Intuitionistic Fuzzy ThetaCompact 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

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.

KEYWORD
Intuitionistic fuzzy topology , Thetacompact

1. Introduction
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.2. Preliminaries
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
μ_{A} :X →I andγ_{A} :X →I denote the degree of membership and the degree of nonmembership, respectively, andμ_{A} +γ_{A} ≤ 1. Obviously, every fuzzy setμ_{A} inX is an intuitionistic fuzzy set of the form (μ_{A} , 1 ？μ_{A} ).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 = (μ_{A} ,γ_{A} ),B = (μ_{B} ,γ_{B} ). 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 = (μ_{A} ,γ_{A} ) inX , denoted byx _{(α,β)} ∈A , ifα ≤μ_{A} (x ) andβ ≥γ_{A} (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 quasicoincident, denoted by UqV, if there exists an element x ∈ X such that μU(x) > γV(x) or γU(x) < μV(x).
The word ‘not quasicoincident’ 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 (qneighborhood ) 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 = (μ_{U} ,γ_{U} ) 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 θcluster 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 θclosure ofA and is denoted by cl_{θ}(A ). An intuitionistic fuzzy setA is called anintuitionistic fuzzy θclosed set ifA = cl_{θ}(A ). The complement of an intuitionistic fuzzyθ closed set is said to be anintuitionistic fuzzy θopen set .Definition 2.12. [11] Let (X,Τ ) andU denote an intuitionistic fuzzy topological space and an intuitionistic fuzzy set inX , respectively. Theintuitionistic fuzzy θinterior 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 θcontinuous 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 θIrresolute and Weakly θContinuity
Definition 3.1. Let (X,Τ ) and (Y,U ) be IF topological spaces. A mappingf : (X,Τ ) → (Y,U ) is said to beintuitionistic fuzzy θirresolute 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. LetΤ_{θ} be an IF topology onX generated using the subbase of all the IFθ open sets inX , and letU_{θ} 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,Τ_{θ} ) → (Y,U_{θ} ) is IF continuous.Proof. Trivial.Recall that a fuzzy set
A is said to be afuzzy θneighborhood of a fuzzy pointx_{α} if there exists a fuzzy closedq neighborhoodU ofx_{α} , such that[13].
Definition 3.3. An intuitionistic fuzzy setA is said to be anintuitionistic fuzzy θneighborhood 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 θcontinuous function if for each fuzzy pointx_{α} inX and each fuzzy openq neighborhoodV off (x_{α} ), there exists a fuzzy openq neighborhoodU ofx_{α} such thatf (U ) ≤ cl(V ) [13].Definition 3.4. A functionf : (X,Τ ) → (Y,T' ) is said to beintuitionistic fuzzy weakly θcontinuous 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)} ) ∈ cl_{θ}(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)} ) ∈ cl_{θ}(B ). Hence we obtainf (cl(int(cl(f ^{？1}(B ))))) ≤ cl_{θ}(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 θcontinuous 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 θCompactness
Definition 4.1. A collection {G_{i} i ∈I } of intuitionistic fuzzyθ open sets in an intuitionistic fuzzy topological space (X,Τ ) is said to be anintuitionistic fuzzy θopen cover of a setA ifA ≤？ {G_{i} i ∈I }.Definition 4.2. An intuitionistic fuzzy topological space (X,Τ ) is said to beintuitionistic fuzzy θcompact 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 θcompact if for every collection {G_{i} i ∈I } of intuitionistic fuzzyθ open sets ofX such thatA ≤？ {G_{i} i ∈I }, there is a finite subsetI _{0} ofI such thatA ≤？ {G_{i} 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 = {F_{i} 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 = {G_{i} 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}, … ,G_{n} } ing such thatThis implies that {
G _{1},G _{2}, … ,G_{n} } is a finite subcover ofX ing . Hence,X is IFθ compact.Theorem 4.6. Let (X,Τ ) denote an IF topological space andΤ_{θ} 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,Τ_{θ} ) is IF compact.Proof. Let (X,Τ_{θ} ) be IF compact and letg = {G_{i} i ∈I } denote an IFθ open cover ofX inT . Since for eachi ∈I, G_{i} ∈Τ_{θ}, g is an IF open cover ofX inΤ_{θ} . Since (X,Τ_{θ} ) is IF compact,g has a finite subcover ofX . Hence, (X,Τ ) is IFθ compact.Let (
X,Τ ) be IFθ compact and letg = {G_{i} G_{i} ∈Τ_{θ}, i ∈I } denote an IF open cover ofX inΤ_{θ} . Since for eachi ∈I, G_{i} ∈Τ_{θ} ,G_{i} 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,Τ_{θ} ) 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 = {G_{i} i ∈I } denote an IFθ open cover ofA . SinceA^{c} is an IFθ open subset ofX, g = {G_{i} i ∈I } ∪A^{c} 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 inΤ_{θ} 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. LetΤ_{θ} denote an IF topology onX generated by the subbase of all IFθ open sets inX and letU_{θ} 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,Τ_{θ} ) → (Y,U_{θ} ) is IF continuous.Proof. Trivial.Recall that a function
f : (X,Τ ) → (Y,T' ) is said to beintuitionistic fuzzy strongly θcontinuous 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 θcontinuous image of an IF θcompact set is IF compact.
(2) Let (X,Τ) and (Y,U) denote IF topological spaces and let f : (X,Τ) → (Y,U) be IF θirresolute. If a subset A of X is IF θcompact, then image f(A) is IF θcompact.
Proof. (1) Letf : (X,Τ ) → (Y,U ) denote an IF stronglyθ continuous mapping from an IFθ compact spaceX onto an IF topological spaceY . Letg = {G_{i} i ∈I } be an IF open cover ofY . Sincef is an IF stronglyθ continuous function,f : (X,Τ_{θ} ) → (Y,U ) is an IF continuous function (Theorem 4.2 of [11]). Therefore, {f ^{？1}(G_{i} ) i ∈I } is an IFθ open cover ofX . SinceX is IFθ compact, there is a finite subsetI _{0} ofI such thatSince
f is onto, {G_{i} i ∈I _{0}} is a finite subcover ofY . Hence,Y is IF compact.(2) Let
g = {G_{i} i ∈I } be an IFθ open cover off (A ) inY . Sincef is an IFθ irresolute, for eachG_{i} ,f ^{？1}(G_{i} ) is an IFθ open set. Moreover, {f ^{？1}(G_{i} ) 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}(G_{i} ) i ∈I _{0}}. Therefore,f (A ) ≤？ {G_{i} 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 = {G_{i} i ∈I } be an IFθ open cover ofA ？ B inX . SinceB^{c} is IFθ open inX , (？ {G_{i} i ∈I }) ∨B^{c} is an IFθ open cover ofA . SinceA is IFθ compact, there is a finite subsetI _{0} ofI such thatA ≤ (？ {G_{i} i ∈I _{0}}) ∨B^{c} . Therefore,A ∧B ≤ (？ {G_{i} i ∈I _{0}}). Hence,A ∧B is IFθ compact.5. Conclusion
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.> Conflict of Interest
No potential conflict of interest relevant to this article was reported.