Soft ω s -irresoluteness and soft pre- ω s -openness insoft topological spaces

Abstract

We use soft ω_s-open sets to define soft ω_s-irresoluteness, soft ω_s-openness, and soft pre-ω_s-openness as three new classes of soft mappings. We give several characterizations for each of them, specially via soft ω_s-closure and soft ω_s-interior soft operators. With the help of examples, we study several relationships regarding these three notions and their related known notions. In particular, we show that soft ω_s-irresoluteness is strictly weaker than soft ω_s-continuity, soft ω_s-openness lies strictly between soft openness and soft semi-openness, pre-ω_s-openness is strictly weaker than ω_s-openness, soft ω_s-irresoluteness is independent of each of soft continuity and soft irresoluteness, soft pre-ω_s-openness is independent of each of soft openness and soft pre-semi-openness, soft ω_s-irresoluteness and soft continuity (resp. soft irresoluteness) are equivalent for soft mappings between soft locally countable (resp. soft anti-locally countable) soft topological spaces, and soft pre-ω_s-openness and soft pre-semi-continuity are equivalent for soft mappings between soft locally countable soft topological spaces. Moreover, we study the relationship between our new concepts in soft topological spaces and their topological analog.

Keywords

soft irresolute soft mapping soft semi-open soft mapping soft pre-semi-open soft mapping

1 Introduction and preliminaries

We confront numerous situations in our daily lives that include uncertainty that we cannot solve with standard mathematical techniques. Some scenarios, such as fuzzy sets and soft sets, have been presented to cope with these sorts of challenges. In 1999, Molodtsov [1] developed the notion of “soft sets” as a novel mathematical technique for dealing with imprecise data and ambiguous circumstances. We confront numerous situations in our daily lives that include uncertainty that we cannot solve with standard mathematical techniques. Some scenarios, such as fuzzy sets and soft sets, have been presented to cope with these sorts of challenges. In 1999, Molodtsov [1] developed the notion of “soft sets” as a novel mathematical technique for dealing with imprecise data and ambiguous circumstances. Molodtsov discussed in detail how soft sets may be utilized in numerous areas and their benefits over fuzzy setsathematical techniques. Some scenarios, such as fuzzy sets and soft sets, have been presented to cope with these sorts of challenges. In 1999, Molodtsov [1] developed the notion of “soft sets” as a novel mathematical technique for dealing with imprecise data and ambiguous circumstances. Molodtsov discussed in detail how soft sets may be utilized in numerous areas and their benefits over fuzzy sets. This technique has sparked the curiosity of many academics and scholars who are interested in uncertainty in both theoretical and applied settings. Maji et al. [2] used soft sets to solve decision-making issues in 2002, and they [3] gave the initial conception for a set of operations between soft sets in 2003. Some of these operations had flaws, prompting some writers to revise their definitions and create new types for diverse purposes, as indicated in the published literature [4, 5]. Other mathematical structures, such as soft group theory [6], soft ring theory [7], and soft category theory [8], have been studied since Maji’s contribution.

Shabir and Naz [9] and Cagman et al. [10] presented two strategies for defining a topology over a family of soft sets in 2011. The distinction was in the form of a set of parameters: Shabir and Naz defined a soft topology over a constant set of parameters, whereas Cagman et al. defined a soft topology over changing subsets of parameters. According to Shabir and Naz’s definition, soft topological notions may be described similarly to their equivalents in classical topology, which encourages and stimulates scholars to pursue this area of research.

Since the introduction of soft topology, many researchers have focused on the generalization of topological concepts in soft topologies, such as “soft compact and soft Lindelof” [11 –15], “soft connected” [16], “soft paracompact” [16], “soft extremely disconnected” [17], “soft separable spaces” [18], “soft separation axioms” [19 –21], “soft metric spaces” [22 –24], and “soft homogeneous spaces” [25, 26]. Soft topology research is still ongoing (for example, [27 –38]), and substantial contributions are yet possible.

Soft continuity of mappings was defined by Nazmul and Samanta [39] in 2013. Then many modifications of “soft continuity” and “soft openness” appeared. For instance, “soft α-continuous mappings” [40], “soft semicontinuous mappings” [41], “soft β-continuous mappings” [42], soft ω-continuous mappings [43], soft ω_s-continuous functions [43], soft somewhat ω-continuous functions [44], soft semi ω-continuous [45], “soft ω-irresolute mappings” [45], “soft semi ω-irresolute mappings” [45], “soft α-open mappings” [40], “soft semi-open mappings” [41], “soft β-open mappings” [42], “soft somewhat ω-open mappings”, and so on.

The author in [46] defined soft ω_s-open sets as a class of soft sets that lies strictly between soft ω_s-open sets and soft semiopen sets, then he introduced in [43] soft ω_s-continuous mappings as a class of soft mappings that lies strictly between soft continuous mappings and soft semicontinuous mappings.

In this paper, We use soft ω_s-open sets to define soft ω_s-irresoluteness, soft ω_s-openness, and soft pre-ω_s-openness as three new classes of soft mappings. We give several characterizations for each of them, specially via soft ω_s-closure and soft ω_s-interior soft operators. With the help of examples, we study several relationships regarding these three notions and their related known notions. In particular, we show that soft ω_s-irresoluteness is strictly weaker than soft ω_s-continuity, soft ω_s-openness lies strictly between soft openness and soft semi-openness, pre-ω_s-openness is strictly weaker than ω_s-openness, soft ω_s-irresoluteness is independent of each of soft continuity and soft irresoluteness, soft pre-ω_s-openness is independent of each of soft openness and soft pre-semi-openness, soft ω_s-irresoluteness and soft continuity (resp. soft irresoluteness) are equivalent for soft mappings between soft locally countable (resp. soft anti-locally countable) soft topological spaces, and soft pre-ω_s-openness and soft pre-semi-continuity are equivalent for soft mappings between soft locally countable soft topological spaces. Moreover, we study the relationship between our new concepts in soft topological spaces and their topological analog.

Now, we shall recollect several notions that will be employed in the sequel of this paper.

From now on, TS will denote topological space. Let (Y, μ) be a TS A ⊆ Y, throughout this paper, Cl_μ (A) and Int_μ (A) will denote the closure of A in (Y, μ) and the interior of A in (Y, μ), respectively.

Definition 1.1. Let (Y, μ) be a TS, Z ⊆ Y, and y ∈ Y. Then

(1) [47] y is a condensation point of Z if for each O ∈ μ with y ∈ O, the set O ∩ Z is uncountable.

(2) [48] Z is called an ω-closed set in (Y, μ) if it contains all its condensation points.

(3) [48] Z is called an ω-open set in (Y, μ) if Y - Z is an ω-closed set in (Y, μ); the family of all ω-open sets in (Y, μ) will be denoted by μ_ω.

(4) [49] Z is called a semi-open set in (Y, μ) if there is O ∈ μ such that O ⊆ Z ⊆ Cl_μ (Z); the family of all semi-open sets in (Y, μ) will be denoted by SO (Y, μ).

(5) [50] Z is called an ω_s-open set in (Y, μ) if there is O ∈ μ such that O ⊆ Z ⊆ Cl_{μ_ω} (Z); the family of all ω_s-open sets in (Y, μ) will be denoted by ω_s (Y, μ).

(6) [50] Z is called an ω_s-closed set in (Y, μ) if Y - Z ∈ ω_s (Y, μ).

Definition 1.2. A mapping p : (V, μ) ⟶ (U, α) between the TSs (V, μ) and (U, α) is said to be

(1) [51] semi-open if p (U) ∈ SO (U, α) for every U ∈ μ.

(2) [52] pre-semi-open if p (U) ∈ SO (U, α) for every U ∈ SO (V, μ).

(3) [53] ω_s-irresolute if p^-1 (O) ∈ ω_s (V, μ) for every O ∈ ω_s (U, α).

(4) [53] ω_s-open if p (U) ∈ ω_s (U, α) for every U ∈ μ.

(5) [53] pre-ω_s-open if p (U) ∈ ω_s (U, α) for every U ∈ ω_s (V, μ).

Definition 1.3. [1] Let V be an initial universe and T be a set of parameters. A soft set over V relative to T is a function $H :, T ⟶ P (V)$ , where $P (V)$ is the powerset of V. The family of all soft sets over V relative to T will be denoted by SS (V, T).

Definition 1.4. [3] Let H ∈ SS (V, T).

(a) H is called a null soft set over V relative to T, denoted by 0_T, if H (t) =∅ for each t ∈ T.

(b) H is called an absolute soft set over V relative to T, denoted by 1_T, if H (t) = V for each t ∈ T.

Definition 1.5. [3] Let H, G ∈ SS (V, T).

(1) H is a soft subset of G, denoted by $H \tilde{\subseteq} G$ , if H (t) ⊆ G (t) for each t ∈ T.

(2) The soft union of H and G is denoted by $H \tilde{\cup} G$ and defined to be the soft set $H \tilde{\cup} G \in SS (V, T)$ where $(H \tilde{\cup} G) (t) = H (t) \cup G (t)$ for each t ∈ T.

(3) The soft intersection of H and G is denoted by $H \tilde{\cap} G$ and defined to be the soft set $H \tilde{\cap} G \in SS (V, T)$ where $(H \tilde{\cap} G) (t) = H (t) \cap G (t)$ for each t ∈ T.

(4) The soft difference of H and G is denoted by H - G and defined to be the soft set H - G ∈ SS (V, T) where (H - G) (t) = H (t) - G (t) for each t ∈ T.

Definition 1.6. [54] Let Γ be an arbitrary index set and {H_r : r ∈ Γ } ⊆ SS (V, T).

(a) The soft union of these soft sets is the soft set denoted by ${\tilde{\cup}}_{r \in Γ} S_{t}$ and defined by $({\tilde{\cup}}_{r \in Γ} H_{r}) (t) = \cup_{r \in Γ} H_{r} (t)$ for each t ∈ T.

(b) The soft intersection of these soft sets is the soft set denoted by ${\tilde{\cap}}_{t \in Γ} S_{t}$ and defined by $({\tilde{\cap}}_{r \in Γ} H_{r}) (t) = \cap_{r \in Γ} H_{r} (t)$ for each t ∈ T.

Definition 1.7. [11] Let f_{p
₁
u
₁} : SS (V, T) ⟶ SS (U, S) and f_{p
₂
u
₂} : SS (U, S) ⟶ SS (W, R) be mappings, then the composition of f_{p
₁
u
₁} and f_{p
₂
u
₂} is soft mapping from SS (V, T) onto SS (W, R) denoted by f_{p
₁
u
₁} ∘ f_{p
₂
u
₂}and defined by f_{p
₁
u
₁} ∘ f_{p
₂
u
₂} = f_{(p₂∘p₁)(u₂∘u₁)}.

Definition 1.8. [22] Let H ∈ SS (V, T). Then H is called a countable soft set if for every t ∈ T, the set H (t) is countable. The collection of all countable soft sets from SS (V, T) will be denoted by CSS (V, T).

Definition 1.9. A soft set K ∈ SS (V, T) defined by

(1) [25] $K (t) = {\begin{matrix} Z & if t = b \\ \emptyset & if t \neq b \end{matrix}$ is denoted by b_Z.

(2) [25] K (t) = T for all t ∈ T is denoted by C_Z.

(3) [22] $K (t) = {\begin{matrix} {y} & if t = b \\ \emptyset & if t \neq b \end{matrix}$ is denoted by b_y and is called a soft point. The set of all soft points in SS (V, T) is denoted by SP (V, T).

Definition 1.10. [22] Let K ∈ SS (V, T) and b_y ∈ SP (V, T). Then b_y is said to belong to K (notation: $b_{y} \tilde{\in} K$ ) if y ∈ K (b).

Definition 1.11. [55] Let SS (V, T) and SS (U, S) be families of soft sets. Let p : V ⟶ U and u : T ⟶ S be functions. Then a soft mapping f_pu : SS (V, T) ⟶ SS (U, S) is defined as:

(a) Let H ∈ SS (V, T). The image of H under f_pu, written as f_pu (H) ∈ SS (U, S) is defined by

$\begin{matrix} (f_{pu} (H)) (s) \\ = {\begin{matrix} \cup {p (H (t)) : u (t) = s} & if u^{- 1} (s) \neq \emptyset \\ \emptyset & if u^{- 1} (s) = \emptyset . \end{matrix} \end{matrix}$ (b) Let G ∈ SS (U, S). The inverse of G under f_pu, written as $f_{pu}^{- 1} (G) \in SS (V, T)$ is defined by

$\begin{matrix} (f_{pu}^{- 1} (G)) (t) = p^{- 1} (G (u (t))) . \end{matrix}$

Definition 1.12. [9] Let ψ ⊆ SS (V, T). Then ψ is called a soft topology on V relative to T if

(1) 0_T, 1_T ∈ ψ,

(2) the soft union of any number of soft sets in ψ belongs to ψ,

(3) the soft intersection of any two soft sets in ψ belongs to ψ.

The triplet (V, ψ, T) is called a soft topological space (STS) over V relative to T. The members of ψ are called soft open sets in (V, ψ, T) and their soft complements are called soft closed sets in (V, ψ, T).

Definition 1.13. [9] Let (V, ψ, T) be a STS and let N ∈ SS (V, T). Then

(1) the soft closure of N in (V, ψ, T) is denoted by Cl_ψ (N) and defined by Cl_ψ (H) =

$\tilde{\cap} {M : M is soft closed in (V, ψ, T) and N \tilde{\subseteq} M}$ .

(2) the soft interior of N in (V, ψ, T) is denoted by Int_ψ (N) and defined by Int_ψ (N) =

$\tilde{\cup} {K : K \in ψ and K \tilde{\subseteq} N}$ .

Definition 1.14. [11] A soft mapping f_pu : (V, ψ, T)⟶

(U, φ, S) is called

(1) soft continuous if $f_{pu}^{- 1} (G) \in ψ$ for every G ∈ φ.

(2) soft open if f_pu (H) ∈ φ for every H ∈ ψ.

Definition 1.15. [56] Let (V, ψ, T) be a STS and let N ∈ SS (V, T). Then N is called a soft ω-open set in (V, ψ, T) if for each $t_{v} \tilde{\in} N$ , there are G ∈ ψ and K ∈ CSS (V, T) such that $t_{v} \tilde{\in} G - K \tilde{\subseteq} N$ . The family of all soft ω-open set in (V, ψ, T) is denoted by ψ_ω.

It is proved in [56] that (V, ψ_ω, T) is a STS, ψ ⊆ ψ_ω, and ψ ≠ ψ_ω in general.

Theorem 1.16. [39]For any TS (V, μ), the family {K∈ SS (V, T) : K (t) ∈ μ foreveryt ∈ T } is a soft topology on V relative to T. This soft topology will be denoted by τ (μ).

Theorem 1.17. [25] For any collection of TSs { (V, ψ_t) : t∈ T }, the family

$\begin{matrix} {H \in SS (V, T) : H (t) \in ψ_{t} every t \in T} \end{matrix}$

forms a soft topology on V relative to T. This soft topology is denoted by ⊕_t∈Tψ_t.

Definition 1.18. Let (V, ψ, T) be a STS and let N ∈ SS (V, T). Then

(1) [57] N is called a soft semi-open set in (V, ψ, T) if there is G ∈ ψ such that $G \tilde{\subseteq} N \tilde{\subseteq} {Cl}_{ψ} (G)$ ; the family of all semi-open sets in (V, ψ, T) will be denoted by SO (V, ψ, T).

(2) [38] N is called a soft ω_s-open set in (V, ψ, T) if there is G ∈ ψ such that $G \tilde{\subseteq} N \tilde{\subseteq} {Cl}_{ψ_{ω}} (G)$ ; the family of all soft ω_s-open sets in (V, ψ, T) will be denoted by ω_s (V, ψ, T).

(3) [38] N is called a soft ω_s-closed set in (V, ψ, T) if 1_T - N ∈ ω_s (V, ψ, T).

(4) [38] the soft ω_s-closure of N in (V, ψ, T) is denoted by ω_s-Cl_ψ (N) and defined by ω_s-Cl_ψ (H) =

$\tilde{\cap} {M : M is soft ω_{s} - closed in (V, ψ, T) and N \tilde{\subseteq} M}$ .

(5) [38] the soft ω_s-interior of N in (V, ψ, T) is denoted by ω_s-int_ψ (N) and defined by ω_s-int_ψ (N) =

$\tilde{\cup} {K : K \in ω_{s} (V, ψ, T) and K \tilde{\subseteq} N}$ .

Definition 1.19. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called

(1) [41] soft irresolute if $f_{pu}^{- 1} (H) \in SO (V, ψ, T)$ for every H ∈ SO (U, φ, S).

(2) [41] soft semi-open if f_pu (K) ∈ SO (U, φ, S) for every K ∈ ψ.

(3) soft pre-semi-open if f_pu (K) ∈ SO (U, φ, S) for every K ∈ SO (V, ψ, T).

(4) [43] soft ω_s-continuous if $f_{pu}^{- 1} (H) \in ω_{s} (V, ψ, T)$ for every H ∈ φ.

Definition 1.20. A STS (V, ψ, T) is called

(1) [56] soft locally countable if for each t_v ∈ SP (V, T), there is H ∈ ψ ∩ CSS (V, T) such that $t_{v} \tilde{\in} H$ .

(2) [56] soft anti-locally countable if for every K∈ ψ - { 0_T }, K ∉ CSS (V, T).

(3) [58] soft ω-regular if for each t_v ∈ SP (V, T) and each G ∈ ψ such that $t_{v} \tilde{\in} G$ , there is M ∈ ψ such that $t_{v} \tilde{\in} M \tilde{\subseteq} {Cl}_{ψ_{ω}} (M) \tilde{\subseteq} G$ .

2 Soft ω_s-irresoluteness

Definition 2.1. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called soft ω_s-irresolute if for every M ∈ ω_s (U, φ, S), $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ .

Soft irresoluteness and soft ω_s-irresoluteness are independent of each other as it can be seen from the following two examples:

Example 2.2. Let $V = ℝ$ , U ={ 1, 2 }, $T = ℤ$ , $ψ = {C_{Z} : Z \in {\emptyset, V, ℕ, ℚ^{c}, ℕ \cup ℚ^{c}}}$ , and φ ={ C_Z : Z ∈ { ∅ , U, { 1 } , { 2 }} }. Define p : V ⟶ U and u : T ⟶ T by

$p (v) = {\begin{matrix} 1 & if v \in ℚ^{c} \\ 2 & if v \in ℚ \end{matrix}$ and u (t) = t for every t ∈ T.

Consider the soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, T). Since $f_{pu}^{- 1} (C_{{1}}) = C_{ℚ^{c}} \in ψ \subseteq SO (V, ψ, T)$ and $f_{pu}^{- 1} (C_{{2}}) = C_{ℚ} \in SO (V, ψ, T) - ω_{s} (V, ψ, T)$ , then f_pu is soft irresolute but not soft ω_s-irresolute.

Example 2.3. Let $V = ℕ$ , T = [0, 1], and ψ ={ C_Z : Z ∈ { ∅ , V, { 1 } , { 2 } , { 1, 2 }} }. Define p : V ⟶ V and u : T ⟶ T by p (v) =1 when v∈ { 1, 2 }, p (v) = v when $v \in ℕ - {1, 2}$ , and u (t) = t for every t ∈ T. Consider the soft mapping f_pu : (V, ψ, T) ⟶ (V, ψ, T). Since (V, ψ, T) is soft locally countable, then by Theorem 2.8 of [38], ω_s (V, ψ, T) = ψ. Since $f_{pu}^{- 1} (C_{{1}}) = C_{{1, 2}} \in ψ$ and $f_{pu}^{- 1} (C_{{2}}) = 0_{T} \in ψ$ , then f_pu is soft ω_s-irresolute. On the other hand, f_pu is not soft irresolute because there is ${Cl}_{ψ} (C_{{2}}) = C_{ℕ - {1}} \in SO (V, ψ, T)$ such that $f_{pu}^{- 1} (C_{ℕ - {1}}) = C_{ℕ - {1, 2}} \notin SO (V, ψ, T)$ .

Theorem 2.4. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft mapping such that (V, ψ, T) and (U, φ, S) are soft anti-locally countable STSs. Then f_pu is soft irresolute if and only if f_pu is soft ω_s-irresolute.

Proof. Follows from the definitions and Theorem 2.6 of [38].

Soft continuity and soft ω_s-irresoluteness are independent notions as the following examples show:

Example 2.5. Let $V = ℝ$ , $T = ℕ$ , and $ψ = {C_{X} : X \in {\emptyset, ℝ, (3, \infty), {2}, {2} \cup (3, \infty)}}$ . Define p : V ⟶ V and u : T ⟶ T by

$p (v) = {\begin{matrix} 2 & if v \in {2} \cup (3, \infty) \\ v & if v \in ℝ - ({2} \cup (3, \infty)) \end{matrix}$ and u (t) = t for every t ∈ T.

Consider the soft mapping f_pu : (V, ψ, T) ⟶ (V, ψ, T). Since $f_{pu}^{- 1} (C_{(3, \infty)}) = 0_{T} \in ψ$ and $f_{pu}^{- 1} (C_{{2}}) = C_{{2} \cup (3, \infty)} \in ψ$ , then f_pu is soft continuous. Since ${Cl}_{ψ_{ω}} (C_{(3, \infty)}) = C_{ℝ - {2}}$ , then $C_{ℝ - {2}} \in ω_{s} (V, ψ, T)$ . Since $f_{pu}^{- 1} (C_{ℝ - {2}}) = C_{ℝ - ({2} \cup (3, \infty))} \notin ω_{s} (V, ψ, T)$ , then f_pu is not soft ω_s-irresolute.

Example 2.6. Let $V = ℝ$ , U ={ 0, 1 }, $T = ℕ$ , ψ ={ C_Z : Z belongstotheusualtopologyon V }, and φ = SS (U, T). Define p : V ⟶ V and u : T ⟶ T by

$p (v) = {\begin{matrix} 1 & if v \in (- \infty, 0) \\ 2 & if v \in [0, \infty) \end{matrix}$ and u (t) = t for every t ∈ T.

Consider the soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, T). Note that ω_s (U, φ, T) = SS (U, T). Since $f_{pu}^{- 1} (C_{{1}}) = C_{(- \infty, 0)} \in ψ \subseteq ω_{s} (V, ψ, T)$ and $f_{pu}^{- 1} (C_{{2}}) = C_{[0, \infty)} \in ω_{s} (V, ψ, T)$ , then f_pu is soft ω_s-irresolute. However, f_pu is not soft continuous because there is C_{2} ∈ φ such that $f_{pu}^{- 1} (C_{{2}}) = C_{[0, \infty)} \notin ψ$ .

Theorem 2.7. Letf_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft mapping such that (V, ψ, T) and (U, φ, S) are soft locally countable STSs. Then f_pu is soft continuous if and only if f_pu is soft ω_s-irresolute.

Proof. Follows from the definitions and Theorem 2.8 of [38].

Theorem 2.8. Let f_pu : SS (V, T) ⟶ SS (U, S) be a soft mapping between two families of soft sets. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft continuous and f_pu : (V, ψ_ω, D) ⟶ (W, φ_ω, E) is soft open, then f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute.

Proof. Let M ∈ ω_s (U, φ, S). Then there is K ∈ φ such that $K \tilde{\subseteq} M \tilde{\subseteq} {Cl}_{φ_{ω}} (K)$ and thus $f_{pu}^{- 1} (K) \tilde{\subseteq} f_{pu}^{- 1} (M)$

$\tilde{\subseteq} f_{pu}^{- 1} ({Cl}_{φ_{ω}} (K))$ . Since f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft continuous, then $f_{pu}^{- 1} (K) \in ψ$ . Since f_pu : (V, ψ_ω, D) ⟶ (W, φ_ω, E) is soft open, then $f_{pu}^{- 1} ({Cl}_{φ_{ω}} (K)) \tilde{\subseteq}$

${Cl}_{ψ_{ω}} (f_{pu}^{- 1} (K))$ . Therefore, $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . It follows that f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute.

Theorem 2.9. Soft ω_s-irresolute soft mappings are soft ω_s-continuous.

Proof. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be soft ω_s-irresolute. Let M ∈ φ. Then by Theorem 2.2 of [38], M ∈ ω_s (U, φ, S). Since f_pu is soft ω_s-irresolute, then $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . Therefore, f_pu is soft ω_s-continuous.

The soft mapping in Example 2.5 is soft continuous and by Theorem 3.2 of [43], it is soft ω_s-continuous. Thus, the converse of Theorem 2.9 need not be true in general.

Theorem 2.10. A soft mappingf_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute if and only if for every soft ω_s-closed set N of (U, φ, S), $f_{pu}^{- 1} (N)$ is soft ω_s-closed in (V, ψ, T).

Proof. Necessity. Suppose that f_pu is soft ω_s-irresolute. Let N be a soft ω_s-closed set in (U, φ, S). Then 1_S - N ∈ ω_s (U, φ, S). Since f_pu is soft ω_s-irresolute, then $f_{pu}^{- 1} (1_{S} - N) = 1_{T} - f_{pu}^{- 1} (N) \in ω_{s} (V, ψ, T)$ . Therefore, $f_{pu}^{- 1} (N)$ is soft ω_s-closed in (V, ψ, T).

Sufficiency. Suppose that for every soft ω_s-closed subset N of (U, φ, S), $f_{pu}^{- 1} (N)$ is soft ω_s-closed in (V, ψ, T). Let M ∈ ω_s (U, φ, S). Then 1_S - M is soft ω_s-closed in (V, ψ, T). By assumption, $f_{pu}^{- 1} (1_{S} - M) = 1_{T} - f_{pu}^{- 1} (M)$ is soft ω_s-closed in (V, ψ, T), and thus $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . Hence, f_pu is soft ω_s-irresolute.

Theorem 2.11. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute if and only if for every K ∈ SS (V, T), $f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \tilde{\subseteq} ω_{s}$ -Cl_φ (f_pu (K)).

Proof. Necessity. Suppose that f_pu is soft ω_s-irresolute and let K ∈ SS (V, T). Then ω_s-Cl_φ (f_pu (K)) is soft ω_s-closed in (U, φ, S). Thus, by Theorem 2.10, $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (f_{pu} (K)))$ is soft ω_s-closed in (V, ψ, T). Since $K \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (f_{pu} (K)))$ , then ω_s- ${Cl}_{ψ} (K) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (f_{pu} (K)))$ . Thus,

$f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \tilde{\subseteq} f_{pu} (f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (f_{pu} (K))))$

$\tilde{\subseteq} ω_{s}$ -Cl_φ (f_pu (K)).

Sufficiency. Suppose that for every K ∈ SS (V, T), $f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \tilde{\subseteq} ω_{s}$ -Cl_φ (f_pu (K)). We will use Theorem 2.10 to show that f_pu is soft ω_s-irresolute. Let N be a soft ω_s-closed subset of (U, φ, S). Then by assumption we have $f_{pu} (ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (N))) \tilde{\subseteq}$

ω_s- ${Cl}_{φ} (f_{pu} (f_{pu}^{- 1} (N))) \tilde{\subseteq} ω_{s}$ -Cl_φ (N) = N, and thus

$\begin{matrix} ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (N)) \tilde{\subseteq} f_{pu}^{- 1} (N) . \end{matrix}$

Therefore, ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (N)) = f_{pu}^{- 1} (N)$ , and hence $f_{pu}^{- 1} (N)$ is soft ω_s-closed in (V, ψ, T). It follows that f_pu is soft ω_s-irresolute.

Theorem 2.12. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute if and only if for every H ∈ SS (U, S), ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (H)) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ .

Proof. Necessity. Suppose that f_pu is soft ω_s-irresolute and let H ∈ SS (U, S). Then by Theorem 2.10, $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ is soft ω_s-closed in (V, ψ, T). Since $f_{pu}^{- 1} (H) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ , then ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (H)) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ .

Sufficiency. Suppose that for every H ∈ SS (U, S), ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (H)) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ . We will use Theorem 2.10 to show that f_pu is soft ω_s-irresolute. Let N be a soft ω_s-closed subset of (U, φ, S). Then ω_s-Cl_φ (N) = N. So by assumption, ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (N)) \tilde{\subseteq}$

$f_{pu}^{- 1} (N)$ , and thus ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (N)) = f_{pu}^{- 1} (N)$ . Therefore, $f_{pu}^{- 1} (N)$ is soft ω_s-closed in (V, ψ, T).

Theorem 2.13. For a soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S), the following are equivalent:

(a) f_pu is soft ω_s-irresolute.

(b) For each t_v ∈ SP (V, T) and each M ∈ ω_s (U, φ, S) such that $f_{pu} (t_{v}) \tilde{\in} M$ , there is K ∈ ω_s (V, ψ, T) such that $t_{v} \tilde{\in} K$ and $f_{pu} (K) \tilde{\subseteq} M$ .

Proof. (a) ⇒ (b): Suppose that f_pu is soft ω_s-irresolute. Let M ∈ ω_s (U, φ, S) such that $f_{pu} (t_{v}) \tilde{\in} M$ . Since f_pu is soft ω_s-irresolute, then $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . Take $K = f_{pu}^{- 1} (M)$ . Then K ∈ ω_s (V, ψ, T) such that $t_{v} \tilde{\in} K$ and $f_{pu} (K) \tilde{\subseteq} M$ .

(b) ⇒ (a): Let M ∈ ω_s (U, φ, S). For every $t_{v} \tilde{\in} f_{pu}^{- 1} (M)$ we have $f_{pu} (t_{v}) \tilde{\in} M$ , and by (b), there is K_{t
_v} ∈ ω_s (V, ψ, T) such that $t_{v} \tilde{\in} K_{t_{v}}$ and $f_{pu} (K_{t_{v}}) \tilde{\subseteq} M$ . Therefore, $f_{pu}^{- 1} (M) = {\tilde{\cup}}_{t_{v} \tilde{\in} f_{pu}^{- 1} (M)} K_{t_{v}}$ . Hence, by Theorem 2.11 of [38], $f_{pu}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . It follows that f_pu is soft ω_s-irresolute.

Theorem 2.14. For a soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S), the following are equivalent:

(a) f_pu is soft ω_s-irresolute.

(b) For each K ∈ SS (V, T), $f_{pu} ({Int}_{ψ_{ω}} ({Cl}_{ψ} (K)) \tilde{\subseteq}$

ω_s-Cl_φ (f_pu (K)).

$\tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ .

Proof. (a) ⇒ (b): Suppose that f_pu is soft ω_s-irresolute. Let K ∈ SS (V, T). Then by Theorem 2.11, $f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \tilde{\subseteq} ω_{s}$ -Cl_φ (f_pu (K)). Thus, by Lemma 3.9 of [43], it follows that

$\begin{matrix} f_{pu} ({Int}_{ψ_{ω}} ({Cl}_{ψ} (K)) & \tilde{\subseteq} & f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \\ \tilde{\subseteq} & ω_{s} - {Cl}_{φ} (f_{pu} (K)) . \end{matrix}$

(b) ⇒ (a): We will apply Theorem 2.11. Let K ∈ SS (V, T). Then by (b),

$\begin{matrix} f_{pu} ({Int}_{ψ_{ω}} ({Cl}_{ψ} (K)) \tilde{\subseteq} ω_{s} - {Cl}_{φ} (f_{pu} (K)) . \end{matrix}$

Thus, by Lemma 3.9 of [43], $\begin{matrix} f_{pu} (ω_{s} - {Cl}_{ψ} (K)) \\ = f_{pu} (K \tilde{\cup} {Int}_{ψ_{ω}} ({Cl}_{ψ} (K))) \\ = f_{pu} (K) \tilde{\cup} f_{pu} ({Int}_{ψ_{ω}} ({Cl}_{ψ} (K))) \\ \tilde{\subseteq} ω_{s} - {Cl}_{φ} (f_{pu} (K)) . \end{matrix}$

(a) ⇒ (c): Suppose that f_pu is soft ω_s-irresolute and let H ∈ SS (U, S). Then by Theorem 2.12, ω_s- ${Cl}_{ψ} (f_{pu}^{- 1} (H)) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ . Therefore, by Lemma 3.9 of [43], we have

$\begin{matrix} {Int}_{ψ_{ω}} ({Cl}_{ψ} (f_{pu}^{- 1} (H))) & \tilde{\subseteq} & ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (H)) \\ \tilde{\subseteq} & f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H)) . \end{matrix}$

${Int}_{ψ_{ω}} ({Cl}_{ψ} (f_{pu}^{- 1} (H))) \tilde{\subseteq} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H))$ . Thus, by Lemma 3.9 of [43], we have

$\begin{matrix} ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (H)) & = & f_{pu}^{- 1} (H) \tilde{\cup} {Int}_{ψ_{ω}} ({Cl}_{ψ} (f_{pu}^{- 1} (H))) \\ \tilde{\subseteq} & f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (H)) . \end{matrix}$

Theorem 2.15. Iff_{p
₁
u
₁} : (V, ψ, T) ⟶ (U, φ, S) and f_{p
₂
u
₂} : (U, φ, S) ⟶ (W, ρ, R) are soft ω_s-irresolute mappings, then f_{(p₂∘p₁)(u₂∘u₁)} : (V, ψ, T) ⟶ (W, ρ, R) is soft ω_s-irresolute.

Proof. Let M ∈ ω_s (W, ρ, R). Since f_{p
₂
u
₂} : (U, φ, S) ⟶ (W, ρ, R) is soft ω_s-irresolute, then $f_{p_{2} u_{2}}^{- 1} (M) \in ω_{s} (U, φ, S)$ . Since f_{p
₁
u
₁} : (V, ψ, T) ⟶ (U, φ, S) is soft ω-irresolute, then $f_{p_{1} u_{1}}^{- 1} (f_{p_{2} u_{2}}^{- 1} (M)) = f_{(p_{2} \circ p_{1}) (u_{2} \circ u_{1})}^{- 1} (M) \in ω_{s} (V, ψ, T)$ . It follows that f_{(p₂∘p₁)(u₂∘u₁)} : (V, ψ, T) ⟶ (W, ρ, R) is soft ω_s-irresolute.

Theorem 2.16. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U and u : T ⟶ S be functions. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-irresolute if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-irresolute for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-irresolute. Let t ∈ T. Let Z ∈ ω_s (U, φ_u(t)). Then by Theorem 4.8 of [38], (u (t)) _Z ∈ ω_s (U, ⊕ _s∈Sφ_s, S) and so $f_{pu}^{- 1} {(u (t))}_{Z} \in ω_{s} (V, \oplus_{t \in T} ψ_{t}, T)$ . Thus, by Theorem 4.8 of [38], $(f_{pu}^{- 1} ({(u (t))}_{Z})) (t) = p^{- 1} (({(u (t))}_{Z}) (u (t))) = p^{- 1} (Z) \in ω_{s} (V, ψ_{t})$ . It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-irresolute.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-irresolute for all t ∈ T. We will apply Theorem 2.13. Let t_v ∈ SP (V, T) and let M ∈ ω_s (U, φ, S) such that $f_{pu} (t_{v}) \tilde{\in} M$ . Then we have ${(u (t))}_{p (v)} \tilde{\in} M$ and so p (v) ∈ M (u (t)). Also, by Theorem 4.8 of [38], M (u (t)) ∈ ω_s (U, φ_u(t)). Since p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-irresolute, then there is X ∈ ω_s (V, ψ_t) such that v ∈ X and $p (X) \tilde{\subseteq} M (u (t))$ . Also, by Theorem 4.8 of [38], we have t_X ∈ ω_s (V, ⊕ _t∈Tψ_t, T). Therefore, we have $t_{v} \tilde{\in} t_{X} \in ω_{s} (V, \oplus_{t \in T} ψ_{t}, T)$ and $f_{pu} (t_{X}) \tilde{\subseteq} M$ . It follows that f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-irresolute.

Corollary 2.17. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be a function. Then p : (V, μ) ⟶ (U, α) is ω_s-irresolute if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft ω_s-irresolute.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈T μ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 2.16, we get the result.

3 Soft ω_s-open soft mappings

Definition 3.1. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called soft ω_s-open if for each K ∈ ψ, f_pu (K) ∈ ω_s (U, φ, S).

Theorem 3.2. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft mapping. If for a soft base $A$ of (V, ψ, T), f_pu (A) ∈ ω_s (U, φ, S) for each $A \in A$ , then f_pu is soft ω_s-open.

Proof. Suppose that for a soft base $A$ of (V, ψ, T), f_pu (A) ∈ ω_s (U, φ, S) for each $A \in A$ . Let K∈ ψ - { 0_T }. Choose $A_{1} \subseteq A$ such that $K = {\tilde{\cup}}_{A \in A_{1}} A$ . Then

$f_{pu} (K) = f_{pu} ({\tilde{\cup}}_{A \in A_{1}} A) = {\tilde{\cup}}_{A \in A_{1}} f_{pu} (A)$ . Since by assumption f_pu (A) ∈ ω_s (U, φ, S) for all $A \in A_{1}$ , then by Theorem 2.11 of [38], $f_{pu} (K) = {\tilde{\cup}}_{A \in A_{1}} f_{pu} (A) \in ω_{s} (U, φ, S)$ . Therefore, f_pu is soft ω_s-open.

Theorem 3.3. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U and u : T ⟶ S be functions. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft open if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is open for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft open. Let t ∈ T. Let X ∈ ψ_t. Then t_X ∈ ⊕ _t∈Tψ_t and so f_pu (t_X) = (u (t)) _p(X) ∈ ⊕ _s∈Sφ_s. Thus, ((u (t)) _p(X)) (u (t)) = p (X) ∈ φ_u(t). It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is open.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is open for all t ∈ T. Let $A = {t_{X} : t \in T and X \in ψ_{t}}$ . By Theorem 3.5 of [25], $A$ is a soft base for (V, ⊕ _t∈Tψ_t, T). We will show that f_pu (A) ∈ ⊕ _s∈Sφ_s for all $A \in A$ . Let $t_{X} \in A$ where t ∈ T and X ∈ ψ_t. Since p : (V, ψ_t) ⟶ (U, φ_u(t)) is open, then p (X) ∈ φ_u(t). Thus, f_pu (t_X) = (u (t)) _p(X) ∈ ⊕ _s∈Sφ_s. It follows that f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft open.

Corollary 3.4. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be a function. Then p : (V, μ) ⟶ (U, α) is open if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft open.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈Tμ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 3.3, we get the result.

Theorem 3.5. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U and u : T ⟶ S be functions. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-open if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-open for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-open. Let t ∈ T. Let X ∈ ψ_t. Then t_X ∈ ⊕ _t∈Tψ_t, and so f_pu (t_X) = (u (t)) _p(X) ∈ ω_s (U, ⊕ _s∈Sφ_s, S). Thus, by Theorem 4.8 of [38], ((u (t)) _p(X)) (u (t)) = p (X) ∈ ω_s (W, φ_u(t)). It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-open.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-open for all t ∈ T. Let $A = {t_{X} : t \in T and X \in ψ_{t}}$ . By Theorem 3.5 of [25], $A$ is a soft base for (V, ⊕ _t∈Tψ_t, T). We will apply Theorem 3.2. Let $t_{X} \in A$ where t ∈ T and X ∈ ψ_t. Since p : (V, ψ_t) ⟶ (U, φ_u(t)) is ω_s-open, then p (X) ∈ ω_s (U, φ_u(t)). Thus, by Theorem 4.8 of [38], f_pu (t_X) = (u (t)) _p(X) ∈ ω_s (U, ⊕ _s∈Sφ_s, S). It follows that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft ω_s-open.

Corollary 3.6. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be a function. Then p : (V, μ) ⟶ (U, α) is ω_s-open if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft ω_s-open.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈T μ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 3.5, we get the result.

Theorem 3.7. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U and u : T ⟶ S be functions. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft semi-open if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is semi-open for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft semi-open. Let t ∈ T. Let X ∈ ψ_t. Then t_X ∈ ⊕ _t∈Tψ_t and so f_pu (t_X) = (u (t)) _p(X) ∈ SO (U, ⊕ _s∈Sφ_s, S). Thus, by Theorem 4.10 of [38], ((u (t)) _p(X)) (u (t)) = p (X) ∈ SO (U, φ_u(t)). It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is semi-open.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is semi-open for all t ∈ T. Let $A = {t_{X} : t \in T and X \in ψ_{t}}$ . By Theorem 3.5 of [25], $A$ is a soft base for (V, ⊕ _t∈Tψ_t, T). We will show that f_pu (A) ∈ SO (U, ⊕ _s∈Sφ_s, S) for all $A \in A$ . Let $t_{X} \in A$ where t ∈ T and X ∈ ψ_t. Since p : (V, ψ_t) ⟶ (U, φ_u(t)) is semi-open, then p (X) ∈ SO (U, φ_u(t)). Thus, by Theorem 4.10 of [38], f_pu (t_X) = (u (t)) _p(X) ∈ SO (U, ⊕ _s∈Sφ_s, S). It follows that f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft semi-open.

Corollary 3.8. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be a function. Then p : (V, μ) ⟶ (U, α) is semi-open if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft semi-open.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈T μ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 3.7, we get the result.

Theorem 3.9. Every soft open soft mapping is soft ω_s-open.

Proof. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft open soft mapping and let K ∈ ψ. Since f_pu is soft open, then f_pu (K) ∈ φ. Thus, by Theorem 2.2 of [38], f_pu (K) ∈ ω_s (U, φ, S). Hence, f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-open.

As the following example proves, the implication in Theorem 3.9 is not reversible in general:

Example 3.10. Let $V = ℝ$ , $T = ℕ$ , and μ be the usual topology on V. Define p : V ⟶ V and u : T ⟶ T by p (v) = v² and u (t) = t for every t ∈ T. Then by Example 3.4 of [53], p : (V, μ) ⟶ (V, μ) is ω_s-open but not open. Therefore, by Corollaries 3.4 and 3.6, f_pu : (V, τ (μ) , D) ⟶ (V, τ (μ) , D) is soft ω_s-open but not soft open.

Theorem 3.11. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is a soft ω_s-open mapping such that (U, φ, S) is soft locally countable, then f_pu is soft open.

Proof. Let K ∈ ψ. Since f_pu is soft ω_s-open, then f_pu (K) ∈ ω_s (U, φ, S). Since (U, φ, S) is soft locally countable, then by Theorem 2.8 of [38], f_pu (K) ∈ φ. Hence, f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft open.

Theorem 3.12. Every soft ω_s-open soft mapping is soft semi-open.

Proof. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft ω_s-open mapping and let K ∈ ψ. Since f_pu is soft ω_s-open, then f_pu (K) ∈ ω_s (U, φ, S). Thus, by Theorem 2.2 of [38], f_pu (K) ∈ SO (U, φ, S). Hence, f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft semi-open.

The converse of Theorem 3.12 need not be true in general:

Example 3.13. Let $V = ℝ$ , $T = ℕ$ , and $μ = . {\emptyset, ℕ, ℝ}$ . Define p : V ⟶ V and u : T ⟶ T by p (v) = v - 1 and u (t) = t for every t ∈ T. Then by Example 3.7 of [53], p : (V, μ) ⟶ (V, μ) is semi-open but not ω_s-open. Therefore, by Corollaries 3.6 and 3.8, f_pu : (V, τ (μ) , T) ⟶ (V, τ (μ) , T) is soft semi-open but not soft ω_s-open.

Theorem 3.14. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is a soft semi-open mapping such that (U, φ, S) is soft anti-locally countable, then f_pu is soft ω_s-open.

Proof. Let K ∈ ψ. Since f_pu is soft semi-open, then f_pu (K) ∈ SO (U, φ, S). Since (U, φ, S) is soft anti-locally countable, then by Theorem 2.6 of [38], f_pu (K) ∈ ω_s (U, φ, S). Hence, f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-open.

Theorem 3.15. For a soft mapping f_pu : (V, ψ, T)

⟶ (U, φ, S), the following conditions are equivalent:

(a) f_pu is soft ω_s-open.

(b) $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq} {Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S).

Proof. (a) ⇒ (b): Suppose that f_pu is soft ω_s-open and let N ∈ SS (U, S). Let $t_{v} \tilde{\in} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N))$ . Then $f_{pu} (t_{v}) \tilde{\in} ω_{s}$ -Cl_φ (N). To show that $t_{v} \tilde{\in} {Cl}_{ψ} (f_{pu}^{- 1} (N))$ , let S ∈ ψ such that $t_{v} \tilde{\in} S$ . Since f_pu is soft ω_s-open, then f_pu (S) ∈ ω_s (U, φ, S). Since $f_{pu} (t_{v}) \tilde{\in} f_{pu} (S) \tilde{\cap} ω_{s}$ -Cl_φ (N), then $f_{pu} (S) \tilde{\cap} N \neq 0_{S}$ . Pick $a_{x} \tilde{\in} S$ such that $f_{pu} (a_{x}) \tilde{\in} N$ . Then $a_{x} \tilde{\in} S \cap f_{pu}^{- 1} (N)$ . Therefore, $t_{v} \tilde{\in} {Cl}_{ψ} (f_{pu}^{- 1} (N))$ .

(b) ⇒ (a): Suppose that $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq}$

${Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S), and suppose to the contrary that f_pu is not soft ω_s-open. Then there is K ∈ ψ such that f_pu (K) ∉ ω_s (U, φ, S) and so, 1_S - f_pu (K) is not soft ω_s-closed. So, we find $e_{w} \tilde{\in} f_{pu} (K) \tilde{\cap} ω_{s}$ -Cl_φ (1_S - f_pu (K)). Choose $b_{y} \tilde{\in} K$ such that e_w = f_pu (b_y). Then $b_{y} \tilde{\in} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - f_{pu} (K)))$ . By assumption, we have $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - f_{pu} (K)))$

$\begin{matrix} \tilde{\subseteq} & {Cl}_{ψ} (f_{pu}^{- 1} (1_{S} - f_{pu} (K))) \\ = & {Cl}_{ψ} (1_{T} - f_{pu}^{- 1} (f_{pu} (K))) \\ \tilde{\subseteq} & {Cl}_{ψ} (1_{T} - K) \\ = & 1_{T} - K . \end{matrix}$

Therefore, $b_{y} \tilde{\in} 1_{T} - K$ . But $b_{y} \tilde{\in} K$ , a contradiction.

(b) ⇒ (c): Suppose that $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq}$

${Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S). Let N ∈ SS (U, S). Then by (b),

$\begin{matrix} f_{pu}^{- 1} (ω_{s} - Int (N)) & = & f_{pu}^{- 1} (1_{S} - ω_{s} - {Cl}_{φ} (1_{S} - N)) \\ = & 1_{T} - f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - N)) \\ \tilde{\supseteq} & 1_{T} - {Cl}_{ψ} (f_{pu}^{- 1} (1_{S} - N)) \\ = & 1_{T} - {Cl}_{ψ} (1_{T} - f_{pu}^{- 1} (N)) \\ = & {Int}_{ψ} (f_{pu}^{- 1} (N)) . \end{matrix}$

$f_{pu}^{- 1} (ω_{s} - {Int}_{φ} (N))$ for every N ∈ SS (U, S). Let N ∈ SS (U, S). Then by (c),

$\begin{matrix} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) & = & f_{pu}^{- 1} (1_{S} - ω_{s} - {Int}_{φ} (1_{S} - N)) \\ = & 1_{T} - f_{pu}^{- 1} (ω_{s} - {Int}_{φ} (1_{S} - N)) \\ \tilde{\subseteq} & 1_{T} - {Int}_{ψ} (f_{pu}^{- 1} (1_{S} - N)) \\ = & 1_{T} - {Int}_{ψ} (1_{T} - f_{pu}^{- 1} (N)) \\ = & {Cl}_{ψ} (f_{pu}^{- 1} (N)) . \end{matrix}$

4 Soft Pre-ω_s-open

Definition 4.1. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called soft pre-ω_s-open if for each K ∈ ω_s (V, ψ, T), f_pu (K) ∈ ω_s (U, φ, S).

Theorem 4.2. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U be a function between two sets and let u : T ⟶ S be an injective function between two sets of parameters. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-ω_s-open if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-ω_s-open for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-ω_s-open. Let t ∈ T. Let X ∈ ω_s (V, ψ_t). Then Theorem 4.8 of [38], t_X ∈ ω_s (V, ⊕ _t∈Tψ_t, T) and so f_pu (t_X) = (u (t)) _p(X) ∈ ω_s (U, ⊕ _s∈Sφ_s, S). Thus, by Theorem 4.8 of [38], ((u (t)) _p(X)) (u (t)) = p (X) ∈ ω_s (W, φ_u(t)). It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-ω_s-open.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-ω_s-open for all t ∈ T. Let K ∈ ω_s (V, ⊕ _t∈Tψ_t, T). Then by Theorem 4.8 of [38], K (t) ∈ ω_s (V, ψ_t) for all t ∈ T. Therefore, p (K (t)) ∈ ω_s (U, φ_u(t)) for all t ∈ T. Thus, we have (f_pu (K)) (e) = ∅ ∈ ω_s (U, φ_s) if u^-1 ({ e }) = ∅ or (f_pu (K)) (e) = p (K (t)) ∈ ω_s (U, φ_s) if u (t) = e. Hence, by Theorem 4.8 of [38], f_pu (K) ∈ ω_s (U, ⊕ _s∈Sφ_s, S). It follows that f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-ω_s-open.

Corollary 4.3. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be an injective function between two sets of parameters. Then p : (V, μ) ⟶ (U, α) is pre-ω_s-open if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft pre-ω_s-open.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈T μ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 4.2, we get the result.

Theorem 4.4. Let { (V, ψ_t) : t∈ T } and

{ (U, φ_s) : s∈ S } be two families of TSs. Let p : V ⟶ U be a function between two sets and let u : T ⟶ S be an injective function between two sets of parameters. Then f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-semi-open if and only if p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-semi-open for all t ∈ T.

Proof. Necessity. Suppose that f_pu : (V, ⊕ _t∈Tψ_t, T)

⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-semi-open. Let t ∈ T. Let X ∈ ω_s (V, ψ_t). Then by Theorem 4.10 of [38], t_X ∈ SO (V, ⊕ _t∈Tψ_t, T) and so f_pu (t_X) = (u (t)) _p(X) ∈ SO (U, ⊕ _s∈Sφ_s, S). Thus, by Theorem 4.10 of [38], ((u (t)) _p(X)) (u (t)) = p (X) ∈ SO (U, φ_u(t)). It follows that p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-semi-open.

Sufficiency. Suppose that p : (V, ψ_t) ⟶ (U, φ_u(t)) is pre-semi-open for all t ∈ T. Let K ∈ SO (V, ⊕ _t∈Tψ_t, T). Then by Theorem 4.10 of [38], K (t) ∈ SO (V, ψ_t) for all t ∈ T. Therefore, p (K (t)) ∈ SO (U, φ_u(t)) for all t ∈ T. Thus, we have (f_pu (K)) (e) = ∅ ∈ SO (U, φ_s) if u^-1 ({ e }) = ∅ or (f_pu (K)) (e) = p (K (t)) ∈ SO (U, φ_s) if u (t) = e. Hence, by Theorem 4.10 of [38], f_pu (K) ∈ SO (U, ⊕ _s∈Sφ_s, S). It follows that f_pu : (V, ⊕ _t∈Tψ_t, T) ⟶ (U, ⊕ _s∈Sφ_s, S) is soft pre-semi-open.

Corollary 4.5. Let p : (V, μ) ⟶ (U, α) be a mapping between two TSs and let u : T ⟶ S be an injective function between two sets of parameters. Then p : (V, μ) ⟶ (U, α) is pre-semi-open if and only if f_pu : (V, τ (μ) , T) ⟶ (U, τ (α) , S) is soft pre-semi-open.

Proof. For each t ∈ T and s ∈ S, put μ_t = μ and α_s = α. Then τ (μ) = ⊕ _t∈T μ_t and τ (α) = ⊕ _s∈S α_s. Thus, by Theorem 4.4, we get the result.

By the next two examples we will show that the notions of soft openness and soft pre-ω_s-openness are independent:

Example 4.6. Let $V = ℝ$ , U ={ a, b, c }, μ ={ ∅ , (– ∞ , 1) , V }, α ={ ∅ , { a } , U }, and $T = ℕ$ . Define p : V ⟶ U and u : T ⟶ T by

$p (v) = {\begin{matrix} a & if v \in (- \infty, 1) \\ b & if v = 1 \\ c & if v \in (1, \infty) \end{matrix}$ and u (t) = t for every t ∈ T.

Then by Example 3.10 of [53], p : (V, μ) ⟶ (U, α) is open but not pre-ω_s-open. Therefore, by Corollaries 3.4 and 4.3, f_pu : (V, τ (μ) , T) ⟶ (V, τ (α) , T) is soft open but not soft pre-ω_s-open.

Example 4.7. Let $V = ℝ$ , $T = ℕ$ and $μ = {\emptyset, ℝ, [1, \infty)}$ . Define p : V ⟶ V and u : T ⟶ T by

p (v) = v - 1 and u (t) = t for every t ∈ T.

Then by Example 3.11 of [53], p : (V, μ) ⟶ (V, μ) is pre-ω_s-open but not open. Therefore, by Corollaries 3.4 and 4.3, f_pu : (V, τ (μ) , T) ⟶ (V, τ (μ) , T) is soft pre-ω_s-open but soft open.

Theorem 4.8. Let (V, ψ, T) and (U, φ, S) be two soft locally countable STSs, and let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft mapping. Then f_pu is soft open if and only if f_pu is soft pre-ω_s-open.

Proof. Follows from the definitions and Theorem 2.8 of [38].

By the next two examples we will show that the notions of soft pre-semi-openness and soft pre-ω_s-openness are independent:

Example 4.9. Let $V = ℝ$ , $T = ℕ$ , and $μ = {\emptyset, ℕ, ℝ}$ . Define p : V ⟶ V and u : T ⟶ T by p (v) = v - 1 and u (t) = t for every t ∈ T. Then by Example 3.13 of [53], p : (V, μ) ⟶ (V, μ) is pre-semi-open but not pre-ω_s-open. Therefore, by Corollaries 4.3 and 4.5, f_pu : (V, τ (μ) , T) ⟶ (V, τ (μ) , T) is soft pre-semi-open but not soft pre-ω_s-open

Example 4.10. Let V = {1, 2, 3, 4},

μ = {∅ , V, {1, 2} , {1} , {2}},

α = {∅ , V, {2, 3, 4} , {1, 2} , {1} , {2}},

and $T = ℝ$ . Let p : V ⟶ V and u : T ⟶ T be the identities functions. By Example 3.14 of [53], p : (V, μ) ⟶ (V, α) is pre-ω_s-open but not pre-semi-open. Therefore, by Corollaries 4.3 and 4.5, f_pu : (V, τ (μ) , T) ⟶ (V, τ (α) , T) is soft pre-ω_s-open but not soft pre-semi-open.

Theorem 4.11. Let (V, ψ, T) and (U, φ, S) be soft locally countable, and let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft mapping. Then f_pu is soft pre-semi-open if and only if f_pu is soft pre-ω_s-open.

Proof. Follows from definitions and Theorem 2.6 of [38].

Theorem 4.12. Every soft pre-ω_s-open mapping is soft ω_s-open.

Proof. Let f_pu : (V, ψ, T) ⟶ (U, φ, S) be a soft pre-ω_s-open soft mapping and let K ∈ ψ. Then by Theorem 2.2 of [38], K ∈ ω_s (V, ψ, T). Since f_pu is soft pre-ω_s-open, then f_pu (K) ∈ ω_s (U, φ, S). Therefore, f_pu is soft ω_s-open.

The soft mapping f_pu in Example 4.6 is soft open but not soft pre-ω_s-open, and by Theorem 3.9, f_pu is soft ω_s-open. Therefore, the converse of the implication in Theorem 4.12 need not be true in general.

Theorem 4.13. For a soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S), the following conditions are equivalent:

(a) f_pu is soft pre-ω_s-open.

(b) $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq} ω_{s}$ - ${Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S).

Proof. (a) ⇒ (b): Suppose that f_pu is soft pre-ω_s-open and let N ∈ SS (U, S). Let $t_{v} \tilde{\in} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N))$ . To show that $t_{v} \tilde{\in} ω_{s}$ - ${Cl}_{ψ} (f_{pu}^{- 1} (N))$ , let H ∈ ω_s (V, ψ, T) such that $t_{v} \tilde{\in} H$ . Then $f_{pu} (t_{v}) \tilde{\in} f_{pu} (H)$ . Since f_pu is soft pre-ω_s-open, then f_pu (H) ∈ ω_s (U, φ, S). Since $f_{pu} (t_{v}) \tilde{\in} f_{pu} (H) \tilde{\cap} ω_{s}$ -Cl_φ (N), then $f_{pu} (H) \tilde{\cap} N \neq 0_{S}$ . Choose $a_{x} \tilde{\in} H$ such that $f_{pu} (a_{x}) \tilde{\in} N$ . Then

$a_{x} \tilde{\in} H \tilde{\cap} f_{pu}^{- 1} (N)$ and thus $H \tilde{\cap} f_{pu}^{- 1} (N) \neq 0_{T}$ .

Therefore, $t_{v} \tilde{\in} ω_{s}$ - ${Cl}_{ψ} (f_{pu}^{- 1} (N))$ .

(b) ⇒ (a): Suppose that $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq} ω_{s}$ - ${Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S), and suppose to the contrary that f_pu is not soft pre-ω_s-open. Then there is K ∈ ω_s (V, ψ, T) such that f_pu (K) ∉ ω_s (U, φ, S). Thus, we have 1_S - f_pu (K) is not soft ω_s-closed, and so there is $b_{y} \tilde{\in} f_{pu} (K) \tilde{\cap} ω_{s}$ -Cl_φ (1_S - f_pu (K)). Choose $a_{x} \tilde{\in} K$ such that b_y = f_pu (a_x). Then $a_{x} \tilde{\in} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - f_{pu} (K)))$ . By (b), we have $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - f_{pu} (K)))$

$\begin{matrix} \tilde{\subseteq} & ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (1_{S} - f_{pu} (K))) \\ = & ω_{s} - {Cl}_{ψ} (1_{T} - f_{pu}^{- 1} (f_{pu} (K))) \\ \tilde{\subseteq} & ω_{s} - {Cl}_{ψ} (1_{T} - K) \\ = & 1_{T} - K . \end{matrix}$

Therefore, $a_{x} \tilde{\in} 1_{T} - K$ but $a_{x} \tilde{\in} K$ , a contradiction.

(b) ⇒ (c): Suppose that $f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (N)) \tilde{\subseteq} ω_{s}$ - ${Cl}_{ψ} (f_{pu}^{- 1} (N))$ for every N ∈ SS (U, S) and let N ∈ SS (U, S). Then

$\begin{matrix} f_{pu}^{- 1} (ω_{s} - {Int}_{φ} (K)) & = & f_{pu}^{- 1} (1_{S} - ω_{s} - {Cl}_{φ} (1_{S} - K)) \\ = & 1_{T} - f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (1_{S} - K)) \\ \tilde{\supseteq} & 1_{T} - ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (1_{S} - K)) \\ = & 1_{T} - ω_{s} - {Cl}_{ψ} (1_{T} - f_{pu}^{- 1} (K)) \\ = & ω_{s} - {Int}_{ψ} (f_{pu}^{- 1} (K)) . \end{matrix}$

$f_{pu}^{- 1} (ω_{s} - {Int}_{φ} (N))$ for every N ∈ SS (U, S) and let N ∈ SS (U, S). Then

$\begin{matrix} f_{pu}^{- 1} (ω_{s} - {Cl}_{φ} (K)) & = & f_{pu}^{- 1} (1_{S} - ω_{s} - {Int}_{φ} (1_{S} - K)) \\ = & 1_{T} - f_{pu}^{- 1} (ω_{s} - {Int}_{φ} (1_{S} - K)) \\ \tilde{\subseteq} & 1_{T} - ω_{s} - {Int}_{ψ} (f_{pu}^{- 1} (1_{S} - K)) \\ = & 1_{T} - ω_{s} - {Int}_{ψ} (1_{T} - f_{pu}^{- 1} (K)) \\ = & ω_{s} - {Cl}_{ψ} (f_{pu}^{- 1} (K)) . \end{matrix}$

Theorem 4.14. Let f_pu : SS (V, T) ⟶ SS (U, S) be a soft mappings between two families of soft sets. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-continuous and f_pu : (V, ψ_ω, T) ⟶ (U, φ_ω, S) is soft open, then f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-irresolute.

Proof. Let K ∈ ω_s (U, φ, S). Choose H ∈ φ such that $H \tilde{\subseteq} K \tilde{\subseteq} {Cl}_{φ_{ω}} (H)$ . Then $f_{pu}^{- 1} (H) \tilde{\subseteq} f_{pu}^{- 1} (K) \tilde{\subseteq}$

$f_{pu}^{- 1} ({Cl}_{φ_{ω}} (H))$ . Since f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft ω_s-continuous, then $f_{pu}^{- 1} (H) \in ω_{s} (V, ψ, T)$ . Since f_pu : (V, ψ_ω, T) ⟶ (U, φ_ω, S) is soft open, then

$f_{pu}^{- 1} ({Cl}_{φ_{ω}} (H)) \tilde{\subseteq} {Cl}_{ψ_{ω}} (f_{pu}^{- 1} (H))$ . Therefore, we have $f_{pu}^{- 1} (H) \tilde{\subseteq} f_{pu}^{- 1} (K) \tilde{\subseteq} {Cl}_{ψ_{ω}} (f_{pu}^{- 1} (H))$ with $f_{pu}^{- 1} (H) \in ω_{s} (V, ψ, T)$ and thus, $f_{pu}^{- 1} (K) \in ω_{s} (V, ψ, T)$ . This ends the proof.

Theorem 4.15. Let f_pu : SS (V, T) ⟶ SS (U, S) be a soft mappings between two families of soft sets. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft open and f_pu : (V, ψ_ω, T) ⟶ (W, φ_ω, S) is soft continuous, then f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft pre-ω_s-open.

Proof. Let M ∈ ω_s (V, ψ, T). Then there is K ∈ ψ such that $K \tilde{\subseteq} M \tilde{\subseteq} {Cl}_{ψ_{ω}} (K)$ , and so $f_{pu} (K) \tilde{\subseteq} f_{pu} (M)$

$\tilde{\subseteq} f_{pu} ({Cl}_{ψ_{ω}} (K))$ . Since f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft open, then f_pu (K) ∈ φ. Since f_pu : (V, ψ_ω, T) ⟶ (U, φ_ω, S) is soft continuous, $f_{pu} ({Cl}_{ψ_{ω}} (K)) \tilde{\subseteq} {Cl}_{φ_{ω}} (f_{pu} (K))$ . Therefore, we have $f_{pu} (K) \tilde{\subseteq} f_{pu} (M) \tilde{\subseteq} {Cl}_{φ_{ω}} (f_{pu} (K))$ with f_pu (K) ∈ φ, and hence, f_pu (M) ∈ ω_s (U, φ, S). This ends the proof.

Definition 4.16. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called soft ω-open if for each K ∈ ψ (V, ψ, T), f_pu (K) ∈ φ_ω.

Definition 4.17. A soft mapping f_pu : (V, ψ, T) ⟶ (U, φ, S) is called soft pre-ω_s-closed if for each soft ω_s-closed of (V, ψ, T), f_pu (M) is soft ω_s-closed set in (U, φ, S).

Theorem 4.18. If f_pu : (V, ψ, T) ⟶ (U, φ, S) is soft pre-ω_s-closed, soft pre-ω_s-open, and soft ω_s-irresolute such that (V, ψ, T) is soft ω-regular, then f_pu is soft ω-open.

Proof. Suppose to the contrary that there is K ∈ ψ such that f_pu (K) ∉ φ_ω. Then we find $t_{v} \tilde{\in} K$ such that $f_{pu} (t_{v}) \tilde{\in} f_{pu} (K) - {Int}_{φ_{ω}} (f_{pu} (K))$ . By soft ω-regularity of (V, ψ, T), there is M ∈ ψ such that $t_{v} \tilde{\in} M \tilde{\subseteq} {Cl}_{ψ_{ω}} (M) \tilde{\subseteq} K$ . Since Cl_{ψ
_ω} (M) is soft ω_s-closed and f_pu is soft ω_s-closed, then 1_S - f_pu (Cl_{ψ
_ω} (M)) ∈ ω_s (U, φ, S) with $f_{pu} (t_{v}) \tilde{\notin} 1_{S} - f_{pu} ({Cl}_{ψ_{ω}} (M))$ . Since $f_{pu} (t_{v}) \tilde{\notin} {Int}_{φ_{ω}} (f_{pu} (K))$ , then $f_{pu} (t_{v}) \tilde{\notin} {Int}_{φ_{ω}} (f_{pu} ({Cl}_{ψ_{ω}} (M)))$ . Thus,

$f_{pu} (t_{v}) \tilde{\in} 1_{S} - {Int}_{φ_{ω}} (f_{pu} ({Cl}_{ψ_{ω}} (M))) =$

Cl_{φ
_ω} (1_S - f_pu (Cl_{ψ
_ω} (M))).

So by Theorem 2.14 of [38],

$(1_{S} - f_{pu} ({Cl}_{ψ_{ω}} (M))) \tilde{\cup} f_{pu} (t_{v}) \in ω_{s} (U, φ, S)$ .

Put $H = f_{pu}^{- 1} ((1_{S} - f_{pu} ({Cl}_{ψ_{ω}} (M))) \tilde{\cup} f_{pu} (t_{v}))$ . Since f_pu is soft ω_s-irresolute, then H ∈ ω_s (V, ψ, T) and by Theorem 2.13 of [38], $M \tilde{\cap} H \in ω_{s} (V, ψ, T)$ . Since f_pu is soft pre-ω_s-open, then $f_{pu} (K \tilde{\cap} H) \in ω_{s} (U, φ, S)$ . Since $f_{pu} (t_{v}) \tilde{\in} f_{pu} (M \tilde{\cap} H)$

$\begin{matrix} \tilde{\subseteq} & f_{pu} (M) \tilde{\cap} f_{pu} (H) \\ \tilde{\subseteq} & f_{pu} (M) \tilde{\cap} ((1_{S} - f_{pu} ({Cl}_{ψ_{ω}} (M))) \tilde{\cup} f_{pu} (t_{v})) \\ = & f_{pu} (t_{v}), \end{matrix}$

then f_pu (t_v) ∈ ω_s (U, φ, S). Thus, there is N ∈ φ such that $N \tilde{\subseteq} f_{pu} (t_{v}) \tilde{\subseteq} {Cl}_{φ_{ω}} (N)$ . Hence, f_pu (t_v) ∈ φ. Since $f_{pu} (t_{v}) \tilde{\in} f_{pu} (K)$ , then

$f_{pu} (t_{v}) \tilde{\in} {Int}_{φ} (f_{pu} (K)) \tilde{\subseteq} {Int}_{φ_{ω}} (f_{pu} (K))$ , a contradiction.

5 Conclusion

Because of its enormous practical utility, soft set theory has been widely applied in numerous real-world applications. Topologists have used soft sets to construct soft topological spaces and soft frames to study traditional topological notions. Abstract topological notions in soft settings, such as soft separation axioms [21], and generalized soft open sets [31], have been effectively utilized to solve real difficulties in information systems, economics, and medicine, as highlighted in the published literature.

In this paper, we have benefited from one of the generalizations of soft open sets called “soft ω_s-open sets” to introduce some modifications of continuity and openness between TSs via the frame of soft topologies. We have presented novel types of soft continuity and soft openness namely, soft ω_s-irresoluteness, soft ω_s-openness, and soft pre-ω_s-openness. We have established the main properties of these concepts and provided some interesting examples to show the relationships between them and other related concepts. Some characterizations of each of these concepts have been obtained. Also, some exciting results behaviors of these concepts and their topologically corresponding notions have been obtained.

The following topics could be considered for future studies: 1) Defining soft ω_s-homeomorphisms; 2) Defining soft define several soft covering properties via soft ω_s-open sets 3) To examine which of the soft topological properties can be preserved under these new types of soft mappings.

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Soft ω s -irresoluteness and soft pre- ω s -openness insoft topological spaces

Abstract

Keywords

1 Introduction and preliminaries

2 Soft ω s -irresoluteness

3 Soft ω s -open soft mappings

4 Soft Pre-ω s -open

5 Conclusion

References

2 Soft ω_s-irresoluteness

3 Soft ω_s-open soft mappings

4 Soft Pre-ω_s-open