general.

Example 2.3. Let Y = ℝ , Z ={ a, b }, μ be the usual topology on Y, E = ℕ , δ =

{F∈ SS(Y, E) : F(e) ∈ μ forall e ∈ E },

and β = SS(Z, E). Define p : Y ⟶ Z and u : E ⟶ E as follows:

p ( y ) = { a if y ≥ 1 b if y < 1 and u(e) = e for all e ∈ E.

Consider the soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, E). Clearly that f _pu is soft somewhat ω-continuous. On the other hand, since C_{a} ∈ β but f pu - 1 ( C { a } ) = C [ 1 , ∞ ) ∉ δ ω , then f _pu is not soft ω-continuous.

Theorem 2.4. Every soft somewhat continuous soft mapping is soft somewhat ω-continuous.

Proof. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be a soft somewhat continuous soft mapping. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E . Since f _pu is soft somewhat continuous, then there exists H∈ δ - { 0 _E  } and H ⊆ ˜ f pu - 1 ( G ) . By Theorem 2 of [2], δ ⊆ δ _ω and so H∈ δ _ω - { 0 _E  }. Therefore, f _pu is soft somewhat ω-continuous.

The converse of Theorem 2.4 need not be true in general.

Example 2.5. Let Y = ℝ and E = [0, 1]. Let δ =

{ F ∈ SS ( Y , E ) : F ( e ) ∈ { ∅ , ℝ , ℕ } for all e ∈ E }

and β =

{ F ∈ SS ( Y , E ) : F ( e ) ∈ { ∅ , ℝ , ℕ , ℝ - ℕ } for all e ∈ E } .

Define p : Y ⟶ Y and u : E ⟶ E as follows:

p(y) = y and u(e) = e for all y ∈ Y and e ∈ E.

Then the soft mapping f _pu  :(Y, δ, E) ⟶(Y, β, E) is soft somewhat ω-continuous but not soft somewhat continuous.

Theorem 2.6. Let p :(Y, ℑ) ⟶(Z, ℵ) be a mapping between two TSs and let u : E ⟶ D be an injective function between two sets of parameters. Then f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat continuous if and only if p :(Y, ℑ) ⟶(Z, ℵ) is somewhat continuous.

Proof. Necessity. Suppose that f _pu  :(Y, τ(ℑ) , E)

⟶(Z, τ(ℵ) , D) is soft somewhat continuous. Let V∈ ℵ such that p^-1(V)≠ ∅. Choose e ∈ E, then u(e)  _V  ∈ τ(ℵ). Also, since ( f pu - 1 ( u ( e ) V ) ) ( e ) = p - 1 ( ( u ( e ) V ) ( u ( e ) ) ) = p - 1 ( V ) ≠ ∅ , then f pu - 1 ( u ( e ) V ) ≠0 _E . Since f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat continuous, then there exists H∈ τ(ℑ) - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( u ( e ) V ) . Since H ≠ 0 _E , then there exists c ∈ E such that H(c)≠ ∅. Since u : E ⟶ D is injective, then f pu - 1 ( u ( e ) V ) ( b ) = p - 1 ( ( u ( e ) V ) ( u ( b ) ) ) = ∅ for all b∈ E - { e }. Since H ⊆ ˜ f pu - 1 ( u ( e ) V ) , then H(c)⊆ ( f pu - 1 ( u ( e ) V ) ) ( c ) , and so c = e. Therefore, we have H(e)∈ ℑ - { ∅ } and H(e) ⊆ V. It follows that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat continuous.

Sufficiency. Suppose that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat continuous. Let G ∈ τ(ℵ) such that f pu - 1 ( G ) ≠ 0 E . Choose e ∈ E such that ( f pu - 1 ( G ) ) ( e ) ≠∅. Since G ∈ τ(ℵ), then G(u(e))∈ ℵ. Since p - 1 ( G ( u ( e ) ) ) = ( f pu - 1 ( G ) ) ( e ) ≠ ∅ and p :(Y, ℑ) ⟶(Z, ℵ) is somewhat continuous, then there exists U∈ ℑ - { ∅ } such that p^-1(U) ⊆ p^-1(G(u(e))). Thus, we have e _U ∈ τ(ℑ) - { 0 _E  } and e U ⊆ ˜ f pu - 1 ( G ) . It follows that f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat continuous.

Theorem 2.7. Let p :(Y, ℑ) ⟶(Z, ℵ) be a mapping between two TSs and let u : E ⟶ D be an injective function between two sets of parameters. Then f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-continuous if and only if p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-continuous.

Proof. Necessity. Suppose that f _pu  :(Y, τ(ℑ) , E)

⟶(Z, τ(ℵ) , D) is soft somewhat ω-continuous. Let V∈ ℵ such that p^-1(V)≠ ∅. Choose e ∈ E, then u(e)  _V  ∈ τ(ℵ). Also, since ( f pu - 1 ( u ( e ) V ) ) ( e ) = p - 1 ( ( u ( e ) V ) ( u ( e ) ) ) = p - 1 ( V ) ≠ ∅ , then f pu - 1 ( u ( e ) V ) ≠0 _E . Since f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-continuous, then there exists H∈(τ(ℑ))  _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( u ( e ) V ) . Since H ≠ 0 _E , then there exists c ∈ E such that H(c)≠ ∅. Since u : E ⟶ D is injective, then f pu - 1 ( u ( e ) V ) ( b ) = p - 1 ( ( u ( e ) V ) ( u ( b ) ) ) = ∅ for all b∈ E - { e }. Since H ⊆ ˜ f pu - 1 ( u ( e ) V ) , then H(c)⊆ ( f pu - 1 ( u ( e ) V ) ) ( c ) , and so c = a. Also, since (τ(ℑ))  _ω = τ(ℑ  _ω), then H∈ τ(ℑ  _ω) - { 0 _E  }. Therefore, we have H(e)∈ ℑ  _ω - { ∅ } and H(e) ⊆ V. It follows that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-continuous.

Sufficiency. Suppose that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-continuous. Let G ∈ τ(ℵ) such that f pu - 1 ( G ) ≠ 0 E . Choose e ∈ E such that ( f pu - 1 ( G ) ) ( e ) ≠ ∅ . Since G ∈ τ(ℵ), then G(u(e))∈ ℵ. Since p - 1 ( G ( u ( e ) ) ) = ( f pu - 1 ( G ) ) ( e ) ≠ ∅ and p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-continuous, then there exists U∈ ℑ  _ω - { ∅ } such that p^-1(U) ⊆ p^-1(G(u(e))). Since (τ(ℑ))  _ω = τ(ℑ  _ω), then we have e _U ∈(τ(ℑ))  _ω - { 0 _E  } and e U ⊆ ˜ f pu - 1 ( G ) . It follows that f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-continuous.

Theorem 2.8. If f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) and f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) are soft somewhat continuous soft mappings and f _{p 1 u 1} (1 _E ) is soft dense in (Z, β, D), then f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat continuous.

Proof. Let K ∈ γ such that f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E . Then f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E and so f p 2 u 2 - 1 ( K ) ≠ 0 D . Since f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft somewhat continuous, then there exists H∈ β - { 0 _D  } such that H ⊆ ˜ f p 2 u 2 - 1 ( K ) . Since f _{p 1 u 1} (1 _E ) is soft dense in (Z, β, D), then H ∩ ˜ f p 1 u 1 ( 1 E ) ≠ 0 D and so f p 1 u 1 - 1 ( H ) ≠ 0 E . Since f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous, then there is M∈ δ - { 0 _E  } such that M ⊆ ˜ f p 1 u 1 - 1 ( H ) ⊆ ˜ f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) . This ends the proof.

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Theorem 2.9. If f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous and f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft continuous, then f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat continuous.

Proof. Let K ∈ γ such that f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E . Since f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft continuous, then f p 2 u 2 - 1 ( K ) ∈ β . Since f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous and f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E , then there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) . This ends the proof.

Not only is the soft composite of two soft somewhat continuous soft mappings not necessarily soft somewhat continuous, but f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) can be soft continuous and f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) does not have to be soft somewhat continuous.

Example 2.10. Let Y ={ 1, 2, 3 }, ℑ =

{∅ , Y, { 2 } , { 1, 3 }}, ℵ ={ ∅ , Y }, R = { ∅ , Y , { 1 , 2 } } , and E ={ a, b }. Consider the functions p₁, p₂ : Y ⟶ Y and u₁, u₂ : E ⟶ E where p₁(1) =1, p₁(2) = p₁(3) =3, and each of p₂, u₁, and u₂ is the identity function. It is not difficult to check that p₁ :(Y, ℑ) ⟶(Y, ℵ) is continuous and p 2 : ( Y , ℵ ) ⟶ ( Y , R ) is somewhat continuous but p 2 ∘ p 1 : ( Y , ℑ ) ⟶ ( Y , R ) is not somewhat continuous. Then by Theorem 5.31 of [1], f _{p 1 u 1}  :(Y, τ(ℑ) , E) ⟶(Y, τ(ℵ) , E) is soft continuous, also by Theorem 2.6, f p 2 u 2 : ( Y , τ ( ℵ ) , E ) ⟶ ( Y , τ ( R ) , E ) is soft somewhat continuous while f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) : ( Y , τ ( ℑ ) , E ) ⟶ ( Y , τ ( R ) , E ) is not soft somewhat continuous.

Theorem 2.11. If f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) and f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) are soft somewhat ω-continuous and f _{p 1 u 1} (1 _E ) is soft dense in (Z, β _ω, D), then f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat ω-continuous.

Proof. Let K ∈ γ such that f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E . Then f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E and so f p 2 u 2 - 1 ( K ) ≠ 0 D . Since f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft somewhat ω-continuous, then there exists H∈ β _ω - { 0 _D  } such that H ⊆ ˜ f p 2 u 2 - 1 ( K ) . Since f _{p 1 u 1} (1 _E ) is soft dense in (Z, β _ω, D), then H ∩ ˜ f p 1 u 1 ( 1 E ) ≠ 0 D and so f p 1 u 1 - 1 ( H ) ≠ 0 E . Since f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there is M∈ δ _ω - { 0 _E  } such that M ⊆ ˜ f p 1 u 1 - 1 ( H ) ⊆ ˜ f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) . This ends the proof.

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Theorem 2.12. If f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous and f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft continuous, then f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat ω-continuous.

Proof. Let K ∈ γ such that f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E . Since f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft continuous, then f p 2 u 2 - 1 ( K ) ∈ β . Since f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous and f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) ≠ 0 E , then there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f p 1 u 1 - 1 ( f p 2 u 2 - 1 ( K ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) - 1 ( K ) . This ends the proof.

The following example shows that the soft composition of two soft somewhat ω-continuous soft mappings need not be soft somewhat ω-continuous:

Example 2.13. Let Y = ℝ , Z ={ a, b, c }, ℑ = { ∅ , Y , ℝ - ℕ } , ℵ ={ ∅ , Z, { b } , { a, c }}, R = { ∅ , Z , { a , b } } , and E = ℕ . Define p₁ :(Y, ℑ) ⟶(Z, ℵ) by

p 1 ( y ) = { a if y ∈ ℕ b if y ∈ ( ℝ - ℕ ) - { - 1 } c if y = - 1 .

Let p 2 : ( Z , ℵ ) ⟶ ( Z , R ) and u₁, u₂ : E ⟶ E be the identities functions. Then p₁ is ω-continuous but not continuous, p₂ is somewhat ω-continuous but the composition p₂ ∘ p₁ : ( Y , ℑ ) ⟶ ( Z , R ) is not somewhat ω-continuous. Thus, by Corollary 2.6 of [5] and Theorem 5.31 of [1] f _{p 1 u 1}  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , E) is soft ω-continuous but not soft continuous. And by Theorem 2.7, f p 2 u 2 : ( Z , τ ( ℵ ) , E ) ⟶ ( Z , τ ( R ) , E ) is soft somewhat ω-continuous while f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) : ( Y , τ ( ℑ ) , E ) ⟶ ( Z , τ ( R ) , E ) is not soft somewhat ω-continuous.

Theorem 2.14. For a soft surjective mapping f _pu  :(Y, δ, E) ⟶(Z, β, D), the following three conditions are equivalent:

(a) f _pu is soft somewhat ω-continuous.

(b) If N is soft closed in (Z, β, D) with f pu - 1 ( N ) ≠ 1 E , then there exists a soft ω-closed set M of (Y, δ, E) such that M ≠ 1 _E and f pu - 1 ( N ) ⊆ ˜ M .

(c) If K is soft dense in (Y, δ _ω, E), then f _pu (K) is soft dense in (Z, β, D).

Proof. (a) ⇒ (b): Let N be soft closed in (Z, β, D) with f pu - 1 ( N ) ≠ 1 E . Then 1 _D  - N ∈ β and f pu - 1 ( 1 D - N ) = 1 E - f pu - 1 ( N ) ≠ 0 E . By (a), there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( 1 D - N ) = 1 E - f pu - 1 ( N ) . Put M = 1 _E  - H. Then M is soft ω-closed in (Y, δ, E), M ≠ 1 _E , and f pu - 1 ( N ) ⊆ ˜ M .

(b) ⇒ (c): Suppose to the contrary that there exists a soft dense set K in (Y, δ _ω, E) but f _pu (K) is soft not soft dense in (Z, β, D). Then Cl _β(f _pu (K)) ≠ 1 _D . Put N = Cl _σ (f _pu (K)). Then N is soft closed in (Z, β, D) with f pu - 1 ( N ) ≠ 1 E . So by (b), there exists a soft ω-closed set M of (Y, δ, E) such that M ≠ 1 _E and f pu - 1 ( N ) ⊆ ˜ M . Thus, we have K ⊆ ˜ f pu - 1 ( ( f pu ( K ) ) ) ⊆ ˜

f pu - 1 ( Cl β ( f pu ( K ) ) ) = f pu - 1 ( N ) ⊆ ˜ M , and so

Cl δ ω ( K ) ⊆ ˜ M ≠ 1 E . But K is soft dense in (Y, δ _ω, E), a contradiction.

(c) ⇒ (a): Suppose to the contrary that f _pu is not soft somewhat continuous. Then there exists G ∈ β such that f pu - 1 ( G ) ≠ 0 E but Int δ ( f pu - 1 ( G ) ) = 0 E . So, 1 E - f pu - 1 ( G ) is soft dense in (Y, δ, E). Thus, by (c), f pu ( 1 E - f pu - 1 ( G ) ) = 1 D - G is soft dense in (Z, β, D) and hence, G ∩ ˜ ( 1 D - G ) ≠ 0 D , a contradiction.

Theorem 2.15. If f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous and K ⊆ Y such that C _K is soft dense in (Y, δ, E), then the soft restriction is soft somewhat continuous.

Proof. Suppose that f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous and let K ⊆ Y such that C _K is soft dense in (Y, δ, E). Let G ∈ β such that , then which means that f pu - 1 ( G ) ≠ 0 E . So by soft somewhat continuity of f _pu , there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since C _K is soft dense in (Y, δ, E) and H∈ δ - { 0 _E  }, then H ∩ ˜ C K ≠ 0 E . Thus, we have H ∩ ˜ C K ∈ δ K - { 0 E } and H ∩ ˜ C K ⊆ ˜ . This shows that is soft somewhat continuous.

The following example shows that the condition ‘C _K is soft dense in (Y, δ, E)’ in Theorem 2.15 cannot be dropped:

Example 2.16. Let Y ={ 1, 2, 3 }, E = ℕ , δ ={ 0 _E , 1 _E , C_{1,2}, C_{3} }, and β ={ 0 _E , 1 _E , C_{2,3} }. Let p : Y ⟶ Y and u : E ⟶ E be the identities functions. Then f _pu  :(Y, δ, E) ⟶(Y, β, E) is soft somewhat continuous. On the other hand, if K ={ 1, 2 }, then is not soft somewhat continuous.

Theorem 2.17. If f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous and K ⊆ Y such that C _K is soft dense in (Y, δ _ω, E), then the soft restriction is soft somewhat ω-continuous.

Proof. Suppose that f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous and let K ⊆ Y such that C _K is soft dense in (Y, δ _ω, E). Let G ∈ β such that , then

= f pu - 1 ( G ) ∩ ˜ C K ≠ 0 E which means that f pu - 1 ( G ) ≠ 0 E . So by soft somewhat ω-continuity of f _pu , there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since C _K is soft dense in (Y, δ _ω, E) and H∈ δ _ω - { 0 _E  }, then H ∩ ˜ C K ≠ 0 E . Thus, we have H ∩ ˜ C K ∈ δ K - { 0 E } and H ∩ ˜ C K ⊆ ˜ . This shows that is soft somewhat ω-continuous.

The following example shows that the condition ’C _K is soft dense in (Y, δ _ω, E)’ in Theorem 2.17 cannot be dropped:

Example 2.18. Let Y = ℝ , E = ℕ , δ = { 0 E , 1 E , C ℚ , C ℝ - ℚ } , and β ={ 0 _E , 1 _E , C_{2,4} }. Define p : Y ⟶ Y by

p ( y ) = { 1 if y ∈ ℚ 2 if y ∈ ( ℝ - ℚ ) - { 3 } 5 if y = 3

and u : E ⟶ E be the identity function. Since f pu - 1 ( C { 2 , 4 } ) = C ( ℝ - ℚ ) - { 3 } ∈ δ ω , then f _pu  :(Y, δ, E) ⟶(Y, β, E) is ω-continuous and by Theorem 2.2, it is soft somewhat ω-continuous. Let K = ℝ - ℚ . If is soft ω-somewhat continuous, then there exists H∈(δ _K )  _ω - { 0 _E  } such that which is impossible.

Theorem 2.19. Let (Y, δ, E) and (Z, β, D) be two STSs and let Y = W ∪ M, where C _W , C _M  ∈ δ. If f _pu  :(Y, δ, E) ⟶(Z, β, D) is a soft mapping such that the restrictions and are soft somewhat continuous, then f _pu is soft somewhat continuous.

Proof. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E , then or . Without loss of generality we may assume that . Then there exists H∈ δ _W  - { 0 _E  } such that H ⊆ ˜ . Since C _W  ∈ δ, then H∈ δ - { 0 _E  } which ends the proof.

In Theorem 2.19, the condition ’C _W , C _M  ∈ δ’ cannot be dropped.

Example 2.20. Let Y ={ 1, 2, 3 }, E = ℕ , δ ={ 0 _E , 1 _E , C_{1}, C_{2,3} }, and β ={ 0 _E , 1 _E , C_{2} }. Let p : Y ⟶ Y and u : E ⟶ E be the identities functions. Let W ={ 1, 2 } and M ={ 1, 3 }. Then and are soft continuous and yet f _pu is not soft somewhat continuous.

Theorem 2.21. Let (Y, δ, E) and (Z, β, D) be two STSs and let Y = W ∪ M, where C _W , C _M  ∈ δ _ω. If f _pu  :(Y, δ, E) ⟶(Z, β, D) is a soft mapping such that the restrictions and are soft somewhat ω-continuous, then f _pu is soft somewhat ω-continuous.

Proof. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E , then or . Without loss of generality we may assume that . Then there exists H∈(δ _W )  _ω - { 0 _E  } and H ⊆ ˜ . Since C _W  ∈ δ _ω, then H∈ δ _ω - { 0 _E  } which ends the proof.

In Theorem 2.21, the condition ’C _W , C _M  ∈ δ _ω’ cannot be dropped:

Example 2.22. Let Y = ℝ , E = ℕ , δ = { 0 E , 1 E , C ℚ , C ℝ - ℚ } , and β = { 0 E , 1 E , C { 3 } } . Define p : Y ⟶ Y by

p ( y ) = { 3 if y = 2 1 if y ∈ ( ℝ - ℚ ) - { 2 } y if y ∈ ℚ

and let u : E ⟶ E be the identity function. Let W = { 2 } and M = ℝ - { 2 } . Then and are soft somewhat ω-continuous and yet f _pu is not soft somewhat ω-continuous.

Theorem 2.23. [24] Let (Y, δ, E) and (Z, β, D) be STSs and W⊆ Y such that C _W ∈ δ - { 0 _E  }. Let q : W ⟶ Z and u : E ⟶ D be functions such that f _qu  :(W, δ _W , E) ⟶(Z, β, D) is soft somewhat continuous and f _qu (C _W ) is soft dense in (Z, β, D). Then for any extension p : Y ⟶ Z of q, the soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous.

The next two examples show that neither C _W  is soft open in (Y, δ, E) nor f _qu (1 _W ) is soft dense in (Z, β, D) can be omitted in Theorem 2.23:

Example 2.24. Let Y ={ 1, 2 }, E = ℕ , δ ={ 0 _E , 1 _E , C_{1} }, β ={ 0 _E , 1 _E , C_{2} }, and W ={ 1 }. Then δ _W  ={ 0 _E , C _W  }. Define q : W ⟶ Y and u : E ⟶ E by q(1) = 1 and u(e) = e for all e ∈ E. Then f _qu  :(W, δ _W , E) ⟶(Y, β, E) is soft continuous and C _W  ∈ δ. Now if we take the extension p : Y ⟶ Y of q, where p(1) =1 and p(2) =2, then the soft mapping f _pu  :(Y, δ, E) ⟶(Y, β, E) is not soft somewhat continuous.

Example 2.25. Let Y ={ 1, 2 }, E = ℕ , δ ={ 0 _E , 1 _E  }, β ={ 0 _E , 1 _E , C_{1} }, and W ={ 1 }. Then δ _W  ={ 0 _E , C _W  }. Define q : W ⟶ Y and u : E ⟶ E by q(1) = 1 and u(e) = e for all e ∈ E. Then f _qu  :(W, δ _W , E) ⟶(Y, β, E) is soft continuous and f _qu (1 _W ) is soft dense in (Y, β, E). Now if we take the extension p : Y ⟶ Y of q, where p(1) =1 and p(2) =2, then the soft mapping f _pu  :(Y, δ, E) ⟶(Y, β, E) is not soft somewhat continuous.

Theorem 2.26. Let (Y, δ, E) and (Z, β, D) be STSs and W⊆ Y such that C _W ∈ δ _ω - { 0 _E  }. Let q : W ⟶ Z and u : E ⟶ D be functions such that f _qu  :(W, δ _W , E) ⟶(Z, β, D) is soft somewhat ω-continuous and f _qu (C _W ) is soft dense in (Z, β, D). Then for any extension p : Y ⟶ Z of q, the soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous.

Proof. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E . Since f _qu (C _W ) is soft dense in (Z, β, D), then G ∩ ˜ f qu ( C W )

≠0 _D . Choose e _w  ∈ SP(Y, E) such that e w ∈ ˜ f qu - 1 ( G ) . Since f _qu  :(W, δ _W , E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there exists H∈(δ _W )  _ω - { 0 _E  } such that H ⊆ ˜ f qu - 1 ( G ) ⊆ ˜ f pu - 1 ( G ) . Since C _W  ∈ δ _ω, then H∈ δ _ω - { 0 _E  } which ends the proof.

The following two examples demonstrate that neither C _W  is soft ω-open in (Y, δ, E) nor f _qu (1 _W ) is soft dense in (Z, β, D) may be deleted from Theorem 2.26:

Example 2.27. Let Y = ℝ , E = ℕ , δ ={ 0 _E , 1 _E , C_{1} }, β ={ 0 _E , 1 _E , C_{2} }, and W ={ 1 }. Then δ _W  ={ 0 _E , C _W  }. Define q : W ⟶ Y and u : E ⟶ E by q(1) = 1 and u(e) = e for all e ∈ E. Then f _qu  :(W, δ _W , E) ⟶(Y, β, E) is soft continuous and C _W  ∈ δ ⊆ δ _ω. Now if we take the extension p : Y ⟶ Y of q, where p(y) = y for all y ∈ Y, then the soft mapping f _pu  :(Y, δ, E) ⟶(Y, β, E) is not soft somewhat ω-continuous.

Example 2.28. Let Y = ℝ , E = ℕ , δ ={ 0 _E , 1 _E  }, β ={ 0 _E , 1 _E , C_{1} }, and W ={ 1 }. Then δ _W  ={ 0 _E , C _W  }. Define q : W ⟶ Y and u : E ⟶ E by q(1) = 1 and u(e) = e for all e ∈ E. Then f _qu  :(W, δ _W , E) ⟶(Y, β, E) is soft continuous and f _qu (1 _W ) is soft dense in (Y, β, E). Now if we take the extension p : Y ⟶ Y of q, where p(y) = y for all y ∈ Y, then the soft mapping f _pu  :(Y, δ, E) ⟶(Y, β, E) is not soft somewhat ω-continuous.

Definition 2.29. Let Y be a nonempty set and let E be a set of parameters. Let δ and β be soft topologies on Y relative to E. Then δ is said to be soft weakly equivalent to δ provided if for any G∈ δ - { 0 _E  }, there exists H∈ β - { 0 _E  } such that H ⊆ ˜ G and for any G∈ β - { 0 _E  }, there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ G .

Theorem 2.30. Let Y be a nonempty set and let E be a set of parameters. Let δ and β be soft topologies on Y relative to E. Let p : Y ⟶ Y and u : E ⟶ E be the identities functions. Then δ is soft weakly equivalent to β if and only if the soft mappings f _pu  :(Y, δ, E) ⟶(Y, β, E) and f _pu  :(Y, β, E) ⟶(Y, δ, E) are both soft somewhat continuous.

Proof. Straightforward.

Theorem 2.31. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat continuous. If (Y, γ, E) is a STS such that δ is soft weakly equivalent to γ, then f _pu  :(Y, γ, E) ⟶(Z, β, D) is soft somewhat continuous.

Proof. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous, then there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since δ is soft weakly equivalent to γ, then there exists M∈ γ - { 0 _E  } such that M ⊆ ˜ H ⊆ ˜ f pu - 1 ( G ) . This ends the proof.

Theorem 2.32. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat ω-continuous. If (Y, γ, E) is a STS such that δ _ω is soft weakly equivalent to γ _ω, then f _pu  :(Y, γ, E) ⟶(Z, β, D) is soft somewhat ω-continuous.

Proof. Let G ∈ β such that f pu - 1 ( G ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since δ _ω is soft weakly equivalent to γ _ω, then there exists M∈ γ _ω - { 0 _E  } such that M ⊆ ˜ H ⊆ ˜ f pu - 1 ( G ) . This ends the proof.

Theorem 2.33. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat continuous and surjective. If (Y, γ, E) and (Z, ρ, D) are STSs such that δ is soft weakly equivalent to γ and β is soft weakly equivalent to ρ, then f _pu  :(Y, γ, E) ⟶(Z, ρ, D) is soft somewhat continuous.

Proof. Let G ∈ ρ such that f pu - 1 ( G ) ≠ 0 E . Since β is soft weakly equivalent to ρ, then there exists M∈ β - { 0 _E  } such that M ⊆ ˜ G . Since f _pu is surjective, then f pu - 1 ( M ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous, then there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( M ) . Since δ is soft weakly equivalent to γ, then there exists K∈ γ - { 0 _E  } such that K ⊆ ˜ H ⊆ ˜ f pu - 1 ( M ) ⊆ ˜ f pu - 1 ( G ) . This ends the proof.

Theorem 2.34. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat ω-continuous and surjective. If (Y, γ, E) and (Z, ρ, D) are STSs such that δ _ω is soft weakly equivalent to γ _ω and β _ω is soft weakly equivalent to ρ _ω, then f _pu  :(Y, γ, E) ⟶(Z, ρ, D) is soft somewhat ω-continuous.

Proof. Let G ∈ ρ such that f pu - 1 ( G ) ≠ 0 E . Since β _ω is soft weakly equivalent to ρ _ω, then there exists M∈ β _ω - { 0 _E  } such that M ⊆ ˜ G . Since f _pu is surjective, then f pu - 1 ( M ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( M ) . Since δ _ω is soft weakly equivalent to γ _ω, then there exists K∈ γ _ω - { 0 _E  } such that K ⊆ ˜ H ⊆ ˜ f pu - 1 ( M ) ⊆ ˜ f pu - 1 ( G ) . This ends the proof.

Theorem 2.35. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat continuous and surjective, where u^-1(d) is countable for all d ∈ D. If (Y, δ, E) is soft separable, then (Z, β, D) is soft separable.

Proof. Suppose that f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous and surjective such that u^-1(d) is countable for all d ∈ D, and (Y, δ, E) is soft separable. Choose M ∈ SS(Y, E) such that M is soft countable and soft dense in (Y, δ, E). For every d ∈ D, u^-1(d) is nonempty and countable, and so (f _pu (M))(d) =∪ { p(M(e)) : u(e) = d } is countable. Hence, f _pu (M) is a countable soft set. To show that f _pu (M) is soft dense in (Z, β, D), let G∈ β - { 0 _D  }. Since f _pu is surjective, then f pu - 1 ( G ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous, then there exists H∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since M is soft dense in (Y, δ, E), then H ∩ ˜ M ≠ 0 E . Choose e y ∈ ˜ H ∩ ˜ M . Then e y ∈ ˜ H and e y ∈ ˜ M . Since e y ∈ ˜ H and H ⊆ ˜ f pu - 1 ( G ) , then e y ∈ ˜ f pu - 1 ( G ) and so f pu ( e y ) ∈ ˜ G . Since e y ∈ ˜ M , then f pu ( e y ) ∈ ˜ f pu ( M ) . Thus, f pu ( e y ) ∈ ˜ G ∩ ˜ f pu ( M ) . It follows that f _pu (M) is soft countable and soft dense in (Z, β, D). Hence, (Z, β, D) is soft separable.

Corollary 2.36. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft continuous and surjective, where u^-1(d) is countable for all d ∈ D. If (Y, δ, E) is soft separable, then (Z, β, D) is soft separable.

Theorem 2.37. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat ω-continuous and surjective, where u^-1(d) is countable for all d ∈ D. If (Y, δ _ω, E) is soft separable, then (Z, β, D) is soft separable.

Proof. Suppose that f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous and surjective such that u^-1(d) is countable for all d ∈ D, and (Y, δ _ω, E) is soft separable. Choose M ∈ SS(Y, E) such that M is soft countable and soft dense in (Y, δ _ω, E). For every d ∈ D, u^-1(d) is nonempty countable, and so (f _pu (M))(d) =∪ { p(M(e)) : u(e) = d } is countable. Hence, f _pu (M) is a countable soft set. To show that f _pu (M) is soft dense in (Z, β, D), let G∈ β - { 0 _D  }. Since f _pu is surjective, then f pu - 1 ( G ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there exists H∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) . Since M is soft dense in (Y, δ _ω, E), then H ∩ ˜ M ≠ 0 E . Choose e y ∈ ˜ H ∩ ˜ M . Then e y ∈ ˜ H and e y ∈ ˜ M . Since e y ∈ ˜ H and H ⊆ ˜ f pu - 1 ( G ) , then e y ∈ ˜ f pu - 1 ( G ) and so f pu ( e y ) ∈ ˜ G . Since e y ∈ ˜ M , then f pu ( e y ) ∈ ˜ f pu ( M ) . Thus, f pu ( e y ) ∈ ˜ G ∩ ˜ f pu ( M ) . It follows that f _pu (M) is soft countable and soft dense in (Z, β, D). Hence, (Z, β, D) is soft separable.

Corollary 2.38. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft ω-continuous and surjective, where u^-1(d) is countable for all d ∈ D. If (Y, δ _ω, E) is soft separable, then (Z, β, D) is soft separable.

Proof. Follows from Theorems 2.2 and 2.37.

Theorem 2.39. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat continuous and surjective. If (Y, δ, E) is a soft D-space, then (Z, β, D) is also a soft D-space.

Proof. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat continuous and surjective with (Y, δ, E) is a soft D-space. Suppose to the contrary that (Z, β, D) is not a soft D-space. Then there exist G, M∈ β - { 0 _D  } such that G ∩ ˜ M = 0 D . Since f _pu is surjective, then f pu - 1 ( G ) ≠ 0 E and f pu - 1 ( M ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat continuous, then there exist H, N∈ δ - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) and N ⊆ ˜ f pu - 1 ( M ) . Thus, H ∩ ˜ N ⊆ ˜ f pu - 1 ( G ) ∩ ˜ f pu - 1 ( M ) = f pu - 1 ( G ∩ ˜ M ) = f pu - 1 ( 0 D )

= 0 _E . This implies that (Y, δ, E) is not a soft D-space, a contradiction.

Corollary 2.40. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft continuous and surjective. If (Y, δ, E) is a soft D-space, then (Z, β, D) is also a soft D-space.

Theorem 2.41. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat ω-continuous and surjective. If (Y, δ _ω, E) is a soft D-space, then (Z, β, D) is also a soft D-space.

Proof. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat ω-continuous and surjective with (Y, δ _ω, E) is a soft D-space. Suppose to the contrary that (Z, β, D) is not a soft D-space. Then there exist G, M∈ β - { 0 _D  } such that G ∩ ˜ M = 0 D . Since f _pu is surjective, then f pu - 1 ( G ) ≠ 0 E and f pu - 1 ( M ) ≠ 0 E . Since f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-continuous, then there exist H, N∈ δ _ω - { 0 _E  } such that H ⊆ ˜ f pu - 1 ( G ) and N ⊆ ˜ f pu - 1 ( M ) . Thus, H ∩ ˜ N ⊆ ˜ f pu - 1 ( G ) ∩ ˜ f pu - 1 ( M ) = f pu - 1 ( G ∩ ˜ M ) = f pu - 1 ( 0 D ) = 0 E . This implies that (Y, δ _ω, E) is not a soft D-space, a contradiction.

Corollary 2.42. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft ω-continuous and surjective. If (Y, δ _ω, E) is a soft D-space, then (Z, β, D) is also a soft D-space.

Proof. Follows from Theorems 2.2 and 2.41.

3 Soft Somewhat ω-Open Soft mappings

Definition 3.1. A soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D) is said to be soft somewhat ω-open if for every H∈ δ - { 0 _E  }, there exists G∈ β _ω - { 0 _D  } such that G ⊆ ˜ f pu ( H ) .

Theorem 3.2. Every soft somewhat open soft mapping is soft somewhat ω-open.

Proof. Let f _pu  :(Y, δ, E) ⟶(Z, β, D) be soft somewhat open and let H∈ δ - { 0 _E  }. Since f _pu is soft somewhat open, then there is G∈ β - { 0 _D  } such that G ⊆ ˜ f pu ( H ) . By Theorem 2 of [2], β ⊆ β _ω and hence G∈ β _ω - { 0 _D  }. Therefore, f _pu  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat ω-open.

Corollary 3.3. Every soft open soft mapping is soft somewhat ω-open.

The converse of Theorem 3.2 need not be true in general.

Example 3.4. Let Y = ℝ , Z ={ a, b, c }, E = [0, 1], δ = { 0 E , 1 E , C ℚ } , and β ={ 0 _E , 1 _E , C_{a} }. Define p : Y ⟶ Y and u : E ⟶ E by

p ( y ) = { a if y ∈ ℚ b if y ∈ ℝ - ℚ and

u(e) = e for all e ∈ E.

Consider the soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, E). Then f _pu is soft somewhat ω-open but is not soft somewhat open.

Theorem 3.5. Let p :(Y, ℑ) ⟶(Z, ℵ) be a mapping between two TSs and let u : E ⟶ D be a function between two sets of parameters. Then f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat open if and only if p :(Y, ℑ) ⟶(Z, ℵ) is somewhat open.

Proof. Necessity. Suppose that f _pu  :(Y, τ(ℑ) , E)

⟶(Z, τ(ℵ) , D) is soft somewhat open. Let U∈ ℑ - { ∅ }. Choose e ∈ E. Then e _U ∈ τ(ℑ) - { 0 _E  }. Since f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat open, then there exists G∈ τ(ℵ) - { 0 _D  } such that G ⊆ ˜ f pu ( e U ) = u ( e ) p ( U ) . Choose d ∈ D such that G(d)≠ ∅. Then d = u(e) and G(d) ⊆ p(U). So we have G(u(e))∈ ℵ - { ∅ } with G(u(e)) ⊆ p(U). It follows that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat open.

Sufficiency. Suppose that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat open. Let H∈ τ(ℑ) - { 0 _E  }. Choose e ∈ E such that H(e)≠ ∅. Then we have H(e)∈ ℑ - { ∅ }. Since p :(Y, ℑ) ⟶(Z, ℵ) is somewhat open, then there exists V∈ ℵ - { ∅ } such that V ⊆ p(H(e)). Thus, we have u(e)  _V ∈ τ(ℵ) - { 0 _E  } and u ( e ) V ⊆ ˜ f pu ( H ) . It follows that f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat open.

Theorem 3.6. Let p :(Y, ℑ) ⟶(Z, ℵ) be a mapping between two TSs and let u : E ⟶ D be a function between two sets of parameters. Then f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-open if and only if p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-open.

Proof. Necessity. Suppose that f _pu  :(Y, τ(ℑ) , E)

⟶(Z, τ(ℵ) , D) is soft somewhat ω-open. Let U∈ ℑ - { ∅ }. Choose e ∈ E. Then e _U ∈ τ(ℑ) - { 0 _E  }. Since f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-open, then there exists G∈(τ(ℵ))  _ω - { 0 _D  } such that G ⊆ ˜ f pu ( e U ) = u ( e ) p ( U ) . Choose d ∈ D such that G(d)≠ ∅. Then d = u(e) and G(d) ⊆ p(U). So we have G(u(e))∈ ℵ  _ω - { ∅ } with G(u(e)) ⊆ p(U). It follows that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat open.

Sufficiency. Suppose that p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-open. Let H∈ τ(ℑ) - { 0 _E  }. Choose e ∈ E such that H(e)≠ ∅. Then we have H(e)∈ ℑ - { ∅ }. Since p :(Y, ℑ) ⟶(Z, ℵ) is somewhat ω-open, then there exists V∈ ℵ  _ω - { ∅ } such that V ⊆ p(H(e)). Thus, we have u(e)  _V ∈(τ(ℵ))  _ω - { 0 _E  } and u ( e ) V ⊆ ˜ f pu ( H ) . It follows that f _pu  :(Y, τ(ℑ) , E) ⟶(Z, τ(ℵ) , D) is soft somewhat ω-open.

Theorem 3.7. If f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat open and f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft somewhat ω-open, then f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat ω-open.

Proof. Let H∈ δ - { 0 _E  }. Since f _{p 1 u 1}  :(Y, δ, E) ⟶(Z, β, D) is soft somewhat open, then there exists G∈ β - { 0 _D  } such that G ⊆ ˜ f p 1 u 1 ( H ) . Since f _{p 2 u 2}  :(Z, β, D) ⟶(W, γ, S) is soft somewhat ω-open, then there exists M∈ γ _ω - { 0 _S  } such that M ⊆ ˜ f p 2 u 2 ( G ) ⊆ ˜ f p 2 u 2 ( f p 1 u 1 ( H ) ) = f ( p 2 ∘ p 1 ) ( u 2 ∘ u 1 ) ( H ) . Therefore, f_{(p2∘p1)(u2∘u1)} :(Y, δ, E) ⟶(W, γ, S) is soft somewhat ω-open.

Theorem 3.8. For a soft mapping f _pu  :(Y, δ, E) ⟶(Z, β, D), the following two conditions are equivalent:

(a) f _pu is soft somewhat ω-open.

(b) If N is a soft dense set in (Z, β _ω, D), then f pu - 1 ( N ) is a soft dense set in (Y, δ, E).

Proof. (a) ⇒ (b): Suppose to the contrary that there exists a soft dense set N in (Z, β _ω, D) such that f pu - 1 ( N ) is not a soft dense set in (Y, δ, E). Then 1 E - Cl δ ( f pu - 1 ( N ) ) ∈ δ - { 0 E } . So by (a), there exists G∈ β _ω - { 0 _D  } such that

G ⊆ ˜ f pu ( 1 E - Cl δ ( f pu - 1 ( N ) ) ) ⊆ ˜ f pu ( 1 E - f pu - 1 ( N ) ) = f pu ( f pu - 1 ( 1 D - N ) ) ⊆ ˜ 1 D - N .