- Research
- Open access
- Published:
Weak forms of continuity in I-double gradation fuzzy topological spaces
SpringerPlus volume 1, Article number: 19 (2012)
Abstract
In this paper, we introduce and characterize double fuzzy weakly preopen and double fuzzy weakly preclosed functions between I-double gradation fuzzy topological spaces and also study these functions in relation to some other types of already known functions.
Introduction
In the history of science, new theories have always been necessary in order for existing scientific theories to progress and this will continue to be true in the future. Two examples of essentially different mathematical theories that deal with the concept of uncertainty are probability theory and the theory of fuzzy sets. Whereas probability theory has a history of around 360 years, the theory of fuzzy sets is little more than 50 years old. Since the 1960s fuzzy methods have entered the scientific and technological world, good theoretical progress (e.g., fuzzy logic, fuzzy probability theory, fuzzy topology, fuzzy algebra) has been made, and there have been technical advances in various areas (e.g., fuzzy control, fuzzy expert systems, fuzzy clustering and data mining).
Chang (1968); Lowen (1976); Šostak (1985); Kubiak (1985); Samanta and Mondal (19972002) and many others contributed a lot to the field of Fuzzy Topology. In recent years Fuzzy Topology has been found to be very useful in solving many practical problems. Shihong Du et. al. (2005) are currently working to fuzzify the 9-intersection Egenhofer model Egenhofer and Franzosa (1991); Herring and Egenhofer (1991) for describing topological relations in Geographic Information Systems (GIS) query. In El-Naschie (19982000), El-Naschie has shown that the notion of Fuzzy Topology is applicable to quantum particle physics and quantum gravity in connection with String Theory and e∞ Theory. Tang (2004) has used a slightly changed version of Chang’s fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.
In this paper, we will introduce the concepts of double fuzzy weakly preopen and double weakly preclosed functions in I-double gradation fuzzy topological spaces. Their properties and the relationships between these functions and other functions introduced previously are investigated.
Preliminaries
Throughout this paper, let X be a nonempty set and I is the closed unit interval [0,1]. I∘=(0,1] and I1=[0,1). The family of all fuzzy subsets on X denoted by IX. By and , we denote the smallest and the greatest fuzzy subsets on X. For a fuzzy subset λ∈IX, denotes its complement. Given a function , f(λ) and f−1(λ) define the direct image and the inverse image of f, defined by and f−1(ν)(x)=ν(f(x)), for each λ∈IX, ν∈IY, and x∈X, respectively. For fuzzy subsets λ and μ in X, we write λqμ to mean that λ is quasi coincident (q-coincident) with μ, that is, there exists at least one point x∈X such that λ(x) + μ(x)>1. Negation of such a statement is denoted as . Notions and notations not described in this paper are standard and usual.
Definition 2.1
[(Samanta and Mondal (19972002); Garcia and Rodabaugh (2005)] An I-double gradation fuzzy topology (τ τ∗) on X is a pair of maps τ, τ∗:IX→I, which satisfies the following properties:
-
(O1)
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)
and for each λ i ∈IX, i∈Γ.
The triplet (X,τ,τ∗) is called an I-double gradation fuzzy topological spaces (I-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. A fuzzy set λ is called an (r,s)-fuzzy closed ((r,s)-fc, for short) set iff is an (r,s)-fo set. Let and be two I-dfts’s. A function is said to be a double fuzzy continuous iff τ1(f−1(ν))≥τ2(ν) and for each ν∈IY.
There was a question we must ask ourselve before starting to present our results, which was: Is it useful to introduce new concepts to I-double gradation fuzzy topological spaces?
We could know that double (initially, intuitionistic) fuzzy sets (and hence double fuzzy topological spaces) deal with ambiguity in a way better than fuzzy sets. In addition to that, double fuzzy topological spaces is a generalization of some other kinds of topological spaces; we can get fuzzy topological spaces in Chang’s sense , where
Also, when the conditions τ∗(λ)=1−τ(λ) and τ(λ) + τ∗(λ)≮1 achieved in Definition 2.1, we get the definition of fuzzy topological spaces in Kubiak- Šostak’s sense Kubiak (1985); Šostak (1985). If we use 2Xinstead of IX, the resulting topological structure will be called double gradation fuzzifying topological spaces (A new structure mentioned for the first time in Bhaumik and Abbas 2008). Besides, we can also get the general topological spaces.
Theorem 2.1
[(Çoker and Demirci1996; Lee and Im (2001)] Let (X τ τ∗) be an I-dfts. Then for each r∈I0, s∈I1 and λ∈IX, we define an operator Cτ,τ∗:IX×I0×I1→IX as follows:
For λ, μ∈IX, r1r2∈I0and s1s2∈I1, the operator Cτ,τ∗satisfies the following statements:
-
(C1)
,
-
(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≤r2and s1≥s2,
-
(C5)
Cτ,τ∗(Cτ,τ∗(λ,r,s),r,s)=Cτ,τ∗(λ,r,s).
Theorem 2.2
[(Çoker and Demirci1996; Lee and Im2001)] Let (X τ τ∗) be an I-dfts. Then for each r∈I0, s∈I1 and λ∈IX, we define an operator Iτ,τ∗:IX×I0×I1→IXas follows:
For λ μ∈IX, r r1r2∈I0and s s1s2∈I1, the operator Iτ,τ∗satisfies the following statements:
-
(I1)
,
-
(I2)
,
-
(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≤r2and s1≥s2,
-
(I6)
Iτ,τ∗(Iτ,τ∗(λ,r,s),r,s)=Iτ,τ∗(λ,r,s),
-
(I7)
If Iτ,τ∗(Cτ,τ∗(λ,r,s),r,s)=λ, then .
Definition 2.2
Let (X,τ,τ∗) be an I-dfts. For λ∈IX, r∈I0and s∈I1.
-
(1)
λ is called (r,s)-fuzzy preopen ((r,s)-fpo, for short) if λ≤I τ,τ ∗(C τ,τ ∗(λ,r,s),r,s). A fuzzy set λ is called (r,s)-fuzzy preclosed ((r,s)-fpc, for short) iff is (r,s)-fpo set. The (r,s)-fuzzy preinterior of λ, denoted by P I τ,τ ∗(λ,r,s) is defined by
The (r,s)-fuzzy preclosure of λ, denoted by P Cτ,τ∗(λ,r,s) is defined by
-
(2)
λ is called (r,s)-fuzzy regular open ((r,s)-fro, for short) if λ=I τ,τ ∗(C τ,τ ∗(λ,r,s),r,s). A fuzzy set λ is called (r,s)-fuzzy regular closed ((r,s)-frc, for short) iff is (r,s)-fro set.
-
(3)
λ is called (r,s)-fuzzy α-open ((r,s)-fα o, for short) if λ≤I τ,τ ∗(C τ,τ ∗(I τ,τ ∗(λ,r,s),r,s),r,s). A fuzzy setλ is called (r,s)-fuzzy α-closed ((r,s)-fα c, for short) iff is (r,s)-fα o set.
Theorem 2.3
Let (X,τ,τ∗) be an I-dfts. For λ∈IX, r∈I0and s∈I1.
-
(1)
λ is (r,s)-fpo (resp. (r,s)-fpc) iff λ=P I τ,τ ∗(λ,r,s) (resp. λ=P C τ,τ ∗(λ,r,s)),
-
(2)
I τ,τ ∗(λ,r,s)≤P I τ,τ ∗(λ,r,s)≤λ≤P C τ,τ ∗(λ,r,s)≤C τ,τ ∗(λ,r,s),
-
(3)
and .
Definition 2.3
Let be a function from an I-dfts into an I-dfts . The function f is called:
-
(1)
double fuzzy preclosed if f(λ) is (r,s)-fpc set in I Yfor each λ∈I X, r∈I 0and s∈I 1; , ,
-
(2)
double fuzzy open if τ 2(f(λ))≥τ 1(λ) and for each λ∈I X, r∈I 0and s∈I 1,
-
(3)
double fuzzy almost open if τ 2(f(λ))≥r and for each (r,s)-fro set λ∈I X, r∈I 0and s∈I 1.
Definition 2.4
Let be a function from an I-dfts into an I-dfts . The function f is called:
-
(1)
double fuzzy weakly open if for each λ∈I X, r∈I 0and s∈I 1; τ 1(λ)≥r and ,
-
(2)
double fuzzy α-open if f(λ) is (r,s)-fα o in I Yfor each λ∈I X, r∈I 0and s∈I 1; τ 1(λ)≥r and .
Definition 2.5
Let (X,τ,τ∗) be an I-dfts, μ∈IX, x t ∈P(X), r∈I0and s∈I1where P(X) is the family of all fuzzy points in X. μ is called an (r,s)-fuzzy open Q-neighborhood of x t if τ(μ)≥r, τ∗(μ)≤s and x t qμ. We denote the set of all (r,s)-fuzzy open Q-neighborhood of x t by Qτ,τ∗(x t ,r,s).
Definition 2.6
Let (X,τ,τ∗) be an I-dfts, λ∈IX, x t ∈P(X), r∈I0and s∈I1. x t is called (r,s)-fuzzy θ-cluster point of λ if for every μ∈Qτ,τ∗(x t ,r,s), we have Cτ,τ∗(μ,r,s)qλ. We denote . Where Dτ,τ∗(λ,r,s) is called (r,s)-fuzzy θ-closure of λ.
Theorem 2.4
Let (X,τ,τ∗) an I-dfts. For λ, μ∈IXand r, s∈I0, we have the following:
-
(1)
,
-
(2)
x t is (r,s)-fuzzy θ-cluster point of λ iff x t ∈D τ,τ ∗(λ,r,s).
-
(3)
C τ,τ ∗(λ,r,s)≤D τ,τ ∗(λ,r,s),
-
(4)
If τ(λ)≥r and τ ∗(λ)≤s, then C τ,τ ∗(λ,r,s)=D τ,τ ∗(λ,r,s),
-
(5)
If λ is (r,s)-fpo, then C τ,τ ∗(λ,r,s)=D τ,τ ∗(λ,r,s),
-
(6)
If λ is (r,s)-fpo and λ=C τ,τ ∗(I τ,τ ∗(λ,r,s),r,s), then D τ,τ ∗(λ,r,s)=λ.
The complement of (r,s)-fuzzy θ-closed set is called (r,s)-fuzzy θ-open and the (r,s)-fuzzy θ-interior operator denoted by Tτ,τ∗(λ,r,s) is defined by .
Remark 2.1
From Theorem 2.4 It is easy to see that:
-
(1)
I τ,τ ∗(λ,r,s)≤T τ,τ ∗(λ,r,s) for any λ∈I X, r∈I 0and s∈I 1,
-
(2)
T τ,τ ∗(λ,r,s)=I τ,τ ∗(λ,r,s) for each λ∈I X, r∈I 0and s∈I 1; τ(λ)≥r and τ ∗(λ)≤s.
Double Fuzzy weakly preopen functions
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. □
Double Fuzzy weakly preclosed functions
Definition 4.13
A function is said to be double fuzzy weakly preclosed function if
for each λ∈IX, r∈I0 and s∈I1; and .
Remark 4.3
Clearly, every double fuzzy preclosed function is double fuzzy weakly preclosed, but the converse need not be true in general, as the next example shows.
Example 4.4
Let X={a,b} and Y={x,y}. Fuzzy sets λ1 and λ2 are defined as:
Let
Then the function defined by f(a)=x, f(b)=y is double fuzzy weakly preclosed but is not double fuzzy preclosed.
Theorem 4.14
For a function . The following statements are equivalent.
-
(1)
f is double fuzzy weakly preclosed;
-
(2)
for each λ∈I X, r∈I 0and s∈I 1; τ 1(λ)≥r and ;
-
(3)
for each λ∈I X, r∈I 0and s∈I 1; and ;
-
(4)
for each (r,s)-fpc set λ∈I X, r∈I 0and s∈I 1;
-
(5)
for each (r,s)-fα c λ∈I X, r∈I 0and s∈I 1.
Proof
Straightforward. □
Theorem 4.15
For a function . The following statements are equivalent.
-
(1)
f is double fuzzy weakly preclosed;
-
(2)
for each (r,s)-fro set λ∈I X, r∈I 0and s∈I 1;
-
(3)
For each ν∈I Y, μ∈I X, r∈I 0and s∈I 1; τ 1(μ)≥r and with f −1(ν)≤μ, there exists (r,s)-fpo set γ∈I Ywith ν≤γ and ;
-
(4)
For each fuzzy point y s ∈ P(Y) and each μ∈I X, r∈I 0and s∈I 1such that τ 1(μ)≥r and with f −1(y s )≤μ, there exists (r,s)-fpo set γ∈I Y; y s ≤γ and ;
-
(5)
for each λ∈I X, r∈I 0and s∈I 1;
-
(6)
for each λ∈I X, r∈I 0and s∈I 1;
-
(7)
for each (r,s)-fpo set λ∈I X, r∈I 0and s∈I 1.
Proof
We will prove (2)⇒(3) and (1)⇒(6).
(2)⇒(3): Let ν∈IY, r∈I0, s∈I1 and let μ∈IX; τ1(μ)≥r and with f−1(ν)≤μ. Then and consequently, . Since is (r,s)-fro, by (2). Let . Then γ is (r,s)-fpo with ν≤γ and
(1)⇒(6): Let ν∈IY, r∈I0 and s∈I1; , and . Since , there exists (r,s)-fpo γ∈IY with y s ≤γ and by (6). Therefore , so that . □
Theorem 4.16
If is double fuzzy weakly preclosed, then for each y s ∈ P(Y) and each , there exists (r,s)-fpo set γ∈IY; , such that .
Proof
Let . Then μ(x) + s>1 and hence there exists t∈(0,1) such that μ(x)>t>1−s. Then . By Theorem 3.7-6 there exists (r,s)-fpo set γ∈IY; y t ≤γ such that . Now, γ(y)>t and hence γ(y)>1−s. Thus γ is (r,s)-fpo neighborhood of y s . □
Definition 4.14
Let (X,τ,τ∗) be an I-dfts. A fuzzy set λ∈IXis called (r,s)-fuzzy pre-Q-neighborhood of x t if there exists (r,s)-fpo set μ∈IXsuch that x t qμ≤λ. We denote the set of all (r,s)-fuzzy pre-Q-neighborhood of x t by PQτ,τ∗(x t ,r,s).
Theorem 4.17
In an I-dfts (X,τ,τ∗). A fuzzy point x t ∈P Cτ,τ∗(λ,r) if and only if for every μ ∈ PQτ,τ∗(x t ,r,s), μqλ is hold.
Proof
Straightforward. □
Theorem 4.18
If is double fuzzy weakly preclosed and if for each ν∈IX, r∈I0and s∈I1; , and each there exists such that . Then f is double fuzzy preclosed.
Proof
Let ν∈IX, r∈I0and s∈I1; , and let . Then , and hence there exists such that . Since f is double fuzzy weakly preclosed by using Theorem 3.12, there exists (r,s)-fuzzy pre-Q-neighborhood γ∈IYwith y s ≤γ and . Therefore, we obtain and hence , this shows that . Therefore, f(ν) is (r,s)-fpc and f is double fuzzy preclosed function. □
Definition 4.15
A function is said to be double fuzzy contra-open (resp. double fuzzy contra-closed) if and (resp. τ2(f(λ))≥r and ) for each λ∈IX, r∈I0and s∈I1; τ1(λ)≥r and (resp. and ).
Theorem 4.19
If is double fuzzy contra-open, then f is double fuzzy weakly preclosed.
Proof
Let λ∈IX, r∈I0 and s∈I1 such that and . Then,
□
Theorem 4.20
If is double fuzzy weakly preclosed, then for every ν∈IYand every λ∈IX, r∈I0, s∈I1such that τ1(λ)≥r and with f−1(ν)≤λ, there exists (r,s)-fpc set γ∈IYsuch that ν≤γ and .
Proof
Let ν∈IYand let λ∈IX, r∈I0and s∈I1such that τ1(λ)≥r and with f−1(ν)≤λ. Put , then γ is (r,s)-fpc set in IYsuch that ν≤γ since . And since f is double fuzzy weakly preclosed, . □
Corollary 4.21
If is double fuzzy weakly preclosed, then for every y s ∈ P(Y) and every λ∈IX, r∈I0and s∈I1such that τ1(λ)≥r and with f−1(y s )≤λ, there exists (r,s)-fpc set γ∈IY; y s ≤γ such that .
Definition 4.16
A fuzzy set λ∈IXis called (r,s)-fuzzy θ-compact if for each family {μ i ∣i∈J} in satisfy for each x∈X, there exist a finite subset J0of J such that .
Theorem 4.22
If is double fuzzy weakly preclosed with all fibers (r,s)-fuzzy θ-closed, then f(λ) is (r,s)-fpc for each (r,s)-fuzzy θ-compact λ∈IX, r∈I0and s∈I1.
Proof
Let λ be (r,s)-fuzzy θ-compact and let . Then and for each x t ≤λ there is with and . Clearly satisfy for each x∈X and since λ is (r,s)-fuzzy θ-compact, there is such that , where . Since f is double fuzzy weakly preclosed, by using Theorem 3.12 there exists with
Therefore y s ≤γ and . Thus . Thus f(λ) is (r,s)-fpc set. □
Definition 4.17
Let (X,τ,τ∗) be an I-dfts. The fuzzy sets λ, μ∈IXare (r,s)-fuzzy strongly separated if there exist ν, γ∈IXsuch that τ(ν)≥r and τ∗(ν)≤s, τ(γ)≥r with λ≤ν, μ≤γ and .
Definition 4.18
An I-dfts (X,τ,τ∗) is called (r,s)-fuzzy pre T2if for each , with different supports there exists (r,s)-fpo sets λ, μ∈IXsuch that , and .
Theorem 4.23
If is double fuzzy weakly preclosed surjection and all fibers are (r,s)-fuzzy strongly separated, then is (r,s)-fuzzy pre-T2.
Proof
Let and let γ,ν∈IX, r∈I0and s∈I1; τ1(γ)≥r, , τ1(ν)≥r and τ1(ν)≤s such that and respectively with . By using Theorem 3.12-4 there are (r,s)-fpo sets λ, μ∈IYsuch that and , and . Therefore , because and f is surjective. Thus is (r,s)-fuzzy pre-T2. □
Definition 4.19
an I-dfts (X,τ,τ∗) is said to be (r,s)-extremally disconnected if τ(Cτ,τ∗(λ,r,s))≥r and τ∗(Cτ,τ∗(λ,r,s))≤s for each λ∈IX; τ(λ)≥r and τ∗(λ)≤s.
Definition 4.20
an I-dfts (X,τ,τ∗) is said to be (r,s)-fuzzy almost compact if for each (r,s)-fuzzy open cover {λ i ∣i∈J} of X, there is a finite subset J0of J such that .
Definition 4.21
A fuzzy set λ in an I-dfts (X,τ,τ∗) is said to be (r,s)-fuzzy p-compact iff for each family of (r,s)-fpo sets {μ i ∣i∈J} satisfies for each x∈X. There exists finite subfamily J0of J such that for each x∈X.
Theorem 4.24
Let be (r,s)-extremally disconnected I-dfts. Let be double fuzzy open and double fuzzy preclosed injective function such that f−1(y s ) is (r,s)-fuzzy almost compact for each y s ∈ P(Y). If λ∈IYis (r,s)-fuzzy P-compact. Then f−1(λ) is (r,s)-fuzzy almost compact.
Proof
Let {ν j ∣æ∈J} be (r,s)-fuzzy open cover of f−1(λ). Then for each y s ≤λ∧f(X), , for some finite subfamily J(y s ) of J. Since is (r,s)-extremally disconnected each and , hence and . So by Corollary 4.21 there exists (r,s)-fpc set such that . Then, is (r,s)-fuzzy preclosed cover of λ, for some finite fuzzy subset K of λ∧f(X). Hence,
so . Therefore f−1 (λ) is (r,s)-fuzzy almost compact. □
References
Bhaumik RN, Abbas SE: (L,M)-intuitionistic fuzzy topological spaces. Bull Kerala Mathematics Assoc 2008, 5: 25-47.
Chang CL: Fuzzy topological spaces. J Math Anal Appl 1968, 24: 39-90. 10.1016/0022-247X(68)90048-6
Çoker D, Demirci M: An introduction to intuitionistic fuzzy topological spaces in Šostak’s sense. Busefal 1996, 67: 67-76.
Du S, Qin Q, Wang Q, Li B: Fuzzy description of topological relations i: a unified fuzzy 9-intersection model. Lecture Notes Comp Sci 2005, 3612: 1261-1273. 10.1007/11539902_161
Egenhofer MJ, Franzosa R: Point-set topological spatial relations. J Geo Info Sys 1991, 2: 161-174.
El-Naschie MS: On the uncertainty of cantorian geometry and the twoslit experiment. Chaos, Solitons Fractals 1998, 9: 517-529. 10.1016/S0960-0779(97)00150-1
El-Naschie, MS: On the certification of heterotic strings, M theory and $e^{\infty }$e∞ theory. Chaos, Solitons and Fractals 2000, 2397-2408.
Garcia JG, Rodabaugh SE: Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets Syst 2005, 156: 445-484. 10.1016/j.fss.2005.05.023
Herring J, Egenhofer MJ: Categorizing binary topological relations between regions, lines and points in geographic databases. 1991.
Kubiak T: On fuzzy topologies. 1985. Ph.D thesis, A. Mickiewicz, poznan
Lee EP, Im YB: Mated fuzzy topological spaces. Int J Fuzzy Logic Intell Syst 2001, 11: 161-165.
Lowen R: Fuzzy topological spaces and fuzzy compactness. J Math Anal Appl 1976, 56: 621-623. 10.1016/0022-247X(76)90029-9
Samanta SK, Mondal TK: Intuitionistic gradation of openness: intuitionistic fuzzy topology. Busefal 1997, 73: 8-17.
Samanta SK, Mondal TK: On intuitionistic gradation of openness. Fuzzy Sets Syst 2002, 131: 323-336. 10.1016/S0165-0114(01)00235-4
Šostak AP: On a fuzzy topological structure. Suppl Rend Circ Matem Palermo-Sir II 1985, 11: 89-103.
Tang X: Spatial object modeling in fuzzy topological spaces with applications to land cover change in China. 2004.
Acknowledgements
The author would like to thank the reviewers for their valuable comments and helpful suggestions for improvement of the original manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ghareeb, A. Weak forms of continuity in I-double gradation fuzzy topological spaces. SpringerPlus 1, 19 (2012). https://doi.org/10.1186/2193-1801-1-19
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/2193-1801-1-19