# Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

Authors: Fatimah M. Mohammed; M. S. M. Noorani; A. Ghareeb
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

The aim of this paper is to introduce the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions in double fuzzy topological spaces. Some interesting properties and characterizations of these functions are introduced and discussed. Furthermore, the relationships among the new concepts are discussed with some necessary examples.


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## Body

## 1. Introduction

In 1968, Chang [1] was the first to introduce the concept of fuzzy topological spaces. These spaces and their generalization are later developed by Goguen [2], who replaced the closed interval [0,1] by more general lattice L. On the other hand, by the independent and parallel generalization of Kubiak and Šostak’s [3, 4], made topology itself fuzzy besides their dependence on fuzzy set in 1985.Various generalizations of the concept of fuzzy set have been done by many authors. In [5–10], Atanassove introduced the notion of intuitionistic fuzzy sets. Later Çoker [11] defined intuitionistic fuzzy topology in Chang’s sense. Then, Mondal and Samanta [12] introduced the intuitionistic gradation of openness of fuzzy sets. Gutiérrez García and Rodabaugh [13], in 2005, replaced the term “intuitionistic” and concluded that the most appropriate work is under the name “double.”In 1980, Jain [14] introduced the notion of slightly continuous functions. On the other hand, Nour [15] defined slightly semicontinuous functions as a weak form of slight continuity and investigated their properties. In [16], Noiri introduced the concept of slightly β-continuous functions. Sudha et al. [17] introduced slightly fuzzy ω-continuous functions. Also in 2004, Ekici and Caldas [18] introduced the notion of slight γ-continuity (slight b-continuity).In this paper, the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions are introduced. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the concepts are obtained and established with some interesting counter examples.


## 2. Preliminaries

Throughout this paper, letX be a nonempty set, I the unit interval [0,1], I0=(0,1], and I1=[0,1). The family of all fuzzy sets in X is denoted by IX. Pt(X) is the family of all fuzzy points in X. By 0_ and 1_ we denote the smallest and the greatest fuzzy sets on X. For a fuzzy set λ∈IX, 1_-λ denotes its complement. Given a function f:X→Y, f(λ) and f-1(λ) defined the direct image and the inverse image of f, defined by f(λ)(y)=⋁f(x)=yλ(x) and f-1(μ)(x)=μ(f(x)) for each λ∈IX, μ∈IY, and x∈X, respectively. All other notations are standard notations of fuzzy set theory.Definition 1 (see [12, 13]).
A double fuzzy topology(τ,τ*) on X is a pair of maps τ, τ*:IX→I, which satisfies the following properties: (O1)
τ(λ)≤1_-τ*(λ) for each λ∈IX;(O2)
τ(λ1∧λ2)≥τ(λ1)∧τ(λ2) and τ*(λ1∧λ2)≤τ*(λ1)∨τ*(λ2) for each λ1,λ2∈IX;(O3)
τ(⋁i∈Γλi)≥⋀i∈Γτ(λi) and τ*(⋁i∈Γλi)≤⋁i∈Γτ*(λi) for each λi∈IX, i∈Γ.The triplet(X,τ,τ*) is called double fuzzy topological spaces (dfts, for short). A fuzzy set λ is called an (r,s)-fuzzy open ((r,s)-fo, for short) if τ(λ)≥r and τ*(λ)≤s, λ is called an (r,s)-fuzzy closed ((r,s)-fc, for short) if and only if 1_-λ is an (r,s)-fo set, and λ is called (r,s)-fuzzy clopen ((r,s)-fco, for short) if and only if λ is (r,s)-fo set and (r,s)-fc set. Let (X,τ1,τ1*) and (Y,τ2,τ2*) be two dfts's. A function f:X→Y is said to be a double fuzzy continuous if and only if τ1(f-1(ν))≥τ2(ν) and τ1*(f-1(ν))≤τ2*(ν) for each ν∈IY.Before starting to present our results, there are two questions that we must ask ourselves. First, what is the difference between classical topology and double fuzzy topology? Secondly, where we can apply our results?To answer the first question, we should know that double fuzzy sets and hence double fuzzy topological spaces deal with obscurities. In addition to that, we observed that the concept of double fuzzy topological spaces is a generalization of fuzzy topological spaces and classical topology. For example, when the first condition in Definition1 does not hold, we get the definition of fuzzy topological spaces in Kubiak-Šostak’s sense [3, 4]. Also, in the same definition, when we replace 2X with IX, we will get results in double gradation fuzzifying topological spaces [19]. Appropriate changes can be made to get results in the classical topological spaces.With regard to applications, since double fuzzy topology forms an extension of fuzzy topology and general topology, we think that our results can be applied in the fuzzy mathematics, which has many applications in different branches of engineering and ICT. For example, recently double fuzzy topological spaces have been applied to study sensor bias [20] and there exist well-established applications of fuzzy topological spaces in the areas of digital topology [21], image processing [22], and geographic information systems (GIS) problems [23].Theorem 2 (see [24, 25]).
Let(X,τ,τ*) be a dfts. Then for each r∈I0, s∈I1, and λ∈IX, one defines an operator Cτ,τ*:IX×I0×I1→IX as follows:
(1)Cτ,τ*(λ,r,s)=⋀{μ∈IX∣λ≤μ,τ(1_-μ)≥r,hh.hμ∈IXτ*(1_-μ)≤s}.
For λ,μ∈IX, r,r1,r2∈I0 and s,s1,s2∈I1, the operator Cτ,τ* satisfies the following statements: (C1)
Cτ,τ*(0_,r,s) = 0_;(C2)
λ≤Cτ,τ*(λ,r,s);(C3)
Cτ,τ*(λ,r,s)∨Cτ,τ*(μ,r,s)=Cτ,τ*(λ∨μ,r,s);(C4)
Cτ,τ*(λ,r1,s1)≤Cτ,τ*(λ,r2,s2) if r1≤r2 and s1≥s2;(C5)
Cτ,τ*(Cτ,τ*(λ,r,s),r,s)=Cτ,τ*(λ,r,s).Theorem 3 (see [24, 25]).
Let(X,τ,τ*) be a dfts. Then for each r∈I0, s∈I1, and λ∈IX, one defines an operator Iτ,τ*:IX×I0×I1→IX as follows:
(2)Iτ,τ*(λ,r,s)=⋁{μ∈IX∣μ≤λ,τ(μ)≥r,τ*(μ)≤s}.
For λ,μ∈IX, r,r1,r2∈I0 and s,s1,s2∈I1, the operator Iτ,τ* satisfies the following statements: (I1)
Iτ,τ*(1_-λ,r,s)=1_-Cτ,τ*(λ,r,s);(I2)
Iτ,τ*(1_,r,s)=1_;(I3)
Iτ,τ*(λ,r,s)≤λ;(I4)
Iτ,τ*(λ,r,s)∧Iτ,τ*(μ,r,s)=Iτ,τ*(λ∧μ,r,s);(I5)
Iτ,τ*(λ,r1,s1)≥Iτ,τ*(λ,r2,s2) if r1≤r2 and s1≥s2;(I6)
Iτ,τ*(Iτ,τ*(λ,r,s),r,s)=Iτ,τ*(λ,r,s);(I7)
IfIτ,τ*(Cτ,τ*(λ,r,s),r,s)=λ, then Cτ,τ*(Iτ,τ*(1_-λ,r,s),r,s)=1_-λ.Definition 4 (see [26]).
Let(X,τ,τ*) be a dfts. For each λ,μ∈IX, r∈I0, and s∈I1. (1)
A fuzzy setλ is called (r,s)-fuzzy semiclosed (briefly, (r,s)-fsc) if Iτ,τ*(Cτ,τ*(λ,r,s),r,s)≤λ. λ is called (r,s)-fuzzy semiopen (briefly, (r,s)-fso) if and only if 1_-λ is an (r,s)-fuzzy semiclosed set.(2)
An(r,s)-fuzzy semiclosure of λ is defined by SCτ,τ*(λ,r,s) = ⋀{μ∈IX∣λ≤μ and μ is (r,s)-fsc}.Definition 5 (see [26]).
Let(X,τ,τ*) be a dfts. For each λ,μ∈IX, r∈I0 and s∈I1. (1)
A fuzzy setλ is called (r,s)-generalized fuzzy semiclosed (briefly, (r,s)-gfsc) if SCτ,τ*(λ,r,s)≤μ, λ≤μ and τ(μ)≥r, τ*(μ)≤s. λ is called (r,s)-generalized fuzzy semiopen (briefly, (r,s)-gfso) if and only if 1_-λ is (r,s)-gfsc set.(2)
An(r,s)-fuzzy generalized semiclosure of λ is defined by GSCτ,τ*(λ,r,s) = ⋀{μ∈IX∣λ≤μ and μ is (r,s)-gfsc}.(3)
An(r,s)-fuzzy generalized semi-interior of λ is defined by GSIτ,τ*(λ,r,s) = ⋁{μ∈IX∣μ≤λ and μ is (r,s)-gfso}.Definition 6 (see [27]).
Let(X,τ1,τ1*) and (Y,τ2,τ2*) be dfts's. A function f:(X,τ1,τ1*)→(Y,τ2,τ2*) is called (1)
slightly double fuzzy continuous (briefly, sdfc) if for everyλ∈IX, μ∈IY, r∈I0, and s∈I1 such that μ is (r,s)-fco set and f(λ)≤μ, there exists ν∈IX such that τ1(ν)≥r, τ1*(ν)≤s, λ≤ν, and
(3)f(ν)≤μ,(2)
slightly generalized double fuzzy semicontinuous (briefly, sgdfsc) if for eachλ∈IX, μ∈IY, r∈I0, and s∈I1 such that μ is (r,s)-fco set and f(λ)≤μ, there exists an (r,s)-gfso set ν∈IX such that λ≤ν and
(4)f(ν)≤μ.


## 3. Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

Definition 7.
Let(X,τ1,τ1*) and (Y,τ2,τ2*) be dfts’s. A function f:(X,τ1,τ1*)→(Y,τ2,τ2*) is called somewhat slightly generalized double fuzzy semicontinuous (briefly, swsgdfsc) if for each fuzzy set λ∈IX, μ∈IY, r∈I0, and s∈I1 such that f-1(μ)≠0_ and f(λ)≤μ, there exists an (r,s)-gfso set 0_≠ν∈IX such that λ≤ν and
(5)ν≤f-1(μ).Definition 8.
A fuzzy setλ in a dfts (X,τ,τ*) is called (r,s)-generalized fuzzy semidense (resp., (r,s)-fuzzy-dense*) set if there exists no (r,s)-gfsc (resp., (r,s)-fco) set μ∈IX, r∈I0, and s∈I1 such that
(6)λ<μ<1_.Example 9.
(1) LetX={a,b}. Define λ1 and λ2 as follows:
(7)λ1(a)=0.1,λ1(b)=0.2,λ2(a)=0.8,λ2(b)=0.7.
And define τ(λ) and τ*(λ) as follows:
(8)τ(λ)={1,ifλ=0_or1_;13,ifλ=λ1;23,ifλ=λ2;0,otherwise;τ*(λ)={0,ifλ=0_or1_;23,ifλ=λ1;13,ifλ=λ2;1,otherwise.
So, if λ(a)=0.9, λ(b)=0.8, then there exists no (1/3,2/3)-gfsc set μ in IX such that λ<μ<1_. Therefore, λ is (1/3,2/3)-generalized fuzzy semidense set in IX.
(2) In (1), letλ1 and λ2 be defined as follows:
(9)λ1(a)=0.1,λ2(b)=0.2,λ2(a)=0.9,λ2(b)=0.8.
So, if λ(a)=0.8, λ(b)=0.9, then there exists no (1/3,2/3)-fco set μ in IX such that λ<μ<1_. Therefore, λ is (1/3,2/3)-fuzzy-dense* set in IX.Definition 10.
Let(X,τ,τ*) be a dfts. For a fuzzy set λ∈IX, r∈I0, and s∈I1, Iτ,τ** and Cτ,τ** are defined as follows: (1)
Iτ,τ**(λ,r,s)=⋁{μ∈IX∣μ≤λ and μ is (r,s)-fco};(2)
Cτ,τ**(λ,r,s)=⋀{μ∈IX∣λ≤μ and μ is (r,s)-fco}.Proposition 11.
Let(X,τ1,τ1*) and (Y,τ2,τ2*) be dfts’s, and let f:(X,τ1,τ1*)→(Y,τ2,τ2*) be any function. Then the following are equivalent. (1)
f is swsgdfsc function.(2)
If  λ is an  (r,s)-fco set such that  f-1(λ)≠1_ and  λ≤f(1_-ν), for each  ν∈IX, r∈I0,  and  s∈I1,  then there exists an  (r,s)-gfsc set  μ≤1_-ν∈IXsuch thatμ≥f-1(λ).(3)
Ifλ is  (r,s)-gfs-dense set in  IX,  thenf(λ) is(r,s)-fuzzy-dense* set inIY such that every  (r,s)-fco set  μ≤f(1_-ν),  for each  ν∈IX,  r∈I0, and  s∈I1.Proof.
(
1
)
⇒
(
2
) Suppose f is swsgdfsc function, and let λ be any (r,s)-fco set in IY such that f-1(λ)≠1_ and λ≤f(1_-ν), for each ν∈IX, r∈I0, and s∈I1. Then, 1_-λ is (r,s)-fco in IY such that f-1(1_-λ)≠0_ and f(ν)≤1_-λ. Then by the hypothesis, there exists an (r,s)-gfso set 0_≠α∈IX, r∈I0, and s∈I1 such that ν≤α and α≤f-1(1_-λ). That is, 1_-α is an (r,s)-gfsc set and
(10)1_-α≥1_-f-1(1_-λ)=f-1(λ).
Put 1_-α=μ. Then μ is an (r,s)-gfsc set in IX such that μ≥f-1(λ).
(
2
)
⇒
(
3
) Let λ be an (r,s)-gfs-dense set in IX, and suppose that f(λ) is not a fuzzy-dense* set in IY, such that each (r,s)-fco set μ≤f(1_-ν), for each ν∈IX, r∈I0, and s∈I1. Then, there exists an (r,s)-fco set α∈IY such that
(11)f(λ)<α<1_,
since
(12)α<1_,f-1(α)≠1_.
Now,α is an (r,s)-fco set such that f-1(α)≠1_ and f(1_-ν)≥α, for each ν∈IX, r∈I0, and s∈I1. Then by the hypothesis, there exists an (r,s)-gfsc set γ≤1_-ν∈IX such that γ≥f-1(α).
But(13)f-1(α)>f-1(f(λ))=λ.
That is,γ≥λ. Therefore, there exists an (r,s)-gfsc set γ∈IX, r∈I0, and s∈I1 such that γ≥λ, which is a contradiction. Therefore, f(λ) is an (r,s)-fuzzy dense* set in IY such that
(14)γ≤f(1_-ν),
for each ν∈IX and (r,s)-fco set γ∈IY.
(
3
)
⇒
(
1
) Let λ be an (r,s)-fco set such that f-1(λ)≠0_ and f(ν)≤λ, for each ν∈IX, r∈I0, and s∈I1. Then, λ≠0_. Now, suppose that ν≤α and GSIτ,τ*(f-1(λ),r,s)=0_∈IX. Then,
(15)GSCτ,τ*(1_-f-1(λ),r,s)=1_∈IX.
That is, 1_-f-1(λ) is an (r,s)-gfs-dense in IX. Then by (3), f(1_-f-1(λ)) is an (r,s)-fuzzy dense* set such that there exists an (r,s)-fco set μ≤f(1_-ν), for each ν∈IX, r∈I0, and s∈I1.
But(16)f(1_-f-1(λ))=f(f-1(1_-λ))≤1_-λ<1_,
since 1_-λ is an (r,s)-fco and
(17)f(1_-f-1(λ))≤1_-λ,Cτ,τ**(f(1_-f-1(λ)),r,s)≤1_-λ.
That is,
(18)1_-λ≥1_⟹λ=0_,
which is a contradiction, since λ≠0_. Therefore, ν≤α and GSIτ,τ*(f-1(λ),r,s)≠0_. So f is swsgdfsc.


## 4. Somewhat Slightly Generalized Double Fuzzy Semiopen Functions

Definition 12.
Let(X,τ1,τ1*) and (Y,τ2,τ2*) be dfts’s. A function f:(X,τ1,τ1*)→(Y,τ2,τ2*) is called  (1)
generalized double fuzzy semiopen (briefly, gdfso) if for each(r,s)-gfso set λ∈IX, r∈I0, and s∈I1, f(λ) is an (r,s)-gfso in IY;(2)
slightly generalized double fuzzy semiopen (briefly, sgdfso) if for each(r,s)-gfso set λ∈IX and each μ∈IX, r∈I0, and s∈I1 such that λ≤μ, f(λ) is an (r,s)-fco set in IY and
(19)f(λ)≤f(μ);(3)
somewhat generalized double fuzzy semiopen (briefly, swgdfso) if for each(r,s)-gfso set 0_≠λ∈IX, r∈I0, and s∈I1, there exists an (r,s)-gfso set 0_≠μ∈IY such that
(20)f(λ)≥μ;(4)
somewhat slightly generalized double fuzzy semiopen (briefly, swsgdfso) if for each(r,s)-gfso set 0_≠λ∈IX such that λ≤ν and for each ν∈IX, r∈I0, and s∈I1, there exists an (r,s)-fco set 0_≠μ∈IY, μ≤f(ν) such that
(21)f(λ)≥μ.That is,Iτ,τ**(f(λ),r,s)≠0_, and there exists an (r,s)-fco set μ such that f(ν)≥μ and λ≤ν, for each ν∈IX, r∈I0, and s∈I1.Proposition 13.
Let(X,τ1,τ1*), (Y,τ2,τ2*), and (Z,τ3,τ3*) be dfts’s. If f:(X,τ1,τ1*)→(Y,τ2,τ2*) and g:(Y,τ2,τ2*)→(Z,τ3,τ3*) are swsgdfso functions, then g∘f:(X,τ1,τ1*)→(Z,τ3,τ3*) is a swsgdfso function.Proof.
Let0_≠λ∈IX be an (r,s)-gfso set r∈I0 and s∈I1 such that λ≤μ, for each fuzzy set μ∈IX, r∈I0, and s∈I1. Since f is swsgdfso, then there exists an (r,s)-fco set 0_≠ν∈IY, and f(μ)≥ν such that f(λ)≥ν.
Now,GSIτ,τ*(f(λ),r,s)) is an (r,s)-gfso in IY such that
(22)GSIτ,τ*(f(λ),r,s)≠0_,GSIτ,τ*(f(λ),r,s)≤f(μ),
for each f(μ)∈IY.
Sinceg is swsgdfso, then there exists an (r,s)-fco set 0_≠γ∈IZ and γ≤g(f(μ)) such that
(23)γ≤g(GSIτ,τ*(f(λ),r,s)).
But
(24)g(GSIτ,τ*(f(λ),r,s))≤g(f(λ)).
Thus, there exists an (r,s)-fco set 0_≠γ∈IZ and
(25)(g∘f)(μ)≥γ,
such that
(26)(g∘f)(λ)≥γ.
Therefore, g∘f is swsgdfso.Proposition 14.
Let(X,τ1,τ1*) and (Y,τ2,τ2*) be dfts’s, and let f:(X,τ1,τ1*)→(Y,τ2,τ2*) be a bijective function. Then the following are equivalent. (1)
fis swsgdfso function.(2)
Ifλ is an  (r,s)-gfsc set in  IX such that  f(λ)≠1_ and  λ≥ν for each  ν∈IX, then there exists an  (r,s)-fco set  μ∈IY,  μ≠1_, and  f(ν)≤μ such that  f(λ)≤μ.Proof.
(
1
)
⇒
(
2
) Let λ be an (r,s)-gfsc set in IX such that f(λ)≠1_ and λ≥ν, for each ν∈IX, r∈I0, and s∈I1. Then, 1_-λ is an (r,s)-gfso set in IX such that f(1_-λ)≠0_ and 1_-λ≤1_-ν, for each ν∈IX. So
(27)1_-λ≠0_.
Since f is a swsgdfso, then there exists an (r,s)-fco set 0_≠δ∈IY and f(1_-ν)≥δ such that
(28)f(1_-λ)≥δ.
Now, 1_-δ is an (r,s)-fco set in IY such that 1_-δ≠1_ and 1_-δ≥f(ν) such that
(29)1_-δ≥f(λ).
Take
(30)1_-δ=μ,
so (2) is proved.
(
2
)
⇒
(
1
) Let λ≠0_ be any (r,s)-gfso set in IX such that λ≤ν, for each ν∈IX. Then, 1_-λ is an (r,s)-gfsc set in IX such that 1_-λ≠1_ and 1_-λ≥1_-ν for each ν∈IXr∈I0 and s∈I1. Now,
(31)f(1_-λ)=1_-f(λ)≠1_.
For, if 1_-f(λ)=1_, then
(32)f(λ)=0_⟹λ=0_.
Hence by the hypothesis, there exists an (r,s)-fco set μ∈IY, 1_≠μ≥f(1_-ν), such that
(33)f(1_-λ)≤μ.
That is,
(34)0_≠1_-μ≤f(ν),
such that
(35)1_-μ≤f(λ).
Let 1_-μ=γ. Then, γ≠0_ is an (r,s)-fco set in IY such that f(ν)≥γ and f(λ)≥γ. Therefore, f is swsgdfso function.


## 5. Interrelations

The following implication illustrates the relationships between different functions in Figure1.(a)
(b)
(c)None of these implications is reversible whereA→B represents A implies B, as shown by the following examples.Example 15.
LetX={a,b}.(1)
Letf:(X,τ1,τ1*)→(X,τ2,τ2*) be the identity function. Define λ1, λ2, μ1, and μ2 as follows:
(36)λ1(a)=0.1,λ1(b)=0.2,λ2(a)=0.9,λ2(b)=0.8,μ1(a)=0.5,μ1(b)=0.5,μ2(a)=1.0,μ2(b)=0.7.
And define (τ1,τ1*) and (τ2,τ2*) as follows:
(37)τ1(λ)={1,ifλ∈{0_,1_},14,ifλ=λ1,18,ifλ=λ2,0,otherwise,τ1*(λ)={0,ifλ∈{0_,1_},18,ifλ=λ1,14,ifλ=λ2,1,otherwise,τ2(λ)={1,ifλ∈{0_,1_},14,ifλ=μ1,18,ifλ=μ2,0,otherwise,τ2*(λ)={0,ifλ∈{0_,1_},18,ifλ=μ1,14,ifλ=μ2,1,otherwise.
Then, f is sgdfsc function but not sdfc.(2)
In (1),f is swsgdfsc function but not sdfc.(3)
Letf:(X,τ1,τ1*)→(X,τ2,τ2*) be a function defined by
(38)f(a)=a,f(b)=a.
Define λ1, μ1, and μ2 as follows:
(39)λ1(a)=1.0,λ1(b)=0.9,μ1(a)=0.0,μ1(b)=0.2,μ2(a)=1.0,μ2(b)=0.8,and define (τ1,τ1*) and (τ2,τ2*) as follows:
(40)τ1(λ)={1,ifλ∈{0_,1_},12,ifλ=λ1,0,otherwise,τ1*(λ)={0,ifλ∈{0_,1_},12,ifλ=λ1,1,otherwise,τ2(λ)={1,ifλ∈{0_,1_},13,ifλ=μ1,23,ifλ=μ2,0,otherwise,τ2*(λ)={0,ifλ∈{0_,1_},23,ifλ=μ1,13,ifλ=μ2,1,otherwise.
Then, f is swsgdfsc function but not sgdfsc.(4)
Letf:(X,τ1,τ1*)→(X,τ2,τ2*) be the identity function. Define λ1, λ2, and μ as follows:
(41)λ1(a)=0.0,λ1(b)=1.0,λ2(a)=1.0,λ2(b)=0.0,μ(a)=0.0,μ(b)=0.3.
And define (τ1,τ1*) and (τ2,τ2*) as follows:
(42)τ1(λ)={1,ifλ∈{0_,1_},13,ifλ=λ1,23,ifλ=λ2,0,otherwise,τ1*(λ)={0,ifλ∈{0_,1_},23,ifλ=λ1,13,ifλ=λ2,1,otherwise,τ2(λ)={1,ifλ∈{0_,1_},13,ifλ=μ,0,otherwise,τ2*(λ)={0,ifλ∈{0_,1_}23,ifλ=μ,1,otherwise.
Then, f is swgdfso function but not gdfso.(5)
Letf:(X,τ1,τ1*)→(X,τ2,τ2*) be the identity function. Define λ1, μ1, and μ2 as follows:
(43)λ1(a)=0.1,λ1(b)=0.1,μ1(a)=0.05,μ1(b)=0.02,μ2(a)=0.95,μ2(b)=0.98.
And define (τ1,τ1*) and (τ2,τ2*) as follows:
(44)τ1(λ)={1,ifλ∈{0_,1_},13,ifλ=λ1,0,otherwise,τ1*(λ)={0,ifλ∈{0_,1_},23,ifλ=λ1,1,otherwise,τ2(λ)={1,ifλ∈{0_,1_},13,ifλ=μ1,23,ifλ=μ2,0,otherwise,τ2*(λ)={0,ifλ∈{0_,1_},23,ifλ=μ1,13,ifλ=μ2,1,otherwise.
Then, f is swsgdfso function but not sgdfso.



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*Source: 756376-2014-06-12.xml*