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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 183-192

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.183

© The Korean Institute of Intelligent Systems

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Samer Al Ghour

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

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

Received: November 28, 2021; Revised: January 16, 2022; Accepted: January 26, 2022

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

In this study, soft ω-continuity and soft ωs-continuity are introduced as two new classes of soft functions, and several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity and that soft ωs-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions are introduced for the equivalence between soft ωs-continuity and soft ω-continuity, as well as that between soft ωs-continuity and soft semi-continuity. Furthermore, composition theorems regarding soft ω-continuity and soft ωs-continuity are given. Finally, the relationships between the generated soft topological spaces and induced topological spaces are studied.

Keywords: Soft semi-continuous functions, Soft ω-open sets, ω-continuity, ωs-continuity, Soft generated soft topological space, Soft induced topological spaces

Herein, we follow the notions and terminology that appear in [13]. Throughout this paper, ST denotes a topological space and STS denotes a soft topological space. To deal with uncertain objects, Molodtsov [4] introduced soft sets in 1999. Let X be a universal set and A be a set of parameters. A function K : A → ℘(X) is called a soft set over X relative to A. SS (X, A) denotes the family of all soft sets over X relative to A. A soft set KSS (X, A) is called a countable soft set if for each aA, the set K (a) is countable. CSS (X, A) denotes the family of all members of SS (X, A) that are countable. The null soft set and absolute soft set are denoted by 0A and 1A, respectively. The concept of STS was defined in [5] as follows: the triplet (X, τ, A), where τSS (X, A), is called an STS if 0A and 1Aτ , τ are closed under a finite soft intersection, and τ is closed under an arbitrary soft union. For an STS (X, τ, A) the members of τ are called soft open sets, and their soft complements are called soft closed sets. The family of all soft closed sets is denoted by τc. Mathematicians have extended many STs to STSs; some recent studies have also been conducted ([13,629]). Chen [30] introduced the concept of soft semi-open sets. Following this, many research papers on soft semi-open sets and their modifications appeared. Soft ω-openness was defined in [2] as a generalization of soft openness. In a previous study [3], the author used soft ω-open sets to define ωs-open sets as a class of soft sets that lies strictly between the soft open sets and soft semi-open sets. In this study, soft ω-continuity and soft ωs-continuity are introduced as two new classes of soft functions using soft ω-open sets and soft ωs-open sets. In addition, several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity, and soft ωs-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions for the equivalence between soft ωs-continuity and soft ω- continuity, as well as that between soft ωs-continuity and soft semi-continuity, are introduced. Composition theorems regarding soft ω-continuity and soft ωs-continuity are also provided. Finally, the relationships regarding generated STSs and induced STs are studied.

Let (X, τ, A) be an STS, (X, ℑ) be a TS, HSS(X, A), and DX. Throughout this paper, Clτ (H), intτ (H), and Cl (D) denotes the soft closure of H in (X, τ, A), the soft interior of H in (X, τ, A), and the closure of D in (X,ℑ), respectively.

The following definitions and results are used in the following:

Definition 1.1

Let X be a universal set and A be a set of parameters. Then GSS(X, A) is defined by

(a) G(a)={Y,if a=e,if ae, is denoted by eY.

(b) [1] G(a) = Y for all aA is denoted by CY.

(c) G(a)={{x},if a=e,,if ae, is denoted by ex, and is called a soft point.

The set of all soft points in SS(X, A) is denoted by SP (X, A).

Definition 1.2 [31]

LetGSS(X, A) and exSP(X, A). Then ex is said to belong to F (notation: ex ∊̃ G) if ex ∊̃ G or, equivalently, ex ∊̃ G if and only if xG(e).

Theorem 1.3 [5]

Let (X, τ, A) be an STS. Then the collection {F(a) : Fτ} defines a topology on X for every aA. This topology is denoted by τa .

Theorem 1.4 [32]

Let (X,ℑ) be a TS. Then the collection

{FSS(X,A):F(a)Ifor all aA}

defines a soft topology on X relative to A. This soft topology is denoted by τ (ℑ).

Theorem 1.5 [1]

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let

τ={FSS(X,A):F(a)Iafor all aA}.

Then τ defines a soft topology on X relative to A. This soft topology is denoted by aAIa.

Definition 1.6

An STS (X, τ, A) is called

(a) [2] soft locally countable if for each axSP(X, A), there exists Gτ such that ax ∊̃ G and G is a countable soft set.

(b) [2] soft anti-locally countable if for every Fτ −{0A}, F is not a countable soft set.

Definition 1.7

Let (X,ℑ) be a ST, and let DX. D is called

(a) [33] semi-open if there exists V ∈ ℑ such that VDCl (V ). SO(X, ℑ) denotes the family of all semi-open sets in (X,ℑ).

(b) [34] ωs-open if there exists V ∈ ℑ such that VDClω (V ). ωs(X, ℑ) denotes the family of all ωs-open sets in (X, ℑ).

Similar types of generalized sets in ideal STs have been discussed in [3539].

Definition 1.8 [40]

A function g : (X, ℑ) → (Y, ℵ) is called ω-continuous at a point xX if for every V ∈ ℵ with g(x) ∈ V, there exists U ∈ ℑω such that xU and g(U) ⊆ V . If g is ω-continuous at each point xX, then g is called ω-continuous.

Definition 1.9 [33]

A function g : (X, ℑ) → (Y, ℵ) is called semi-continuous if for each V ∈ ℵ, g−1(V ) ∈ SO(X, ℑ).

Definition 1.10 [34]

A function g : (X, ℑ) → (Y, ℵ) is called ωs-continuous if for each V ∈ ℵ, g−1(V ) ∈ ωs(X, ℑ).

Definition 1.11

Let (X, τ, A) be an STS and let KSS (X, A). Then

(a) [30] K is called a soft semi-open set in (X, τ, A) if there exists Fτ such that F ⊂̃ K ⊂̃ Clτ (F). SO(X, τ, A) denotes the family of all soft semi-open sets in (X, τ, A).

(b) [30] K is called a soft semi-closed set in (X, τ, A) if 1AKSO(X, τ, A).

(c) [3] K is called a soft ωs-open set in (X, τ, A) if there exists Fτ such that F ⊂̃ K ⊂̃ Clτω (F). ωs(X, τ, A) denotes the family of all soft ωs-open sets in (X, τ, A).

According to Definition 1.11, a soft semi-open set always contains a soft open set. Therefore, this is a particular type of generalized soft open set. One can specify another type of generalized set in the STS that does not necessarily contain a soft open set. In the ST, two different types of sets have been discussed in [41].

Definition 1.12 [42]

A soft function fpu : (X, τ, A) → (Y, σ, B) is called soft semi-continuous if for every Gσ, fpu-1(G)SO(X,τ,A).

Definition 1.13 [3]

Let (X, τ, A) be an STS and HSS (X, A).

(a) The soft ωs-closure of H in (X, τ, A) is denoted by ωs-Clτ (H) and defined by

ωs-Clτ(H)=˜{M:Mis soft ωs-closed in (X,τ,A)and H˜M}.

(b) The soft ωs-interior of H in (X, τ,A) is denoted by ωsintτ (H) and defined by

ωs-intτ(H)=˜{K:Kis soft ωs-open in (X,τ,A)and K˜H}.

In this section, we introduce the concept of soft ω-continuous function and establish its main properties. With the help of examples, we investigate the interaction between these concepts within the frameworks of general topology and soft topology.

Definition 2.1

A soft function fpu : (X, τ,A) → (Y, σ,B) is called soft ω-continuous at a soft point axSP(X,A), if Gσ with fpu(ax)∊̃G, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. If fpu is soft ω-continuous at each soft point axSP(X,A), then fpu is called soft ω-continuous.

Theorem 2.2

For a soft function fpu : (X, τ,A) → (Y, σ,B), the following conditions are equivalent:

(a) fpu is soft ω-continuous.

(b) For each Gσ, fpu-1(G)τω.

(c) For a soft base ℳ of (Y, σ,B), fpu-1(M)τω for all M ∈ ℳ.

(d) For a soft subbase of (Y, σ,B), fpu-1(S)τω for all .

(e) For each Nσc, fpu-1(N)(τω)c.

(f) For each KSS(X,A), fpu(Clτω (K))⊂̃Clσ (fpu(K)).

(g) For each LSS(Y,B), Clτω(fpu-1(L))˜fpu-1(Clσ(L)).

Proof

(a) =⇒ (b): Let Gσ and let ax˜fpu-1(G). Then fpu (ax) ∊̃G. By (a), fpu is soft ω-continuous at ax. Thus, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. Because fpu(H)⊂̃G, then we have ax˜H˜fpu-1(fpu(H))˜fpu-1(G). Therefore, fpu-1(G)τω.

(b) =⇒ (c): Obvious.

(c) ⇒ (d): Obvious.

(d) ⇒ (e): Let Nσc. We will show that 1A-fpu-1(N)τω. Let ax˜1A-fpu-1(N)=fpu-1(1B-N). Then fpu (ax) ∊̃; 1BNσ. Since in (d) is a soft subbase for (Y, σ,B), then there exist . such that

fpu(ax)˜S1˜S2˜˜Sn˜1B-N.

So,

ax˜fpu-1(S1)˜fpu-1(S2)˜˜fpu-1(Sn)˜fpu-1(1B-N)=1A-fpu-1(N).

By (d), fpu-1(S1)˜fpu-1(S2)˜˜fpu-1(Sn)τω, and thus, 1A-fpu-1(N)τω.

(e) ⇒ (f): Let KSS(X,A). Then Clσ (fpu(K)) ∈ σc. Thus, by (e), fpu-1(Clσ(fpu(K)))(τω)c. Because K˜fpu-1(fpu(K))˜fpu-1(Clσ(fpu(K))),

Clτω(K)˜fpu-1(Clσ(fpu(K))),

and so,

fpu(Clτω(K))˜fpu(fpu-1(Clσ(fpu(K))))˜Clσ(fpu(K)).

(f) ⇒ (g): Let LSS(Y,B). Then fpu-1(L)SS(X,A). Thus, by (f),

fpu(Clτω(fpu-1(L)))˜Clσ(fpu(fpu-1(L)))˜Clσ(L).

Therefore,

Clτω(fpu-1(L))˜fpu-1(fpu(Clτω(fpu-1(L))))˜fpu-1(Clσ(L)).

(g) ⇒ (a): Let axSP(X,A) and Gσ such that fpu(ax) ∈ G. Then 1BGσc. By (g), Clτω(1A-fpu-1(G))=Clτω(fpu-1(1B-G))˜fpu-1(Clσ(1B-G))=˜fpu-1(1B-G)=1A-fpu-1(G). Thus, 1A-fpu-1(G)(τω)c and hence fpu-1(G)τω. Put H=fpu-1(G). Then ax∊̃;Hτω and fpu(H)=fpu(fpu-1(G))˜G. Therefore, f is soft ω-continuous at ax.

Theorem 2.3

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at axSP(X,A), then p : (X, τa) → (Y, σu(a)) is ω-continuous at x.

Proof

Suppose that fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax. Let Vσu(a) such that p (x) ∈ V . Choose Gσ such that G(u(a)) = V . Then p (x) ∈ G(u(a)), so fpu(ax) = (u(a))p(x)G. Because fpu is soft ω-continuous at ax, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. Thus, we have xH(a) ∈ (τω)a. Also, by Theorem 7 of [2], (τω)a = (τa)ω. To see that p(H(a)) ⊆ V , let zH (a); then, az∊̃H. Because fpu(H)⊂̃G, fpu(az) = (u (a))p(z) ∊̃G and so p(z) ∈ G(u(a)) = V . This shows that p : (X, τa) → (Y, σu(a)) is ω-continuous at x.

Theorem 2.4

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then p : (X, τa) → (Y, σu(a)) is ω-continuous for all aA.

Proof

Let aA. To demonstrate that p : (X, τa) → (Y, σu(a)) is ω-continuous, let xX. Then fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax. So, by Theorem 2.3, p : (X, τa) → (Y, σu(a)) is ω-continuous at x. This shows that p : (X, τa) → (Y, σu(a)) is ω-continuous.

Theorem 2.5

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ω-continuous at a point xX if and only if fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at ax for all aA.

Proof
Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ω-continuous at x. Let aA and let Gτ (ℵ) such that fpu(ax) ∊̃G. Then p(x) ∈ G(u (a)) ∈ ℵ. Because p : (X, ℑ) → (Y, ℵ) is ω-continuous at x, there exists U ∈ ℑω such that xU and p(U) ⊆ G(u (a)). Thus, we have ax∊̃aUτ (ℑω) = (τ (ℑ))ω and fpu(aU) = (u(a))p(U) ⊂̃G. Hence, fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at ax.

Sufficiency

Suppose that fpu : (X, τ(ℑ),A) → (Y, τ(ℵ), B) is soft ωp-continuous at ax. Then by Theorem 2.3,

p : (X, (τ (ℑ))a) → (Y, (τ (ℵ))u(a)) is ω-continuous at x. Because (τ (ℑ))a = ℑ and (τ (ℵ))u(a) = ℵ, p : (X, ℑ) → (Y, ℵ) is ω-continuous at x.

Corollary 2.6

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ω-continuous if and only if fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous.

Theorem 2.7

Soft continuous soft functions are soft ω-continuous.

Proof

Follows from Theorem 2.2 and Theorem 2(a) of [2].

The following example shows that the converse of Theorem 2.7 is not true in general:

Example 2.8

Let X = Y = ℤ, and A = B = ℝ. Let ℑ = {∅︀,X,ℕ} and ℵ be the cofinite topology on Y . Define p : XY and u : AB by p (x) = x and u(a) = a for all xX and aA. Then fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous but not soft continuous.

Theorem 2.9

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and if fqv : (Y, σ,B) → (Z, δ,C) is soft continuous, then f(qp)(vu) : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Proof

Let Kδ. Because fqv : (Y, σ, ,B) → (Z, δ,C) is soft continuous, then fqu-1(K)σ. Because fpu is soft ω-continuous, fqu-1(fqv-1(K))τω. However, fpu-1(fqv-1(K))=f(qp)(vu)-1(K). Therefore, f(qp)(vu)-1(K)τω. Hence, f(qp)(vu) : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Theorem 2.10

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and Z is a non-empty subset of X, then the soft restriction f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous.

Proof

Let Kσ. Since fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then fpu-1(K)τω. So, f(pZ)u-1(K)=fpu-1(K)˜CY(τω)Y. Hence, by Theorem 15 of [2], f(pZ)u-1(K)(τY)ω. Therefore, f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous.

Theorem 2.11

Let fpu : (X, τ,A) → (Y, σ,B) be a soft function and let 1A = CZ ∪̃CW, where CZ, CWτc − {0A}. If f(p|Z)u : (Z, τZ,A) → (Y, σ,B) and f(p|W)u : (W, τW,A) → (Y, σ,B) are soft ω-continuous functions, then fpu is soft ω-continuous.

Proof

We apply statement (e) of Theorem 2.2. Let Nσc. Then

fpu-1(N)=fpu-1(N)˜1A=fpu-1(N)˜(CZ˜CW)=(fpu-1(N)˜CZ)˜(fpu-1(N)˜CW)=(f(pZ)u-1(N))˜(f(pW)u-1(N)).

Because f(p|Z)u and f(p|W)u are soft ω-continuous, then

f(pZ.)u-1(N)((τZ)ω)cand f(pW.)u-1(N)((τW)ω)c.

Therefore, by Theorem 17 of [2], f(pZ.)u-1(N)((τω)Z)c and f(pW.)u-1(N)((τω)W)c. Since CZ, CWτ c, then f(pZ.)u-1(N),f(pW.)u-1(N)(τω)c. Therefore, fpu-1(N)(τω)c.

Theorem 2.12

Let fpu : (X, τ,A) → (Y, σ,B) be a soft function and let axSP(X,A). If there exists ZX such that CZτω such that ax∊̃CZ and f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous at ax, then fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax.

Proof

Let Gσ such that fpu(ax) ∈ G. Because f(p|Z)u is soft ω-continuous at ax, there exists H ∈ (τZ)ω such that ax∊̃H and f(p|Z)u(H) ⊂̃G. BecauseH ∈ (τZ)ω and CZτω, Hτω. Therefore, fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax.

Corollary 2.13

fpu : (X, τ, A) → (Y, σ, B) be a soft function. Let {CXα : α ∈ Δ} ⊆ τ such that 1A = ∪̃{CXα : α ∈ Δ}. If for each α ∈ Δ, f(p|Xα)u : (Xα, τXα, A) → (Y, σ, B) is soft ω-continuous, then fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

Proof

Let axSP(X, A). We show that fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous at ax. Because 1A = ∪̃{CXα : α ∈ Δ}, then there exists α ∈ Δsuch that ax∊̃CXα. Therefore, by Theorem 2.12, it follows that fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

In this section, we introduce the concept of a soft ωs-continuous function and establish its main properties. We also provide several characterizations of this concept, and study the relationships between soft ωs-continuity and some other types of soft continuity.

Definition 3.1

A soft function fpu : (X, τ, A) → (Y, σ, B) is called soft ωs-continuous if for every Gσ, fpu-1(G)ωs(X,τ,A).

Theorem 3.2

Soft continuous functions are soft ωs-continuous.

Proof

Let fpu : (X, τ, A) → (Y, σ, B) be soft continuous and let Gσ. By soft continuity of fpu, fpu-1(G)τ. So by Theorem 2.2 of [3], fpu-1(G)ωs(X,τ,A). Hence, fpu is soft ωs-continuous.

Theorem 3.3

Soft ωs-continuous functions are soft semi-continuous.

Proof

Let fpu : (X, τ, A) → (Y, σ, B) be soft ωs-continuous and let Gσ. By soft ωs-continuity of fpu, fpu-1(G)ωs(X,τ,A). Thus, by Theorem 2.2 of [3], fpu-1(G)SO(X,τ,A). Hence, fpu is soft semi-continuous.

The following example shows that the converses of Theorems 3.2 and 3.3 are not true in general:

Example 3.4

Let X = ℝ, Y = {a, b}, A = B = ℤ, τ = {0A, 1A, C, Cc, C(ℕc)}, and σ = {0B, 1B, C{a}, C{b}}. It is not difficult to see that Clτω (C) = C, Clτ (C) = C, and Clτω (Cc) = Cℝ−ℕ. Thus, CSO(X, τ, A) − ωs(X, τ, A) and Cℝ−ℕωs(X, τ, A)−τ. Let p, q : XY and u, v : AB be defined by

p(x)={a,if x,b,if x-,q(x)={a,if xc,b,if x,u(a)=v(a)=afor all aA.

Because fpu-1(C{a})=Cτωs(X,τ,A) and fpu-1(C{b})=C-ωs(X,τ,A)-τ, then fpu is soft ωs-continuous but not soft continuous. Additionally, because fqv-1(C{a})=CcτSO(X,τ,A) and fqv-1(C{b})=CSO(X,τ,A)-ωs(X,τ,A), fqv is soft semi-continuous but not soft ωs-continuous.

Theorem 3.5

If fpu : (X, τ, A) → (Y, σ, B) is soft ωs-continuous such that (X, τ, A) is soft locally countable, then fpu is soft continuous.

Proof

Let Gσ. By the soft ωs-continuity of fpu, fpu-1(G)ωs(X,τ,A). Thus, by Theorem 2.8 of [3], fpu-1(G)τ. Hence, fpu is soft continuous.

Theorem 3.6

If fpu : (X, τ, A) → (Y, σ, B) is soft semi-continuous such that (X, τ, A) is soft anti-locally countable, then fpu is soft ωs-continuous.

Proof

Let Gσ. By the soft semi-continuity of fpu, fpu-1(G)SO(X,τ,A). Thus, by Theorem 2.6 of [3], fpu-1(G)ωs(X,τ,A). Hence, fpu is soft ωs-continuous.

Theorem 3.7

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For each axSP(X, A) and each Gσ such that fpu(ax)∊̃G, there exists Hωs (X, τ, A) such that ax∊̃H and fpu(H)⊂̃G.

Proof

(a) ⇒ (b): Assume that fpu is soft ωs-continuous. Let Gσ such that fpu(ax)∊̃G. Because fpu is soft ωs-continuous, fpu-1(G)ωs(X,τ,A). Take H=fpu-1(G). Then Hωs (X, τ, A) such that ax∊̃H and fpu(H)⊂̃G.

(b) ⇒ (a): Let Gσ. For each ax˜fpu-1(G), we have fpu(ax)∊̃G, and by (b), there exists Haxωs (X, τ, A) such that ax∊̃Hax and fpu(Hax )⊂̃G, and thus, ax˜Hax˜fpu-1(G). Therefore, fpu-1(G)=˜{Hax:ax˜fpu-1(G)}, and hence fpu-1(G)ωs(X,τ,A).

Theorem 3.8

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For a soft base of (Y, σ, B), fpu-1(M)ωs(X,τ,A) for all M.

(c) For each Nσc, fpu-1(N) is soft ωs-closed.

(d) For each HSS(X, A),

fpu(ωs-Clτ(H))˜Clσ(fpu(H)).

(e) For each KSS(Y, B),

ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)).

(f) For each KSS(Y, B),

fpu-1(intσ(K))˜ωs-intτ(fpu-1(K)).
Proof

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let Nσc. Then 1BNσ – {0B}. Choose ℳ1 ⊆ ℳ such that 1BN = ∪̃{M : B ∈ ℳ1}. Then

1A-fpu-1(N)=fpu-1(1B-N)=fpu-1(˜{M:B1})=˜{fpu-1(M):M1}.

By (b), fpu-1(M)ωs(X,τ,A) for all M ∈ ℳ. Thus, 1A-fpu-1(N)ωs(X,τ,A), and hence fpu-1(N) is soft ωs-closed.

(c) ⇒ (d): Let HSS(X, A). Then Clσ (fpu (H)) ∈ σc. So, by (c) fpu-1(Clσ(fpu(H))) is soft ωs-closed. Because H˜fpu-1(fpu(H))˜fpu-1(Clσ(fpu(H))) and fpu-1(Clσ(fpu(H))) is soft ωs-closed,

ωs-Clτ(H)˜fpu-1(Clσ(fpu(H))).

Thus,

fpu(ωs-Clτ(H))˜fpu(fpu-1(Clσ(fpu(H))))˜Clσ(fpu(H))).

(d) ⇒ (e): Let KSS(Y, B). Then by (d),

fpu(ωs-Clτ(fpu-1(K)))˜Clσ(fpu(fpu-1(K)))˜Clσ(K).

Therefore,

ωs-Clτ(fpu-1(K))˜fpu-1(fpu(ωs-Clτ(fpu-1(K))))˜fpu-1(Clσ(K)).

(e) ⇒ (f): Let KSS(Y, B). Then by (e), ωs-Clτ(fpu-1(1B-K))˜fpu-1(Clσ(1B-K)). In addition, by Theorem 3.33(c) of [3],

1A-ωs-Clτ(1A-fpu-1(K))=ωs-intτ(fpu-1(K)).

Thus,

fpu-1(intσ(K)))=fpu-1(1B-Clσ(1B-K))=1A-fpu-1(Clσ(1B-K))˜1A-ωs-Clτ(fpu-1(1B-K))=1A-ωs-Clτ(1A-fpu-1(K))=ωs-intτ(fpu-1(K)).

(f) ⇒ (a): Let Gσ. Then G = intσ (G). Thus, by (f), fpu-1(G)˜ωs-intτ(fpu-1(G)). Therefore, fpu-1(G)=ωs-intτ(fpu-1(G)) and hence fpu-1(G)fpu-1(G)ωs(X,τ,A).

Lemma 3.9

Let (X, τ, A) be an STS and let HSS(X, A). Then

ωs-Clτ(H)=H˜intτω(Clτ(H)).
Proof

Because ωsClτ (H) is soft ωs-closed, then by Theorem 2.21 of [3], intτω (Clτ (ωs-Clτ (H))) ⊂̃ ωs-Clτ (H). Thus, intτω(Clτ (H)) ⊂̃ intτω(Clτ (ωs-Clτ (H))) ⊂̃ ωs-Clτ (H). Hence, H ∪̃ intτω (Clτ (H)) ⊂̃ ωs-Clτ (H). To show that ωs-Clτ (H) ⊂̃ H ∪̃ intτω (Clτ (H)), it is sufficient to show that H ∪̃ intτω (Clτ (H)) is soft ωs-closed. Because intτω (Clτ (H)) ⊂̃ Clτ (H),

Clτ(intτω(Clτ(H)))˜Clτ(H).

Thus,

intτω(Clτ(H˜intτω(Clτ(H))))=intτω(Clτ(H)˜Clτ(intτω(Clτ(H))))=intτω(Clτ(H))˜H˜intτω(Clτ(H)).

Therefore, by Theorem 2.21 of [3], H ∪̃ intτω (Clτ (H)) is soft ωs-closed.

Theorem 3.10

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For each HSS(X, A),

fpu(intτω(Clτ(H))˜Clσ(fpu(H)).

(c) For each KSS(Y, B),

intτω(Clτ(fpu-1(K)))fpu-1(Clσ(H)).
Proof

(a) ⇒ (b): Suppose that fpu is soft ωs-continuous. Let HSS(X, A). Then by part (d) of Theorem 3.8,

fpu (ωs-Clτ (H)) ⊂̃ Clσ (fpu (H)). Thus, by Lemma 3.9, it follows that

fpu(intτω(Clτ(H))˜fpu(ωs-Clτ(H))˜Clσ(fpu(H)).

(b) ⇒(a): We apply Theorem 3.8(d). Let HSS(X, A). Then by (b),

fpu(intτω(Clτ(H))˜Clσ(fpu(H)).

Thus, by Lemma 3.9,

fpu(ωs-Clτ(H))=fpu(H˜intτω(Clτ(H)))=fpu(H)˜fpu(intτω(Clτ(H)))˜Clσ(fpu(H)).

(a) ⇒ (c): Suppose that fpu is soft ωs-continuous and let KSS(Y, B). Then by Theorem 3.8(e), ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)). Therefore, by Lemma 3.9, it follows that

intτω(Clτ(fpu-1(K)))˜ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)).

(c) ⇒ (a): We apply Theorem 3.8(e). Let KSS(Y, B). Then by (c), we have intτω(Clτ(fpu-1(K)))˜fpu-1(Clσ(K)). Thus, by Lemma 3.9, it follows that

ωs-Clτ(fpu-1(K))=fpu-1(K)˜intτω(Clτ(fpu-1(K)))˜fpu-1(Clσ(K)).

Theorem 3.11

If fpu : (X, τ, A) → (Y, σ, B) is soft ωs-continuous and if fqv : (Y, σ, B) → (Z, δ, C) is soft continuous, then f(qp)(vu) : (X, τ, A) → (Z, δ, C) is soft ωs-continuous.

Proof

Let Kδ. Because fqv : (Y, σ,, B) → (Z, δ, C) is soft continuous, fqv-1(K)σ. Because fpu is soft ωs-continuous, fpu-1(fqv-1(K))ωs(X,τ,A). However, fpu-1(fqv-1(K))=f(qp)(vu)-1(K). Therefore, f(qp)(vu)-1(K)ωs(X,τ,A). Hence, f(qp)(vu) : (X, τ, A) → (Z, δ, C) is soft ωs-continuous.

The composition of two soft ωs-continuous functions is not required to be soft ωs-continuous, in general:

Example 3.12

Let ℑ be the usual topology on ℝ and let τ = {FSS(ℝ, ℕ) : F (a) ∈ ℑ for all a ∈ ℕ}. Define p, q :

ℝ → ℝ and u, v : ℕ → ℕ as follows:

p(x)={x,if x1,0,if x>1,q(x)={0,if x<1,3,if x1,u(a)=v(a)=afor all a.

Then

(qp)(x)={0,if x1,3,if x=1,

and

(vu)(a)=afor all a.

fpu and fqv are obviously soft semi-continuous, and (ℝ, τu, ℕ) is soft anti-locally countable; thus, by Theorem 3.6, fpu and fqv are soft ωs-continuous. Let GSS(ℝ, ℕ), where G = C(2, ∞). Then Gτ, but (fqvfpu)-1(G)=(f(qp)(vu)-1)   (G)=H, where H(a) = {1} for all a ∈ ℕ. Thus, (fqvfpu)−1 (G) ∉ ωs(ℝ, τ, ℕ). Hence, fqvfpu is not soft ωs-continuous.

Lemma 3.13

Let (X, ℑ) be a TS and let A be a set of parameters. Let HSS(X, A). Then Hωs(X, τ (ℑ), A) if and only if H (a) ∈ ωs(X, ℑ) for all aA.

Proof

For each aA, set ℑa = ℑ. Then τ(I)=aAIa. Then by Theorem 4.8 of [3], we obtain the result.

Theorem 3.14

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ωs-continuous if and only if fpu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ωs-continuous.

Proof
Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ωs-continuous. Let Gτ (ℵ). Then for each aA, G(u (a)) ∈ ℵ, and so p−1(G(u (a))) ∈ ωs(X, ℑ). Therefore, (fpu-1(G))   (a)=p-1(G(u(a)))ωs(X,I) for all aA. Hence, by Lemma 3.13, fpu-1(G)ωs(X,τ(I),A).

Sufficiency

Suppose that fpu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ωp-continuous. Let V ∈ ℵ. Then CVτ (ℵ), and so fpu-1(CV)ωs(X,τ(I),A). Pick aA. Then, by Lemma 3.13, (fpu-1(CV))   (a)=p-1(V)ωs(X,I).

The classes of ω-continuous functions and soft ωs-continuous functions were extended to include STSs. Also, soft. Several characterizations, relationships, and examples were given. The following topics could be considered in future studies: defining soft ω-open functions and defining soft ωs-open functions.

No potential conflict of interest relevant to this article was reported.

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Samer Al Ghour received his Ph.D. in Mathematics from the University of Jordan, Jordan in 1999. Currently, he is a professor in the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo


Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 183-192

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.183

Copyright © The Korean Institute of Intelligent Systems.

Soft -Continuity and Soft -Continuity in Soft Topological Spaces

Samer Al Ghour

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

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

Received: November 28, 2021; Revised: January 16, 2022; Accepted: January 26, 2022

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

Abstract

In this study, soft ω-continuity and soft ωs-continuity are introduced as two new classes of soft functions, and several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity and that soft ωs-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions are introduced for the equivalence between soft ωs-continuity and soft ω-continuity, as well as that between soft ωs-continuity and soft semi-continuity. Furthermore, composition theorems regarding soft ω-continuity and soft ωs-continuity are given. Finally, the relationships between the generated soft topological spaces and induced topological spaces are studied.

Keywords: Soft semi-continuous functions, Soft ω-open sets, ω-continuity, ωs-continuity, Soft generated soft topological space, Soft induced topological spaces

1. Introduction and Preliminaries

Herein, we follow the notions and terminology that appear in [13]. Throughout this paper, ST denotes a topological space and STS denotes a soft topological space. To deal with uncertain objects, Molodtsov [4] introduced soft sets in 1999. Let X be a universal set and A be a set of parameters. A function K : A → ℘(X) is called a soft set over X relative to A. SS (X, A) denotes the family of all soft sets over X relative to A. A soft set KSS (X, A) is called a countable soft set if for each aA, the set K (a) is countable. CSS (X, A) denotes the family of all members of SS (X, A) that are countable. The null soft set and absolute soft set are denoted by 0A and 1A, respectively. The concept of STS was defined in [5] as follows: the triplet (X, τ, A), where τSS (X, A), is called an STS if 0A and 1Aτ , τ are closed under a finite soft intersection, and τ is closed under an arbitrary soft union. For an STS (X, τ, A) the members of τ are called soft open sets, and their soft complements are called soft closed sets. The family of all soft closed sets is denoted by τc. Mathematicians have extended many STs to STSs; some recent studies have also been conducted ([13,629]). Chen [30] introduced the concept of soft semi-open sets. Following this, many research papers on soft semi-open sets and their modifications appeared. Soft ω-openness was defined in [2] as a generalization of soft openness. In a previous study [3], the author used soft ω-open sets to define ωs-open sets as a class of soft sets that lies strictly between the soft open sets and soft semi-open sets. In this study, soft ω-continuity and soft ωs-continuity are introduced as two new classes of soft functions using soft ω-open sets and soft ωs-open sets. In addition, several characterizations of these concepts are given. It is proven that soft ω-continuity is weaker than soft continuity, and soft ωs-continuity lies strictly between soft ω-continuity and soft semi-continuity. Sufficient conditions for the equivalence between soft ωs-continuity and soft ω- continuity, as well as that between soft ωs-continuity and soft semi-continuity, are introduced. Composition theorems regarding soft ω-continuity and soft ωs-continuity are also provided. Finally, the relationships regarding generated STSs and induced STs are studied.

Let (X, τ, A) be an STS, (X, ℑ) be a TS, HSS(X, A), and DX. Throughout this paper, Clτ (H), intτ (H), and Cl (D) denotes the soft closure of H in (X, τ, A), the soft interior of H in (X, τ, A), and the closure of D in (X,ℑ), respectively.

The following definitions and results are used in the following:

Definition 1.1

Let X be a universal set and A be a set of parameters. Then GSS(X, A) is defined by

(a) G(a)={Y,if a=e,if ae, is denoted by eY.

(b) [1] G(a) = Y for all aA is denoted by CY.

(c) G(a)={{x},if a=e,,if ae, is denoted by ex, and is called a soft point.

The set of all soft points in SS(X, A) is denoted by SP (X, A).

Definition 1.2 [31]

LetGSS(X, A) and exSP(X, A). Then ex is said to belong to F (notation: ex ∊̃ G) if ex ∊̃ G or, equivalently, ex ∊̃ G if and only if xG(e).

Theorem 1.3 [5]

Let (X, τ, A) be an STS. Then the collection {F(a) : Fτ} defines a topology on X for every aA. This topology is denoted by τa .

Theorem 1.4 [32]

Let (X,ℑ) be a TS. Then the collection

{FSS(X,A):F(a)Ifor all aA}

defines a soft topology on X relative to A. This soft topology is denoted by τ (ℑ).

Theorem 1.5 [1]

Let X be an initial universe and let A be a set of parameters. Let {ℑa : aA} be an indexed family of topologies on X and let

τ={FSS(X,A):F(a)Iafor all aA}.

Then τ defines a soft topology on X relative to A. This soft topology is denoted by aAIa.

Definition 1.6

An STS (X, τ, A) is called

(a) [2] soft locally countable if for each axSP(X, A), there exists Gτ such that ax ∊̃ G and G is a countable soft set.

(b) [2] soft anti-locally countable if for every Fτ −{0A}, F is not a countable soft set.

Definition 1.7

Let (X,ℑ) be a ST, and let DX. D is called

(a) [33] semi-open if there exists V ∈ ℑ such that VDCl (V ). SO(X, ℑ) denotes the family of all semi-open sets in (X,ℑ).

(b) [34] ωs-open if there exists V ∈ ℑ such that VDClω (V ). ωs(X, ℑ) denotes the family of all ωs-open sets in (X, ℑ).

Similar types of generalized sets in ideal STs have been discussed in [3539].

Definition 1.8 [40]

A function g : (X, ℑ) → (Y, ℵ) is called ω-continuous at a point xX if for every V ∈ ℵ with g(x) ∈ V, there exists U ∈ ℑω such that xU and g(U) ⊆ V . If g is ω-continuous at each point xX, then g is called ω-continuous.

Definition 1.9 [33]

A function g : (X, ℑ) → (Y, ℵ) is called semi-continuous if for each V ∈ ℵ, g−1(V ) ∈ SO(X, ℑ).

Definition 1.10 [34]

A function g : (X, ℑ) → (Y, ℵ) is called ωs-continuous if for each V ∈ ℵ, g−1(V ) ∈ ωs(X, ℑ).

Definition 1.11

Let (X, τ, A) be an STS and let KSS (X, A). Then

(a) [30] K is called a soft semi-open set in (X, τ, A) if there exists Fτ such that F ⊂̃ K ⊂̃ Clτ (F). SO(X, τ, A) denotes the family of all soft semi-open sets in (X, τ, A).

(b) [30] K is called a soft semi-closed set in (X, τ, A) if 1AKSO(X, τ, A).

(c) [3] K is called a soft ωs-open set in (X, τ, A) if there exists Fτ such that F ⊂̃ K ⊂̃ Clτω (F). ωs(X, τ, A) denotes the family of all soft ωs-open sets in (X, τ, A).

According to Definition 1.11, a soft semi-open set always contains a soft open set. Therefore, this is a particular type of generalized soft open set. One can specify another type of generalized set in the STS that does not necessarily contain a soft open set. In the ST, two different types of sets have been discussed in [41].

Definition 1.12 [42]

A soft function fpu : (X, τ, A) → (Y, σ, B) is called soft semi-continuous if for every Gσ, fpu-1(G)SO(X,τ,A).

Definition 1.13 [3]

Let (X, τ, A) be an STS and HSS (X, A).

(a) The soft ωs-closure of H in (X, τ, A) is denoted by ωs-Clτ (H) and defined by

ωs-Clτ(H)=˜{M:Mis soft ωs-closed in (X,τ,A)and H˜M}.

(b) The soft ωs-interior of H in (X, τ,A) is denoted by ωsintτ (H) and defined by

ωs-intτ(H)=˜{K:Kis soft ωs-open in (X,τ,A)and K˜H}.

2. Soft ω-Continuity

In this section, we introduce the concept of soft ω-continuous function and establish its main properties. With the help of examples, we investigate the interaction between these concepts within the frameworks of general topology and soft topology.

Definition 2.1

A soft function fpu : (X, τ,A) → (Y, σ,B) is called soft ω-continuous at a soft point axSP(X,A), if Gσ with fpu(ax)∊̃G, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. If fpu is soft ω-continuous at each soft point axSP(X,A), then fpu is called soft ω-continuous.

Theorem 2.2

For a soft function fpu : (X, τ,A) → (Y, σ,B), the following conditions are equivalent:

(a) fpu is soft ω-continuous.

(b) For each Gσ, fpu-1(G)τω.

(c) For a soft base ℳ of (Y, σ,B), fpu-1(M)τω for all M ∈ ℳ.

(d) For a soft subbase of (Y, σ,B), fpu-1(S)τω for all .

(e) For each Nσc, fpu-1(N)(τω)c.

(f) For each KSS(X,A), fpu(Clτω (K))⊂̃Clσ (fpu(K)).

(g) For each LSS(Y,B), Clτω(fpu-1(L))˜fpu-1(Clσ(L)).

Proof

(a) =⇒ (b): Let Gσ and let ax˜fpu-1(G). Then fpu (ax) ∊̃G. By (a), fpu is soft ω-continuous at ax. Thus, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. Because fpu(H)⊂̃G, then we have ax˜H˜fpu-1(fpu(H))˜fpu-1(G). Therefore, fpu-1(G)τω.

(b) =⇒ (c): Obvious.

(c) ⇒ (d): Obvious.

(d) ⇒ (e): Let Nσc. We will show that 1A-fpu-1(N)τω. Let ax˜1A-fpu-1(N)=fpu-1(1B-N). Then fpu (ax) ∊̃; 1BNσ. Since in (d) is a soft subbase for (Y, σ,B), then there exist . such that

fpu(ax)˜S1˜S2˜˜Sn˜1B-N.

So,

ax˜fpu-1(S1)˜fpu-1(S2)˜˜fpu-1(Sn)˜fpu-1(1B-N)=1A-fpu-1(N).

By (d), fpu-1(S1)˜fpu-1(S2)˜˜fpu-1(Sn)τω, and thus, 1A-fpu-1(N)τω.

(e) ⇒ (f): Let KSS(X,A). Then Clσ (fpu(K)) ∈ σc. Thus, by (e), fpu-1(Clσ(fpu(K)))(τω)c. Because K˜fpu-1(fpu(K))˜fpu-1(Clσ(fpu(K))),

Clτω(K)˜fpu-1(Clσ(fpu(K))),

and so,

fpu(Clτω(K))˜fpu(fpu-1(Clσ(fpu(K))))˜Clσ(fpu(K)).

(f) ⇒ (g): Let LSS(Y,B). Then fpu-1(L)SS(X,A). Thus, by (f),

fpu(Clτω(fpu-1(L)))˜Clσ(fpu(fpu-1(L)))˜Clσ(L).

Therefore,

Clτω(fpu-1(L))˜fpu-1(fpu(Clτω(fpu-1(L))))˜fpu-1(Clσ(L)).

(g) ⇒ (a): Let axSP(X,A) and Gσ such that fpu(ax) ∈ G. Then 1BGσc. By (g), Clτω(1A-fpu-1(G))=Clτω(fpu-1(1B-G))˜fpu-1(Clσ(1B-G))=˜fpu-1(1B-G)=1A-fpu-1(G). Thus, 1A-fpu-1(G)(τω)c and hence fpu-1(G)τω. Put H=fpu-1(G). Then ax∊̃;Hτω and fpu(H)=fpu(fpu-1(G))˜G. Therefore, f is soft ω-continuous at ax.

Theorem 2.3

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at axSP(X,A), then p : (X, τa) → (Y, σu(a)) is ω-continuous at x.

Proof

Suppose that fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax. Let Vσu(a) such that p (x) ∈ V . Choose Gσ such that G(u(a)) = V . Then p (x) ∈ G(u(a)), so fpu(ax) = (u(a))p(x)G. Because fpu is soft ω-continuous at ax, there exists Hτω such that ax∊̃H and fpu(H)⊂̃G. Thus, we have xH(a) ∈ (τω)a. Also, by Theorem 7 of [2], (τω)a = (τa)ω. To see that p(H(a)) ⊆ V , let zH (a); then, az∊̃H. Because fpu(H)⊂̃G, fpu(az) = (u (a))p(z) ∊̃G and so p(z) ∈ G(u(a)) = V . This shows that p : (X, τa) → (Y, σu(a)) is ω-continuous at x.

Theorem 2.4

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then p : (X, τa) → (Y, σu(a)) is ω-continuous for all aA.

Proof

Let aA. To demonstrate that p : (X, τa) → (Y, σu(a)) is ω-continuous, let xX. Then fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax. So, by Theorem 2.3, p : (X, τa) → (Y, σu(a)) is ω-continuous at x. This shows that p : (X, τa) → (Y, σu(a)) is ω-continuous.

Theorem 2.5

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ω-continuous at a point xX if and only if fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at ax for all aA.

Proof
Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ω-continuous at x. Let aA and let Gτ (ℵ) such that fpu(ax) ∊̃G. Then p(x) ∈ G(u (a)) ∈ ℵ. Because p : (X, ℑ) → (Y, ℵ) is ω-continuous at x, there exists U ∈ ℑω such that xU and p(U) ⊆ G(u (a)). Thus, we have ax∊̃aUτ (ℑω) = (τ (ℑ))ω and fpu(aU) = (u(a))p(U) ⊂̃G. Hence, fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous at ax.

Sufficiency

Suppose that fpu : (X, τ(ℑ),A) → (Y, τ(ℵ), B) is soft ωp-continuous at ax. Then by Theorem 2.3,

p : (X, (τ (ℑ))a) → (Y, (τ (ℵ))u(a)) is ω-continuous at x. Because (τ (ℑ))a = ℑ and (τ (ℵ))u(a) = ℵ, p : (X, ℑ) → (Y, ℵ) is ω-continuous at x.

Corollary 2.6

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ω-continuous if and only if fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous.

Theorem 2.7

Soft continuous soft functions are soft ω-continuous.

Proof

Follows from Theorem 2.2 and Theorem 2(a) of [2].

The following example shows that the converse of Theorem 2.7 is not true in general:

Example 2.8

Let X = Y = ℤ, and A = B = ℝ. Let ℑ = {∅︀,X,ℕ} and ℵ be the cofinite topology on Y . Define p : XY and u : AB by p (x) = x and u(a) = a for all xX and aA. Then fpu : (X, τ (ℑ),A) → (Y, τ (ℵ),B) is soft ω-continuous but not soft continuous.

Theorem 2.9

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and if fqv : (Y, σ,B) → (Z, δ,C) is soft continuous, then f(qp)(vu) : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Proof

Let Kδ. Because fqv : (Y, σ, ,B) → (Z, δ,C) is soft continuous, then fqu-1(K)σ. Because fpu is soft ω-continuous, fqu-1(fqv-1(K))τω. However, fpu-1(fqv-1(K))=f(qp)(vu)-1(K). Therefore, f(qp)(vu)-1(K)τω. Hence, f(qp)(vu) : (X, τ,A) → (Z, δ,C) is soft ω-continuous.

Theorem 2.10

If fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous and Z is a non-empty subset of X, then the soft restriction f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous.

Proof

Let Kσ. Since fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous, then fpu-1(K)τω. So, f(pZ)u-1(K)=fpu-1(K)˜CY(τω)Y. Hence, by Theorem 15 of [2], f(pZ)u-1(K)(τY)ω. Therefore, f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous.

Theorem 2.11

Let fpu : (X, τ,A) → (Y, σ,B) be a soft function and let 1A = CZ ∪̃CW, where CZ, CWτc − {0A}. If f(p|Z)u : (Z, τZ,A) → (Y, σ,B) and f(p|W)u : (W, τW,A) → (Y, σ,B) are soft ω-continuous functions, then fpu is soft ω-continuous.

Proof

We apply statement (e) of Theorem 2.2. Let Nσc. Then

fpu-1(N)=fpu-1(N)˜1A=fpu-1(N)˜(CZ˜CW)=(fpu-1(N)˜CZ)˜(fpu-1(N)˜CW)=(f(pZ)u-1(N))˜(f(pW)u-1(N)).

Because f(p|Z)u and f(p|W)u are soft ω-continuous, then

f(pZ.)u-1(N)((τZ)ω)cand f(pW.)u-1(N)((τW)ω)c.

Therefore, by Theorem 17 of [2], f(pZ.)u-1(N)((τω)Z)c and f(pW.)u-1(N)((τω)W)c. Since CZ, CWτ c, then f(pZ.)u-1(N),f(pW.)u-1(N)(τω)c. Therefore, fpu-1(N)(τω)c.

Theorem 2.12

Let fpu : (X, τ,A) → (Y, σ,B) be a soft function and let axSP(X,A). If there exists ZX such that CZτω such that ax∊̃CZ and f(p|Z)u : (Z, τZ,A) → (Y, σ,B) is soft ω-continuous at ax, then fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax.

Proof

Let Gσ such that fpu(ax) ∈ G. Because f(p|Z)u is soft ω-continuous at ax, there exists H ∈ (τZ)ω such that ax∊̃H and f(p|Z)u(H) ⊂̃G. BecauseH ∈ (τZ)ω and CZτω, Hτω. Therefore, fpu : (X, τ,A) → (Y, σ,B) is soft ω-continuous at ax.

Corollary 2.13

fpu : (X, τ, A) → (Y, σ, B) be a soft function. Let {CXα : α ∈ Δ} ⊆ τ such that 1A = ∪̃{CXα : α ∈ Δ}. If for each α ∈ Δ, f(p|Xα)u : (Xα, τXα, A) → (Y, σ, B) is soft ω-continuous, then fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

Proof

Let axSP(X, A). We show that fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous at ax. Because 1A = ∪̃{CXα : α ∈ Δ}, then there exists α ∈ Δsuch that ax∊̃CXα. Therefore, by Theorem 2.12, it follows that fpu : (X, τ, A) → (Y, σ, B) is soft ω-continuous.

3. Soft ωs-Continuity

In this section, we introduce the concept of a soft ωs-continuous function and establish its main properties. We also provide several characterizations of this concept, and study the relationships between soft ωs-continuity and some other types of soft continuity.

Definition 3.1

A soft function fpu : (X, τ, A) → (Y, σ, B) is called soft ωs-continuous if for every Gσ, fpu-1(G)ωs(X,τ,A).

Theorem 3.2

Soft continuous functions are soft ωs-continuous.

Proof

Let fpu : (X, τ, A) → (Y, σ, B) be soft continuous and let Gσ. By soft continuity of fpu, fpu-1(G)τ. So by Theorem 2.2 of [3], fpu-1(G)ωs(X,τ,A). Hence, fpu is soft ωs-continuous.

Theorem 3.3

Soft ωs-continuous functions are soft semi-continuous.

Proof

Let fpu : (X, τ, A) → (Y, σ, B) be soft ωs-continuous and let Gσ. By soft ωs-continuity of fpu, fpu-1(G)ωs(X,τ,A). Thus, by Theorem 2.2 of [3], fpu-1(G)SO(X,τ,A). Hence, fpu is soft semi-continuous.

The following example shows that the converses of Theorems 3.2 and 3.3 are not true in general:

Example 3.4

Let X = ℝ, Y = {a, b}, A = B = ℤ, τ = {0A, 1A, C, Cc, C(ℕc)}, and σ = {0B, 1B, C{a}, C{b}}. It is not difficult to see that Clτω (C) = C, Clτ (C) = C, and Clτω (Cc) = Cℝ−ℕ. Thus, CSO(X, τ, A) − ωs(X, τ, A) and Cℝ−ℕωs(X, τ, A)−τ. Let p, q : XY and u, v : AB be defined by

p(x)={a,if x,b,if x-,q(x)={a,if xc,b,if x,u(a)=v(a)=afor all aA.

Because fpu-1(C{a})=Cτωs(X,τ,A) and fpu-1(C{b})=C-ωs(X,τ,A)-τ, then fpu is soft ωs-continuous but not soft continuous. Additionally, because fqv-1(C{a})=CcτSO(X,τ,A) and fqv-1(C{b})=CSO(X,τ,A)-ωs(X,τ,A), fqv is soft semi-continuous but not soft ωs-continuous.

Theorem 3.5

If fpu : (X, τ, A) → (Y, σ, B) is soft ωs-continuous such that (X, τ, A) is soft locally countable, then fpu is soft continuous.

Proof

Let Gσ. By the soft ωs-continuity of fpu, fpu-1(G)ωs(X,τ,A). Thus, by Theorem 2.8 of [3], fpu-1(G)τ. Hence, fpu is soft continuous.

Theorem 3.6

If fpu : (X, τ, A) → (Y, σ, B) is soft semi-continuous such that (X, τ, A) is soft anti-locally countable, then fpu is soft ωs-continuous.

Proof

Let Gσ. By the soft semi-continuity of fpu, fpu-1(G)SO(X,τ,A). Thus, by Theorem 2.6 of [3], fpu-1(G)ωs(X,τ,A). Hence, fpu is soft ωs-continuous.

Theorem 3.7

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For each axSP(X, A) and each Gσ such that fpu(ax)∊̃G, there exists Hωs (X, τ, A) such that ax∊̃H and fpu(H)⊂̃G.

Proof

(a) ⇒ (b): Assume that fpu is soft ωs-continuous. Let Gσ such that fpu(ax)∊̃G. Because fpu is soft ωs-continuous, fpu-1(G)ωs(X,τ,A). Take H=fpu-1(G). Then Hωs (X, τ, A) such that ax∊̃H and fpu(H)⊂̃G.

(b) ⇒ (a): Let Gσ. For each ax˜fpu-1(G), we have fpu(ax)∊̃G, and by (b), there exists Haxωs (X, τ, A) such that ax∊̃Hax and fpu(Hax )⊂̃G, and thus, ax˜Hax˜fpu-1(G). Therefore, fpu-1(G)=˜{Hax:ax˜fpu-1(G)}, and hence fpu-1(G)ωs(X,τ,A).

Theorem 3.8

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For a soft base of (Y, σ, B), fpu-1(M)ωs(X,τ,A) for all M.

(c) For each Nσc, fpu-1(N) is soft ωs-closed.

(d) For each HSS(X, A),

fpu(ωs-Clτ(H))˜Clσ(fpu(H)).

(e) For each KSS(Y, B),

ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)).

(f) For each KSS(Y, B),

fpu-1(intσ(K))˜ωs-intτ(fpu-1(K)).
Proof

(a) ⇒ (b): It is obvious.

(b) ⇒(c): Let Nσc. Then 1BNσ – {0B}. Choose ℳ1 ⊆ ℳ such that 1BN = ∪̃{M : B ∈ ℳ1}. Then

1A-fpu-1(N)=fpu-1(1B-N)=fpu-1(˜{M:B1})=˜{fpu-1(M):M1}.

By (b), fpu-1(M)ωs(X,τ,A) for all M ∈ ℳ. Thus, 1A-fpu-1(N)ωs(X,τ,A), and hence fpu-1(N) is soft ωs-closed.

(c) ⇒ (d): Let HSS(X, A). Then Clσ (fpu (H)) ∈ σc. So, by (c) fpu-1(Clσ(fpu(H))) is soft ωs-closed. Because H˜fpu-1(fpu(H))˜fpu-1(Clσ(fpu(H))) and fpu-1(Clσ(fpu(H))) is soft ωs-closed,

ωs-Clτ(H)˜fpu-1(Clσ(fpu(H))).

Thus,

fpu(ωs-Clτ(H))˜fpu(fpu-1(Clσ(fpu(H))))˜Clσ(fpu(H))).

(d) ⇒ (e): Let KSS(Y, B). Then by (d),

fpu(ωs-Clτ(fpu-1(K)))˜Clσ(fpu(fpu-1(K)))˜Clσ(K).

Therefore,

ωs-Clτ(fpu-1(K))˜fpu-1(fpu(ωs-Clτ(fpu-1(K))))˜fpu-1(Clσ(K)).

(e) ⇒ (f): Let KSS(Y, B). Then by (e), ωs-Clτ(fpu-1(1B-K))˜fpu-1(Clσ(1B-K)). In addition, by Theorem 3.33(c) of [3],

1A-ωs-Clτ(1A-fpu-1(K))=ωs-intτ(fpu-1(K)).

Thus,

fpu-1(intσ(K)))=fpu-1(1B-Clσ(1B-K))=1A-fpu-1(Clσ(1B-K))˜1A-ωs-Clτ(fpu-1(1B-K))=1A-ωs-Clτ(1A-fpu-1(K))=ωs-intτ(fpu-1(K)).

(f) ⇒ (a): Let Gσ. Then G = intσ (G). Thus, by (f), fpu-1(G)˜ωs-intτ(fpu-1(G)). Therefore, fpu-1(G)=ωs-intτ(fpu-1(G)) and hence fpu-1(G)fpu-1(G)ωs(X,τ,A).

Lemma 3.9

Let (X, τ, A) be an STS and let HSS(X, A). Then

ωs-Clτ(H)=H˜intτω(Clτ(H)).
Proof

Because ωsClτ (H) is soft ωs-closed, then by Theorem 2.21 of [3], intτω (Clτ (ωs-Clτ (H))) ⊂̃ ωs-Clτ (H). Thus, intτω(Clτ (H)) ⊂̃ intτω(Clτ (ωs-Clτ (H))) ⊂̃ ωs-Clτ (H). Hence, H ∪̃ intτω (Clτ (H)) ⊂̃ ωs-Clτ (H). To show that ωs-Clτ (H) ⊂̃ H ∪̃ intτω (Clτ (H)), it is sufficient to show that H ∪̃ intτω (Clτ (H)) is soft ωs-closed. Because intτω (Clτ (H)) ⊂̃ Clτ (H),

Clτ(intτω(Clτ(H)))˜Clτ(H).

Thus,

intτω(Clτ(H˜intτω(Clτ(H))))=intτω(Clτ(H)˜Clτ(intτω(Clτ(H))))=intτω(Clτ(H))˜H˜intτω(Clτ(H)).

Therefore, by Theorem 2.21 of [3], H ∪̃ intτω (Clτ (H)) is soft ωs-closed.

Theorem 3.10

For a soft function fpu : (X, τ, A) → (Y, σ, B), the following are equivalent:

(a) fpu is soft ωs-continuous.

(b) For each HSS(X, A),

fpu(intτω(Clτ(H))˜Clσ(fpu(H)).

(c) For each KSS(Y, B),

intτω(Clτ(fpu-1(K)))fpu-1(Clσ(H)).
Proof

(a) ⇒ (b): Suppose that fpu is soft ωs-continuous. Let HSS(X, A). Then by part (d) of Theorem 3.8,

fpu (ωs-Clτ (H)) ⊂̃ Clσ (fpu (H)). Thus, by Lemma 3.9, it follows that

fpu(intτω(Clτ(H))˜fpu(ωs-Clτ(H))˜Clσ(fpu(H)).

(b) ⇒(a): We apply Theorem 3.8(d). Let HSS(X, A). Then by (b),

fpu(intτω(Clτ(H))˜Clσ(fpu(H)).

Thus, by Lemma 3.9,

fpu(ωs-Clτ(H))=fpu(H˜intτω(Clτ(H)))=fpu(H)˜fpu(intτω(Clτ(H)))˜Clσ(fpu(H)).

(a) ⇒ (c): Suppose that fpu is soft ωs-continuous and let KSS(Y, B). Then by Theorem 3.8(e), ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)). Therefore, by Lemma 3.9, it follows that

intτω(Clτ(fpu-1(K)))˜ωs-Clτ(fpu-1(K))˜fpu-1(Clσ(K)).

(c) ⇒ (a): We apply Theorem 3.8(e). Let KSS(Y, B). Then by (c), we have intτω(Clτ(fpu-1(K)))˜fpu-1(Clσ(K)). Thus, by Lemma 3.9, it follows that

ωs-Clτ(fpu-1(K))=fpu-1(K)˜intτω(Clτ(fpu-1(K)))˜fpu-1(Clσ(K)).

Theorem 3.11

If fpu : (X, τ, A) → (Y, σ, B) is soft ωs-continuous and if fqv : (Y, σ, B) → (Z, δ, C) is soft continuous, then f(qp)(vu) : (X, τ, A) → (Z, δ, C) is soft ωs-continuous.

Proof

Let Kδ. Because fqv : (Y, σ,, B) → (Z, δ, C) is soft continuous, fqv-1(K)σ. Because fpu is soft ωs-continuous, fpu-1(fqv-1(K))ωs(X,τ,A). However, fpu-1(fqv-1(K))=f(qp)(vu)-1(K). Therefore, f(qp)(vu)-1(K)ωs(X,τ,A). Hence, f(qp)(vu) : (X, τ, A) → (Z, δ, C) is soft ωs-continuous.

The composition of two soft ωs-continuous functions is not required to be soft ωs-continuous, in general:

Example 3.12

Let ℑ be the usual topology on ℝ and let τ = {FSS(ℝ, ℕ) : F (a) ∈ ℑ for all a ∈ ℕ}. Define p, q :

ℝ → ℝ and u, v : ℕ → ℕ as follows:

p(x)={x,if x1,0,if x>1,q(x)={0,if x<1,3,if x1,u(a)=v(a)=afor all a.

Then

(qp)(x)={0,if x1,3,if x=1,

and

(vu)(a)=afor all a.

fpu and fqv are obviously soft semi-continuous, and (ℝ, τu, ℕ) is soft anti-locally countable; thus, by Theorem 3.6, fpu and fqv are soft ωs-continuous. Let GSS(ℝ, ℕ), where G = C(2, ∞). Then Gτ, but (fqvfpu)-1(G)=(f(qp)(vu)-1)   (G)=H, where H(a) = {1} for all a ∈ ℕ. Thus, (fqvfpu)−1 (G) ∉ ωs(ℝ, τ, ℕ). Hence, fqvfpu is not soft ωs-continuous.

Lemma 3.13

Let (X, ℑ) be a TS and let A be a set of parameters. Let HSS(X, A). Then Hωs(X, τ (ℑ), A) if and only if H (a) ∈ ωs(X, ℑ) for all aA.

Proof

For each aA, set ℑa = ℑ. Then τ(I)=aAIa. Then by Theorem 4.8 of [3], we obtain the result.

Theorem 3.14

Let p : (X, ℑ) → (Y, ℵ) be a function between two TSs and let u : AB be a function between two sets of parameters. Then p : (X, ℑ) → (Y, ℵ) is ωs-continuous if and only if fpu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ωs-continuous.

Proof
Necessity

Suppose that p : (X, ℑ) → (Y, ℵ) is ωs-continuous. Let Gτ (ℵ). Then for each aA, G(u (a)) ∈ ℵ, and so p−1(G(u (a))) ∈ ωs(X, ℑ). Therefore, (fpu-1(G))   (a)=p-1(G(u(a)))ωs(X,I) for all aA. Hence, by Lemma 3.13, fpu-1(G)ωs(X,τ(I),A).

Sufficiency

Suppose that fpu : (X, τ (ℑ), A) → (Y, τ (ℵ), B) is soft ωp-continuous. Let V ∈ ℵ. Then CVτ (ℵ), and so fpu-1(CV)ωs(X,τ(I),A). Pick aA. Then, by Lemma 3.13, (fpu-1(CV))   (a)=p-1(V)ωs(X,I).

4. Conclusion

The classes of ω-continuous functions and soft ωs-continuous functions were extended to include STSs. Also, soft. Several characterizations, relationships, and examples were given. The following topics could be considered in future studies: defining soft ω-open functions and defining soft ωs-open functions.

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