Definition 3.7
A function is said to be double fuzzy weakly preopen if
for each λ∈IX, r∈I0 and s∈I1; τ1(λ)≥r and .
Remark 3.2
Every double fuzzy weakly open function is double fuzzy preopen and every double fuzzy preopen function is double fuzzy weakly preopen, but the converse need not be true in general.
Example 3.1
Let X={a,b,c} and Y={x,y,z}. Fuzzy sets λ1, λ2 and λ3 are defined as:
Define τ1and τ2 as follows:
Then the mapping defined by f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy preopen. Where , and f(λ) is not -fpo.
Example 3.2
Let X={a,b,c} and Y={x,y,z}. Fuzzy sets λ1, λ2 and λ3 are defined as:
Let and defined as follows:
Then the mapping defined by f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy weakly open. Since .
Theorem 3.5
For a function , . The following statements are equivalent:
-
(1)
f is double fuzzy weakly preopen,
-
(2)
for each λ∈I X, r∈I 0and s∈I 1,
-
(3)
for each ν∈I Y, r∈I 0and s∈I 1,
-
(4)
for each ν∈I Y, r∈I 0and s∈I 1.
Proof
(1)⇒(2) Let λ∈IX and . Then there exists such that . Thus and hence
Since f is double fuzzy weakly preopen,
and hence . This shows that . Thus (f(λ),r,s)) and so, .
(2)⇒(1) Let μ∈IX; τ1(μ)≥r and . Since , then
Hence f is double fuzzy weakly preopen.
(2)⇒(3) Let ν∈IY. By using (2), , . Therefore, .
(3)⇒(2) Trivial.
(3)⇒(4) Let ν∈IY. Using (3), we have
Therefore, we obtain , r,s).
(4)⇒(3) Similarly we obtain, , for every ν∈IY, r∈I0 and s∈I1, i.e., . □
Theorem 3.6
For the function . The following statements are equivalent:
-
(1)
f is double fuzzy weakly preopen,
-
(2)
For each x
t
∈ P(X) and each μ∈I X; τ 1(μ)≥r and with x
t
≤μ, there exists (r,s)-fpo set γ such that f(x
t
)≤γ and .
Proof
(1)⇒(2) Let x
t
∈ P(X) and μ∈IXsuch that τ1(μ)≥r, and x
t
≤μ. Since f is double fuzzy weakly preopen, then . Let . Hence (μ,r,s)), with f(x
t
)≤γ.
(2)⇒(1) Let μ∈IX; τ1(μ)≥r, and y
s
≤f(μ). It follows from (2) that for some (r,s)-fpo γ∈IYand y
s
≤γ. Hence we have, . This shows that , i.e. f is double fuzzy weakly preopen function. □
Theorem 3.7
Let be a bijective function. Then the following statements are equivalent:
-
(1)
f is double fuzzy weakly preopen;
-
(2)
for each λ∈I X, r∈I 0and s∈I 1; τ 1(λ)≥r and ;
-
(3)
for each ν∈I X, r∈I 0and s∈I 1; and .
Proof
(1)⇒(2) Let ν∈IX; τ1(ν)≥r and . Then we have,
and so . Hence .
(2)⇒(3) Let λ∈IX; τ1(λ)≥r and . Since is (r,s)-fc set and r,s) by (3) we have .
(3)⇒(2) Trivial.
(2)⇒(1) Trivial. □
Theorem 3.8
For a function . The following statements are equivalent:
-
(1)
f is double fuzzy weakly preopen;
-
(2)
for each ν∈I X, r∈I 0and s∈I 1; τ 1(ν)≥r and ;
-
(3)
for each λ∈I X, r∈I 0and s∈I 1; τ 1(λ)≥r and ;
-
(4)
, for each (r,s)-fpo set λ∈I X;
-
(5)
, for each (r,s)-fα o set λ∈I X.
Proof
(1)⇒(2) Let ν∈IX, r∈I0 and s∈I1; and . By (1),
(2)⇒(3) It is clear.
(3)⇒(4) Let λ be (r,s)-fpo set. Hence by (3),
(4)⇒(5) and (5)⇒(1) are clear. □
Definition 3.8
A function is said to be double fuzzy strongly continuous, if for each λ∈IX, r∈I0and s∈I1.
Theorem 3.9
If is double fuzzy weakly preopen and double fuzzy strongly continuous function, then f is double fuzzy preopen.
Proof
Let λ∈IX such that τ1(λ)≥r and . Since f is double fuzzy weakly preopen
However, since f is double fuzzy strongly continuous, then and therefore f(λ) is (r,s)-fpo. □
Definition 3.9
A function is said to be double fuzzy contra-preclosed if f(λ) is (r,s)-fpo for each λ∈IX, r∈I0and s∈I1; and .
Theorem 3.10
If is double fuzzy contra-preclosed, then f is double fuzzy weakly preopen function.
Proof
Let λ∈IX; τ1(λ)≥r and . Then, we have
□
The converse of the above theorem need not be true in general as in the following Example.
Example 3.3
Let X={a,b,c} and Y={x,y,z}. Define fuzzy sets λ1, λ2 as follows:
Let and defined as follows:
Then the function defined as f(a)=x, f(b)=y and f(c)=z is double fuzzy weakly preopen but it isn’t double fuzzy contra-preclosed.
Definition 3.10
An I-dfts (X,τ,τ∗) is said to be (r,s)-fuzzy regular space if for each λ∈IX; τ(λ)≥r and τ∗(λ)≤s is a union of (r,s)-fo sets μ
i
∈IXsuch that Cτ,τ∗(μ
i
,r,s)≤λ for each i∈J.
Theorem 3.11
Let (X,τ,τ∗) be (r,s)-regular fuzzy topological space. Then, is double fuzzy weakly preopen if and only if f is double fuzzy preopen.
Proof
The sufficiency is clear. For the necessity, let λ∈IX, r∈I0, s∈I1; , τ1(λ)≥r and . For each x
t
≤λ, let . Hence we obtain that and,
Thus f is double fuzzy preopen. □
Theorem 3.12
If is double fuzzy almost open function, then it is double fuzzy weakly preopen.
Proof
Let λ∈IX; τ1(λ)≥r and τ 1∗(λ)≤s. Since f is double fuzzy almost open and is (r,s)-fro, then
and hence
This shows that f is double fuzzy weakly preopen. □
Definition 3.11
Let (X,τ,τ∗) be an I-dfts, r∈I0and s∈I1. The two fuzzy sets λ, μ∈IXare said to be (r,s)-fuzzy separated iff and . A fuzzy set which cannot be expressed as a union of two (r,s)-fuzzy separated sets is said to be (r,s)-fuzzy connected.
Definition 3.12
Let (X,τ,τ∗) an I-dfts. The fuzzy sets λ, μ∈IXsuch that , , are said to be fuzzy (r,s)-pre-separated if and or equivalently if there exist two (r,s)-fpo sets ν, γ such that λ≤ν, μ≤γ, and . An I-dfts which can not be expressed as a union of two fuzzy (r,s)-pre-separated sets is said to be fuzzy (r,s)-pre-connected space.
Theorem 3.13
If is an injective double fuzzy weakly preopen and strongly double fuzzy continuous function from the space onto an (r,s)-fuzzy pre-connected space , then is (r,s)-fuzzy connected.
Proof
Let be not (r,s)-fuzzy connected. Then there exist (r,s)-fuzzy separated sets β, γ∈IX such that . Since β and γ are (r,s)-fuzzy separated, there exists λ, μ∈IX; τ1(λ)≥r, τ1(μ)≥r and , such that β≤λ, γ≤μ, and . Hence we have f(β)≤f(λ), f(γ)≤f(μ), and . Since f is double fuzzy weakly preopen and double fuzzy strongly continuous function, from Theorem 3.10 we have f(λ) and f(μ) are (r,s)-fpo sets. Therefore, f(β) and f(γ) are (r,s)-fuzzy pre-separated and
which is contradiction with is (r,s)-fuzzy pre-connected. Thus is (r,s)-fuzzy connected. □