Intuitionistic Fuzzy Theta-Compact Spaces
- Author: Eom Yeon Seok, Lee Seok Jong
- Organization: Eom Yeon Seok; Lee Seok Jong
- Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue3, p224~230, 25 Sep 2013
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ABSTRACT
In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy
θ -compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzyθ -compactness is strictly weaker than intuitionistic fuzzy compactness. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzyθ -compactness in the original intuitionistic fuzzy topology. This characterization shows that intuitionistic fuzzyθ -compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.
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KEYWORD
Intuitionistic fuzzy topology , Theta-compact
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The concept of an intuitionistic fuzzy set as a generalization of fuzzy sets was introduced by Atanassov [1]. Coker and his colleagues [2?4] introduced an intuitionistic fuzzy topology using intuitionistic fuzzy sets.
Many researchers studied continuity and compactness in fuzzy topological spaces and intuitionistic fuzzy topological spaces [5?8]. Recently, Hanafy et al. [9] introduced an intuitionistic fuzzy
θ -closure operator and intuitionistic fuzzyθ -continuity.In this paper, we introduce certain types of continuous functions and intuitionistic fuzzy
θ -compactness in intuitionistic fuzzy topological spaces. We show that intuitionistic fuzzyθ -compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we show that the sufficient condition in Theorem 4.5 holds for intuitionistic fuzzyθ -compact spaces; however, in general, it fails for intuitionistic fuzzy compact spaces. Furthermore, we show that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to the intuitionistic fuzzyθ -compactness in the original intuitionistic fuzzy topology described in Theorem 4.6. This characterization shows that the intuitionistic fuzzyθ -compactness can be related to an appropriated notion of intuitionistic fuzzy convergence.Let
X andI denote a nonempty set and unit interval [0, 1], respectively. Anintuitionistic fuzzy set A inX is an object of the formA = (μA, γA),
where the functions
μA :X →I andγA :X →I denote the degree of membership and the degree of non-membership, respectively, andμA +γA ≤ 1. Obviously, every fuzzy setμA inX is an intuitionistic fuzzy set of the form (μA , 1 ?μA ).Throughout this paper,
I (X ) denotes the family of all intuitionistic fuzzy sets inX and intuitionistic fuzzy is abbreviated as IF.Definition 2.1. [1] LetX denote a nonempty set and let intuitionistic fuzzy setsA andB be of the formA = (μA ,γA ),B = (μB ,γB ). Then,(1) A ≤ B iff μA(x) ≤ μB(x) and γA(x) ≥ γB(x) for all x ∈ X,
(2) A = B iff A ≤ B and B ≤ A,
(3) Ac = (γA,μA),
(4) A ∩ B = (μA ∧ μB, γA ∨ γB),
(5) A ∪ B = (μA ∨ μB, γA ∧ γB),
Definition 2.2. [2] Anintuitionistic fuzzy topology onX is a familyΤ of intuitionistic fuzzy sets inX that satisfy the following axioms.(2) G1 ∩ G2 ∈ Τ for any G1, G2 ∈ Τ,
(3) ?Gi ∈ Τ for any {Gi : i ∈ J} ⊆ Τ.
In this case, the pair (
X, Τ ) is called anintuitionistic fuzzy topological space and any intuitionistic fuzzy set inΤ is known as anintuitionistic fuzzy open set inX .Definition 2.3. [2] Let (X, Τ ) andA denote an intuitionistic fuzzy topological space and intuitionistic fuzzy set inX , respectively. Then, theintuitionistic fuzzy interior ofA and theintuitionistic fuzzy closure ofA are defined bycl(A) = ?{K | A ≤ K,Kc ∈ Τ}
and
int(A) = ?{G | G ≤ A,G ∈ Τ}
Theorem 2.4. [2] For any IF setA in an IF topological space (X, Τ ), we havecl(Ac) = (int(A))c and int(Ac) = (cl(A))c.
Definition 2.5. [3, 4] Letα,β ∈ [0, 1] andα +β ≤ 1. Anintuitionistic fuzzy point x (α,β ) ofX is an intuitionistic fuzzy set inX defined byIn this case,
x,α, andβ are called thesupport, value, and nonvalue ofx (α,β ), respectively. An intuitionistic fuzzy pointx (α,β ) is said tobelong to an intuitionistic fuzzy setA = (μA ,γA ) inX , denoted byx (α,β ) ∈A , ifα ≤μA (x ) andβ ≥γA (x ).Remark 2.6. If we consider an IF pointx (α,β ) as an IF set, then we have the relationx (α,β ) ∈A if and only ifx (α,β ) ≤A .Definition 2.7. [4,10] Let (X,Τ ) denote an intuitionistic fuzzy topological space.(1) An intuitionistic fuzzy point x(α,β) is said to be quasicoincident with the intuitionistic fuzzy set U = (μU,γU), denoted by x(α,β)qU, if α > γU(x) or β < μU(x).
(2) Let U = (μU,γU) and V = (μV,γV) denote two intuitionistic fuzzy sets in X. Then, U and V are said to be quasi-coincident, denoted by UqV, if there exists an element x ∈ X such that μU(x) > γV(x) or γU(x) < μV(x).
The word ‘not quasi-coincident’ will be abbreviated as
herein.
Proposition 2.8. [4] LetU, V , andx (α,β ) denote IF sets and an IF point inX , respectively. Then,Definition 2.9. [4] Let (X,Τ ) denote an intuitionistic fuzzy topological space and letx (α,β ) denote an intuitionistic fuzzy point inX . An intuitionistic fuzzy setA is said to be anintuitionistic fuzzy ?-neighborhood (q-neighborhood ) ofx (α,β ) if there exists an intuitionistic fuzzy open setU inX such thatx (α,β ) ∈U ≤A (x (α,β )qU ≤A , respectively).Theorem 2.10. [10] Letx (α,β ) andU = (μU ,γU ) denote an IF point inX and an IF set inX , respectively. Then,x (α,β ) ∈ cl(U ) if and only ifUqN , for any IFq -neighborhoodN ofx (α,β ).Definition 2.11. [9] An intuitionistic fuzzy pointx (α,β ) is said to be anintuitionistic fuzzy θ-cluster point of an intuitionistic fuzzy setA if for each intuitionistic fuzzyq -neighborhoodU ofx (α,β ),Aq cl(U ). The set of all intuitionistic fuzzyθ -cluster points ofA is calledintuitionistic fuzzy θ-closure ofA and is denoted by clθ (A ). An intuitionistic fuzzy setA is called anintuitionistic fuzzy θ-closed set ifA = clθ (A ). The complement of an intuitionistic fuzzyθ -closed set is said to be anintuitionistic fuzzy θ-open set .Definition 2.12. [11] Let (X,Τ ) andU denote an intuitionistic fuzzy topological space and an intuitionistic fuzzy set inX , respectively. Theintuitionistic fuzzy θ-interior ofU is denoted and defined byintθ(U) = (clθ(Uc))c.
Definition 2.13. [2] Let (X,Τ ) and (Y,U ) denote two intuitionistic fuzzy topological spaces and letf :X →Y denote a function. Then,f is said to beintuitionistic fuzzy continuous if the inverse image of an intuitionistic fuzzy open set inY is an intuitionistic fuzzy open set inX .Definition 2.14. [2] An intuitionistic fuzzy topological space (X,Τ ) is said to beintuitionistic fuzzy compact if every open cover ofX has a finite subcover.Definition 2.15. [9] A functionf :X →Y is said to beintuitionistic fuzzy θ-continuous if for each intuitionistic fuzzy pointx (a,b ) inX and each intuitionistic fuzzy openq -neighborhoodV off (x (a,b )), there exists an intuitionistic fuzzy openq -neighborhoodU ofx (a,b ) such thatf (cl(U )) ≤ cl(V ).Proposition 2.16. [12] Letf : (X,Τ ) → (Y,T' ) andx (α,β ) denote a function and an IF point inX , respectively.(1) If f(x(α,β))qV, then x(α,β)qf?1(V) for any IF set V in Y.
(2) If x(α,β)qU, then f(x(α,β))qf(U) for any IF set U in X.
Remark 2.17. Intuitionistic fuzzy sets have some different properties compared to fuzzy sets, and these properties are shown in the subsequent examples.1. x(α,β) ∈ A ∪ B ? x(α,β) ∈ A or x(α,β) ∈ B.
2. x(α,β)qA and x(α,β)qB ? x(α,β)q(A ∩ B).
Thus, we have considerably different results in generalizing concepts of fuzzy topological spaces to the intuitionistic fuzzy topological space.
Example 2.18. LetA, B denote IF sets on the unit interval [0, 1] defined byIn addition, let
x = ¼, (α,β ) = (¼,½). Then,x (α,β ) ∈A ∪B . However,x (α,β ) ?A andx (α,β ) ?B .Example 2.19. LetA, B denote IF sets on the unit interval [0, 1] defined byIn addition, let
x = ¼, (α,β ) = (½,¼). Then,x (α,β )qA andx (α,β )qB ; however,For the notions that are not mentioned in this section, refer to [11].
3. Intuitionistic Fuzzy θ-Irresolute and Weakly θ-Continuity
Definition 3.1. Let (X,Τ ) and (Y,U ) be IF topological spaces. A mappingf : (X,Τ ) → (Y,U ) is said to beintuitionistic fuzzy θ-irresolute if the inverse image of each IFθ -open set inY is IFθ -open inX .Theorem 3.2. Let (X,Τ ) and (Y,U ) be IF topological spaces. LetΤθ be an IF topology onX generated using the subbase of all the IFθ -open sets inX , and letUθ be an IF topology onY generated using the subbase of all the IFθ -open sets inY . Then a functionf : (X,Τ ) → (Y,U ) is IFθ -irresolute if and only iff : (X,Τθ ) → (Y,Uθ ) is IF continuous.Proof. Trivial.Recall that a fuzzy set
A is said to be afuzzy θ-neighborhood of a fuzzy pointxα if there exists a fuzzy closedq -neighborhoodU ofxα , such that[13].
Definition 3.3. An intuitionistic fuzzy setA is said to be anintuitionistic fuzzy θ-neighborhood of intuitionistic fuzzy pointx (α,β ) if there exists an intuitionistic fuzzy openq -neighborhoodU ofx (α,β ) such that cl(U ) ≤A .Recall that a function
f : (X,Τ ) → (Y,T' ) is said to be afuzzy weakly θ-continuous function if for each fuzzy pointxα inX and each fuzzy openq -neighborhoodV off (xα ), there exists a fuzzy openq -neighborhoodU ofxα such thatf (U ) ≤ cl(V ) [13].Definition 3.4. A functionf : (X,Τ ) → (Y,T' ) is said to beintuitionistic fuzzy weakly θ-continuous if for each intuitionistic fuzzy pointx(α,β) inX and each intuitionistic fuzzy openq -neighborhoodV off (x(α,β) ), there exists an intuitionistic fuzzy openq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ).Theorem 3.5. A functionf : (X,Τ ) → (Y,T' ) is IF weaklyθ -continuous if and only if for each IF pointx(α,β) inX and each IF openθ -neighborhoodN off (x(α,β) ) inY ,f ?1(N ) is an IFq -neighborhood ofx(α,β) .Proof. Letf be an IF weaklyθ -continuous function, and letx(α,β) be an IF point inX . LetN be an IFθ -neighborhood off (x(α,β) ) inY . Then there exists an IF openq -neighborhoodV off (x(α,β) ) such that cl(V ) ≤N . Sincef is IF weaklyθ -continuous, there exists an IFq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ) ≤N . ThusU ≤f ?1(N ). Therefore, there exists an IFq -neighborhoodU ofx(α,β) such thatU ≤f ?1(N ). Hencef ?1(N ) is an IFq -neighborhood ofx(α,β) .Conversely, let
x(α,β) be an IF point inX , and letV be an IF openq -neighborhood off (x(α,β) ). Then cl(V ) is an IFθ -neighborhood off (x(α,β) ). By the hypothesis,f ?1(cl(V )) is an an IFq -neighborhood ofx(α,β) . Then there exists an IF open setU such thatx(α,β) qU ≤f ?1(cl(V )). Thusf (U ) ≤ cl(V ). Therefore there exists an IF openq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ). Hencef is an IF weaklyθ -continuous function.Theorem 3.6. If a functionf : (X,Τ ) → (Y,T' ) is IF weaklyθ -continuous, then(1) f(cl(A)) ≤ clθ(f(A)) for each IF set A in X,
(2) f(cl(int(cl(f?1(B))))) ≤ clθ(B) for each IF set B in Y .
Proof. (1) Letx(α,β) ∈ cl(A ), and letV be an IF openq -neighborhood off (x(α,β) ). Sincef is IF weaklyθ -continuous, there exists an IF openq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ). Sincex(α,β) ∈ cl(A ),UqA . Thusf (U )qf (A ). Sincef (U ) ≤ cl(V ), we have cl(V )qf (A ). Thus for each IF openq -neighborhoodV off (x(α,β) ), cl(V )qf (A ). Hencef (x(α,β) ) ∈ clθ (f (A )).(2) Let
B be an IF set inY andx(α,β) ∈ cl(int(cl(f ?1(B )))). LetV be an IF openq -neighborhood off (x(α,β) ). Sincef is IF weaklyθ -continuous, there exists an IF openq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ). Since int(cl(f ?1(B ))) ≤ cl(f ?1(B )),cl(int(cl(f?1(B)))) ≤ cl(cl(f?1(B))) = cl(f?1(B)).
Since
x(α,β) ∈ cl(int(cl(f ?1(B )))),x(α,β) ∈ cl(f ?1(B )). Thusf ?1(B )qU , orBqf (U ). Sincef (U ) ≤ cl(V ), we have cl(V )qB . Thereforef (x(α,β) ) ∈ clθ (B ). Hence we obtainf (cl(int(cl(f ?1(B ))))) ≤ clθ (B ), for each IF setB inY .Theorem 3.7. Letf : (X,Τ ) → (Y,T' ) be a function. Then the following statements are equivalent:(1) f is an IF weakly θ-continuous function.
(2) For each IF open set U with x(α,β)qf?1(U), x(α,β)q int(f?1(cl(U))).
Proof. (1) ⇒ (2). Let
f be an IF weaklyθ -continuous function, and letU be an IF open set withx(α,β) qf ?1(U ). Thenf (x(α,β) )qU . By the definition of IF weaklyθ -continuous, there exists an IF openq -neighborhoodV ofx(α,β) such thatf (V ) ≤ cl(U ). ThusV ≤f ?1(cl(U )), i.e.Therefore,
x(α,β) ? cl((f ?1(cl(U )))c ) = (int(f ?1(cl(U ))))c . Hence we havex(α,β) q (int(f ?1(cl(U )))).(2) ⇒ (1). Let the condition hold, and let
x(α,β) be any IF point inX andV an IF openq -neighborhood off (x(α,β) ). Thenx(α,β) qf ?1(V ). By the hypothesis,x(α,β)qint(f?1(cl(V))).
Put
U = int(f ?1(cl(V ))). ThenU is an IF openq -neighborhood ofx(α,β) . Since int(f ?1(cl(V ))) ≤f ?1(cl(V )),f(int(f?1(cl(V)))) ≤ f(f?1(cl(V))) ≤ cl(V).
Thus
f (U ) ≤ cl(V ). Therefore there exists an IF openq -neighborhoodU ofx(α,β) such thatf (U ) ≤ cl(V ). Hencef is an IF weaklyθ -continuous function.4. Intuitionistic Fuzzy θ-Compactness
Definition 4.1. A collection {Gi |i ∈I } of intuitionistic fuzzyθ -open sets in an intuitionistic fuzzy topological space (X,Τ ) is said to be anintuitionistic fuzzy θ-open cover of a setA ifA ≤? {Gi |i ∈I }.Definition 4.2. An intuitionistic fuzzy topological space (X,Τ ) is said to beintuitionistic fuzzy θ-compact if every intuitionistic fuzzyθ -open cover ofX has a finite subcover.Definition 4.3. A subsetA of an intuitionistic fuzzy topological space (X,Τ ) is said to beintuitionistic fuzzy θ-compact if for every collection {Gi |i ∈I } of intuitionistic fuzzyθ -open sets ofX such thatA ≤? {Gi |i ∈I }, there is a finite subsetI 0 ofI such thatA ≤? {Gi |i ∈I 0}.Remark 4.4. Since every IFθ -open set is IF open, it follows that every IF compact space is IFθ -compact.Theorem 4.5. An IF topological space (X,Τ ) is IFθ -compact if and only if every family of IFθ -closed subsets ofX with the finite intersection property has a nonempty intersection.Proof. LetX be IFθ -compact and letF = {Fi |i ∈I } denote any family of IFθ -closed subsets ofX with the finite intersection property. Suppose thatThen,
is an IF
θ -open cover ofX . SinceX is IFθ -compact, there is a finite subsetI 0 ofI such thatThis implies that
which contradicts the assumption that
F has a finite intersection property. Hence,Let
g = {Gi |i ∈I } denote an IFθ -open cover ofX and consider the familyof an IF
θ -closed set. Sinceg is a cover ofX ,Hence,
g' does not have the finite intersection property, i.e., there are finite numbers of IFθ -open sets {G 1,G 2, … ,Gn } ing such thatThis implies that {
G 1,G 2, … ,Gn } is a finite subcover ofX ing . Hence,X is IFθ -compact.Theorem 4.6. Let (X,Τ ) denote an IF topological space andΤθ denote the IF topology onX generated using the subbase of all IFθ -open sets inX . Then, (X,Τ ) is IFθ -compact if and only if (X,Τθ ) is IF compact.Proof. Let (X,Τθ ) be IF compact and letg = {Gi |i ∈I } denote an IFθ -open cover ofX inT . Since for eachi ∈I, Gi ∈Τθ, g is an IF open cover ofX inΤθ . Since (X,Τθ ) is IF compact,g has a finite subcover ofX . Hence, (X,Τ ) is IFθ -compact.Let (
X,Τ ) be IFθ -compact and letg = {Gi |Gi ∈Τθ, i ∈I } denote an IF open cover ofX inΤθ . Since for eachi ∈I, Gi ∈Τθ ,Gi is an IFθ -open set in (X,Τ ). Therefore,g is an IFθ -open cover ofX inT . Since (X,Τ ) is IFθ -compact,g has a finite subcover ofX . Hence, (X,Τθ ) is IF compact.Theorem 4.7. LetA be an IFθ -closed subset of an IFθ -compact spaceX . Then,A is also IFθ -compact.Proof. LetA denote an IFθ -closed subset ofX and letg = {Gi |i ∈I } denote an IFθ -open cover ofA . SinceAc is an IFθ -open subset ofX, g = {Gi |i ∈I } ∪Ac is an IFθ -open cover ofX . SinceX is IFθ -compact, there is a finite subsetI 0 ofI such thatHence,
A is IFθ -compact relative toX .Theorem 4.8. An IF topological space (X,Τ ) is IFθ -compact if and only if every family of IF closed subsets ofX inΤθ with the finite intersection property has a nonempty intersection.Proof. Trivial by Theorem 4.5.Theorem 4.9. Let (X,Τ ) and (Y,U ) denote IF topological spaces. LetΤθ denote an IF topology onX generated by the subbase of all IFθ -open sets inX and letUθ denote an IF topology onY generated by the subbase of all IFθ -open sets inY . Then, a functionf : (X,Τ ) → (Y,U ) is IFθ -irresolute if and only iff : (X,Τθ ) → (Y,Uθ ) is IF continuous.Proof. Trivial.Recall that a function
f : (X,Τ ) → (Y,T' ) is said to beintuitionistic fuzzy strongly θ-continuous if for each IF pointx(α,β) inX and for each IF openq -neighborhoodV off (x(α,β) ), there exists an IF openq -neighborhoodU ofx(α,β) such thatf (cl(U )) ≤V ([9]).Theorem 4.10. (1) An IF strongly θ-continuous image of an IF θ-compact set is IF compact.
(2) Let (X,Τ) and (Y,U) denote IF topological spaces and let f : (X,Τ) → (Y,U) be IF θ-irresolute. If a subset A of X is IF θ-compact, then image f(A) is IF θ-compact.
Proof. (1) Letf : (X,Τ ) → (Y,U ) denote an IF stronglyθ -continuous mapping from an IFθ -compact spaceX onto an IF topological spaceY . Letg = {Gi |i ∈I } be an IF open cover ofY . Sincef is an IF stronglyθ -continuous function,f : (X,Τθ ) → (Y,U ) is an IF continuous function (Theorem 4.2 of [11]). Therefore, {f ?1(Gi ) |i ∈I } is an IFθ -open cover ofX . SinceX is IFθ -compact, there is a finite subsetI 0 ofI such thatSince
f is onto, {Gi |i ∈I 0} is a finite subcover ofY . Hence,Y is IF compact.(2) Let
g = {Gi |i ∈I } be an IFθ -open cover off (A ) inY . Sincef is an IFθ -irresolute, for eachGi ,f ?1(Gi ) is an IFθ -open set. Moreover, {f ?1(Gi ) |i ∈I } is an IFθ -open cover ofA . SinceA is IFθ -compact relative toX , there exists a finite subsetI 0 ofI such thatA ≤? {f ?1(Gi ) |i ∈I 0}. Therefore,f (A ) ≤? {Gi |i ∈I 0}. Hence,f (A ) is IFθ -compact relative toY .Theorem 4.11. LetA andB be subsets of an IF topological space (X,Τ ). IfA is IFθ -compact andB is IFθ -closed inX , thenA ∧B is IFθ -compact.Proof. Letg = {Gi |i ∈I } be an IFθ -open cover ofA ? B inX . SinceBc is IFθ -open inX , (? {Gi |i ∈I }) ∨Bc is an IFθ -open cover ofA . SinceA is IFθ -compact, there is a finite subsetI 0 ofI such thatA ≤ (? {Gi |i ∈I 0}) ∨Bc . Therefore,A ∧B ≤ (? {Gi |i ∈I 0}). Hence,A ∧B is IFθ -compact.We introduced IF
θ -irresolute and weaklyθ -continuous functions, and intuitionistic fuzzyθ -compactness in intuitionistic fuzzy topological spaces. We showed that intuitionistic fuzzyθ -compactness is strictly weaker than intuitionistic fuzzy compactness. Moreover, we showed that if a topological space is intuitionistic fuzzy retopologized, then intuitionistic fuzzy compactness in the new intuitionistic fuzzy topology is equivalent to intuitionistic fuzzyθ -compactness in the original intuitionistic fuzzy topology.No potential conflict of interest relevant to this article was reported.