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Weakly generalized compactness in intuitionistic fuzzy topological spaces.

[section]1. Introduction

Fuzzy set (FS) as proposed by Zadeh [13] in 1965, is a framework to encounter uncertainty, vagueness and partial truth and it represents a degree of membership for each member of the universe of discourse to a subset of it. After the introduction of fuzzy topology by Chang [31 in 1968, there have been several generalizations of notions of fuzzy sets and fuzzy topology. By adding the degree of non-membership to FS, Atanassov [1] proposed intuitionistic fuzzy set (IFS) in 1986 which looks more accurate to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. In 1997, Coker [4] introduced the concept of intuitionistic fuzzy topological space. In this paper, we introduce a new class of intuitionistic fuzzy topological space called weakly generalized compact space, almost weakly generalized compact space using intuitionistic fuzzy weakly generalized open sets and nearly weakly generalized compact space using intuitionistic fuzzy weakly generalized closed sets and study some of their properties.

[section]2. Preliminaries

Definition 2.1.[1] Let X be a non empty fixed set. An intuitionistic fuzzy set (IFS in short) A in X is an object having the form A = {{x, [[mu].sub.A](x), [v.sub.A](x)) : x [member of] X} where the functions [[mu].sub.A](x) : X [right arrow] [0,1] and [v.sub.A](x) : X [right arrow] [0,1] denote the degree of membership (namely [[mu].sub.A](x)) and the degree of non-membership (namely [v.sub.A](x)) of each element x [member of] X to the set A, respectively, and 0 [less than or equal to] [[mu].sub.A](x) + [v.sub.A](x) [less than or equal to] 1 for each x [member of] X.

Definition 2.2.[1] Let A and B be IFSs of the forms A = {{x, [[mu].sub.A](x), [v.sub.A](x)) : x [member of] X} and B = {{x, [[mu].sub.B](x),[v.sub.B](x)) : x [member of] X}. Then

(i) A [subset or equal to] B if and only if [[mu].sub.A](x) [less than or equal to] [[mu].sub.B] (x) and [v.sub.A](x) [greater than or equal to] [v.sub.B](x) for all x [member of] X.

(ii) A = B if and only if A [subset or equal to] B and B [subset or equal to] A.

(iii) [A.sup.c] = {{x, [v.sub.A](x), [[mu].sub.A](x)) : x [member of] X}.

(iv) A [intersection] B = {{x, [[mu].sub.A](x) [and] [[mu].sub.B](x), [v.sub.A](x) [disjunction] [v.sub.B] (x)) : x [member of] X} .

(v) A [union] B = {{x, [[mu].sub.A](x) [disjunction] [[mu].sub.B] (x), [v.sub.A](x) [and] [v.sub.B](x)) : x [member of] X} .

For the sake of simplicity, the notation A = {x, [[mu].sub.A], [v.sub.A]) shall be used instead of A = {{x, [[mu].sub.A](x), [v.sub.A](x)) : x [member of] X}. Also for the sake of simplicity, we shall use the notation A = {x, ([[mu].sub.A],[[mu].sub.B]), ([v.sub.A], [v.sub.B])) instead of A = {x, (A/[[mu].sub.A],B/[[mu].sub.B]), (A/[v.sub.A],B/[v.sub.B])).

The intuitionistic fuzzy sets [0.sub.~] = {{x, 0,1) : x [member of] X} and [1.sub.~] = {{x, 1,0) : x [member of] X} are the empty set and the whole set of X, respectively.

Definition 2.3.[4] An intuitionistic fuzzy topology (IFT in short) on a non empty set X is a family [tau] of IFSs in X satisfying the following axioms:

(i) [0.sub.~], [1.sub.~] U [member of] [tau].

(ii) [G.sub.1] [intersection] [G.sub.2] [member of] [tau] for any [G.sub.1], [G.sub.2] [member of] [tau].

(iii) [union][G.sub.i] [member of] [tau] for any arbitrary family {[G.sub.i] : i [member of] J} [subset or equal to] [tau].

In this case, the pair (X, [tau]) is called an intuitionistic fuzzy topological space (IFTS in short) and any IFS in [tau] is known as an intuitionistic fuzzy open set (IFOS in short) in X.

The complement [A.sup.c] of an IFOS A in an IFTS (X, [tau]) is called an intuitionistic fuzzy closed set (IFCS in short) in X.

Definition 2.4. [4] Let (X, [tau]) be an IFTS and A = {x, [[mu].sub.A], [v.sub.A]) be an IFS in X. Then the intuitionistic fuzzy interior and an intuitionistic fuzzy closure are defined by

int(A) = [union]{G/G is an IFOS in X and G [subset or equal to] A},

cl(A) = [intersection] {K/K is an IFCS in X and A [subset or equal to] K} .

Note that for any IFS A in (X, [tau]), we have cl([A.sup.c]) = [(int(A)).sup.c] and int([A.sup.c]) = [(cl(A)).sup.c].

Definition 2.5. [5] An IFS A = {{x, [[mu].sub.A](x), [v.sub.A](x)) : x [member of] X} in an IFTS (X, [tau]) is said to be an intuitionistic fuzzy weakly generalized closed set (IFWGCS in short) if cl(int(A)) [subset or equal to] U whenever A C [subset or equal to] U and U is an IFOS in X.

The family of all IFWGCSs of an IFTS (X, [tau]) is denoted by IFWGC(X).

Definition 2.6. [5] An IFS A = {{x, [[mu].sub.A](x), [v.sub.A](x)) : x [member of] X} is said to be an intuitionistic fuzzy weakly generalized open set (IFWGOS in short) in (X, [tau]) if the complement [A.sup.c] is an IFWGCS in (X, [tau]).

The family of all IFWGOS s of an IFTS (X, [tau]) is denoted by IFWGO(X).

Result 2.1. [5] Every IFCS, IF[alpha]CS, IFGCS, IFRCS, IF PCS, IF[alpha]GCS is an IFWGCS but the converses need not be true in general.

Definition 2.7. [6] Let (X, [tau]) be an IFTS and A = {x, [[mu].sub.A], [v.sub.A]) be an IFS in X. Then the intuitionistic fuzzy weakly generalized interior and an intuitionistic fuzzy weakly generalized closure are defined by

wgint(A) = [union] {G/G is an IFWGOS in X and G [subset or equal to] A} ,

wgcl(A) = [intersection] {K/K is an IFWGCS in X and A [subset or equal to] K}.

Definition 2.8. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be a mapping from an IFTS (X, [tau]) into an IFTS (Y, [sigma]). Then f is said to be

(i) Intuitionistic fuzzy weakly generalized continuous [7] (IFWG continuous in short) if [f.sup.-1](B) is an IFWGOS in X for every IFOS B in Y.

(ii) Intuitionistic fuzzy quasi weakly generalized continuous [8] (IF quasi WG continuous in short) if [f.sup.-1](B) is an IFOS in X for every IFWGOS B in Y.

(iii) Intuitionistic fuzzy weakly generalized iresolute [6] (IFWG irresolute in short) if [f.sup.-1](B) is an IFWGOS in X for every IFWGOS B in Y.

(iv) Intuitionistic fuzzy perfectly weakly genralized continuous continuous [10] (IF perfectly WG continuous in short) if [f.sup.-1](B) is an intuitionistic fuzzy clopen set in X for every IFWGCS B in Y.

(v) Intuitionistic fuzzy weakly generalized * open mapping [9] (IFWG*OM in short) if f(B) is an IFWGOS in Y for every IFWGOS B in X.

(vi) Intuitionistic fuzzy contra weakly generalized continuous [11] (IF contra WG continuous in short) if [f.sup.-1](B) is an IFWGCS in X for every IFOS B in Y.

(vii) Intuitionistic fuzzy contra weakly generalized irresolute [12] (IF contra WG irresolute in short) if [f.sup.-1](B) is an IFWGCS in X for every IFWGOS B in Y.

Definition 2.9.[2] Let (X, [tau]) be an IFTS. A family {{x,[micro][G.sub.i], [v.sub.Gi]) : i [member of] I} of IFOSs in X satisfying the condition [1.sub.~] = [union] {{x, [[mu].sub.Qi], [V.sub.Gi]) : i G I} is called an intuitionistic fuzzy open cover of X.

Definition 2.10. [2] A finite sub family of an intuitionistic fuzzy open cover {{x, [[mu].sub.Gi], [V.sub.Gi]) : i [member of] I} of X which is also an intuitionistic fuzzy open cover of X is called a finite sub cover of {{x, [[mu].sub.Gi], [V.sub.Gi]) : i [member of] I}.

Definition 2.11. [2] An IFTS (X, [tau]) is called intuitionistic fuzzy compact if every intuitionistic fuzzy open cover of X has a finite sub cover.

Definition 2.12. [2] An IFTS (X, [tau]) is called intuitionistic fuzzy Lindelof if each intuitionistic fuzzy open cover of X has a countable sub cover for X.

Definition 2.13. [2] An IFTS (X, [tau]) is called intuitionistic fuzzy countable compact if each countable intuitionistic fuzzy open cover of X has a finite sub cover for X.

Definition 2.14. An IFTS (X, [tau]) is said to be

(i) Intuitionistic fuzzy S-closed [2]if each intuitionistic fuzzy regular closed cover of X has a finite sub cover for X.

(ii) Intuitionistic fuzzy S-Lindelof [2] if each intuitionistic fuzzy regular closed cover of X has a countable sub cover for X.

(iii) Intuitionistic fuzzy countable S-closed [21 if each countable intuitionistic fuzzy regular closed cover of X has a finite sub cover for X.

Definition 2.15. An IFTS (X, [tau]) is said to be

(i) Intuitionistic fuzzy strongly S-closed [2] if each intuitionistic fuzzy closed cover of X has a finite sub cover for X.

(ii) Intuitionistic fuzzy strongly S-Lindelof [2] if each intuitionistic fuzzy closed cover of X has a countable sub cover for X.

(iii) Intuitionistic fuzzy countable strongly S-closed [21 if each countable intuitionistic fuzzy closed cover of X has a finite sub cover for X.

Definition 2.16. An IFTS (X, [tau]) is sad to be

(i) Intuitionistic fuzzy almost compact [2] if each intuitionistic fuzzy open cover of X has a finite sub cover the closure of whose members cover X.

(ii) Intuitionistic fuzzy almost Lindelof [21 if each intuitionistic fuzzy open cover of X has a countable sub cover the closure of whose members cover X.

(iii) Intuitionistic fuzzy countable almost compact [2] if each countable intuitionistic fuzzy open cover of X has a finite sub cover the closure of whose members cover X.

[section]3. Intuitionistic fuzzy weakly generalized compact spaces

In this section, we introduce a new class of intuitionistic fuzzy topological spaces called weakly generalized compact spaces, almost weakly generalized compact spaces using intuitionistic fuzzy weakly generalized open sets and nearly weakly generalized compact spaces using intuitionistic fuzzy weakly generalized closed sets and study some of their properties.

Definition 3.1. Let (X, [tau]) be an IFTS. A family {{x, [[mu].sub.Gi], [V.sub.Gi]) : i [member of] I} of IFWGOSs in X satisfying the condition [1.sub.~] = [union] {{x, [[mu].sub.Gi], [v.sub.Gi]) : i [member of] I} is called an intuitionistic fuzzy weakly generalized open cover of X.

Definition 3.2. A finite sub family of an intuitionistic fuzzy weakly generalized open cover {{x, [[mu].sub.Gi], [v.sub.Gi]) : i [member of] I} of X which is also an intuitionistic fuzzy weakly generalized open cover of X is called a finite sub cover of {{x, [[mu].sub.Gi], [v.sub.Gi]) : i [member of] I}.

Definition 3.3. An intuitionistic fuzzy set A of an IFTS(X, [tau]) is said to be intuitionistic fuzzy weakly generalized compact relative to X if every collection {[A.sub.i] : i [member of] I} of intuitionistic fuzzy weakly generalized open subset of X such that A [subset or equal to] [union] {[A.sub.i] : i [member of] I}, there exists a finite subset [I.sub.0] of I such that A [subset or equal to] [union] {[A.sub.i] : i [member of] [I.sub.0]} .

Definition 3.4. Let (X, [tau]) be an IFTS. A family {{x, [[mu].sub.Gi], [v.sub.Gi]) : i [member of] I} of IFWGCSs in X satisfying the condition [1.sub.~] = [union] {{x, [[mu].sub.Gi], [v.sub.Gi]) : i [member of] I} is called an intuitionistic fuzzy weakly generalized closed cover of X.

Definition 3.5. An IFTS (X, [tau]) is said to be intuitionistic fuzzy weakly generalized compact if every intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover.

Definition 3.6. An IFTS (X, [tau]) is said to be intuitionistic fuzzy weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized open cover of X has a countable sub cover for X.

Definition 3.7. An IFTS (X, [tau]) is said to be intuitionistic fuzzy countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover for X.

Definition 3.8. An IFTS (X, [tau]) is said to be intuitionistic fuzzy nearly weakly generalized compact if each intuitionistic fuzzy weakly generalized closed cover of X has a finite sub cover.

Definition 3.9. An IFTS (X, [tau]) is said to be intuitionistic fuzzy nearly weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized closed cover of X has a countable sub cover for X.

Definition 3.10. An IFTS (X, [tau]) is said to be intuitionistic fuzzy nearly countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized closed cover of X has a finite sub cover for X.

Definition 3.11. An IFTS (X, [tau]) is said to be intuitionistic fuzzy almost weakly generalized compact if each intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover the closure of whose members cover X.

Definition 3.12. An IFTS (X, [tau]) is said to be intuitionistic fuzzy almost weakly generalized Lindelof if each intuitionistic fuzzy weakly generalized open cover of X has a countable sub cover the closure of whose members cover X.

Definition 3.13. An IFTS (X, [tau]) is said to be intuitionistic fuzzy almost countable weakly generalized compact if each countable intuitionistic fuzzy weakly generalized open cover of X has a finite sub cover the closure of whose members cover X.

Definition 3.14. A crisp subset B of an IFTS (X, [tau]) is said to be intuitionistic fuzzy weakly generalized compact if B is intuitionistic fuzzy weakly generalized compact as intuitionistic fuzzy subspace of X.

Theorem 3.1. An intuitionistic fuzzy weakly generalized closed crisp subset of an intuitionistic fuzzy weakly generalized compact space is intuitionistic fuzzy weakly generalized compact relative to X.

Proof. Let A be an intuitionistic fuzzy weakly generalized closed crisp subset of an intuitionistic fuzzy weakly generalized compact space (X, [tau]). Then [A.sup.c] is an IFWGOS in X. Let S be a cover of A by IFWGOS in X. Then, the family {S, [A.sup.c]} is an intuitionistic fuzzy weakly generalized open cover of X. Since X is an intuitionistic fuzzy weakly generalized compact space, it has finite sub cover say {[G.sub.1], [G.sub.2] ... [G.sub.n]}. If this sub cover contains [A.sup.c], we discard it. Otherwise leave the sub cover as it is. Thus we obtained a finite intuitionistic fuzzy weakly generalized open sub cover of A. Therefore A is intuitionistic fuzzy weakly generalized compact relative to X.

Theorem 3.2. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IFWG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = [f.sup.-1] ([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1] ([A.sub.i]), so {[f.sup.-1] ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized open cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]([A.sub.1]), [f.sup.-1]([A.sub.2]), ... [f.sup.-1]([A.sub.n])}.

Therefore [1.sub.~] = [[union].sup.n.sub.i=1] [f.sup.-1] ([A.sub.i]) Hence

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That is, {[A.sub.1], [A.sub.2],... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy compact.

Corollary 3.1. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IFWG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy Lindelof.

Proof. Obvious.

Theorem 3.3. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF quasi WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy compact, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy weakly generalized open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I]. From the relation, [1.sub.~] = [f.sup.-1] ([[union].sub.I[member of]I]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1] ([A.sub.i]), so {[f.sup.-1] ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy open cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy compact, there exists a finite sub cover say, {[f.sup.-1] ([A.sub.1]) ,[f.sup.-1] ([A.sub.2]), ... [f.sup.-1] ([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=1] [f.sup.-1] ([A.sub.i]). Hence

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That is, {[A.sub.1], [A.sub.2], ... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

Corollary 3.2. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF perfectly WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy compact, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

Proof. Obvious.

Theorem 3.4. If f : (X, [tau]) [right arrow] (Y, [sigma]) is an IFWG irrresolute mapping and an intuitionistic fuzzy subset B of an IFTS (X, [tau]) is intuitionistic fuzzy weakly generalized compact relative to an IFTS (X, [tau]), then the image f (B) is intuitionistic fuzzy weakly generalized compact relative to (Y, [sigma]).

Proof. Let {[A.sub.i] : i [member of] I} be any collection IFWGOSs of (Y, [sigma]) such that f (B) [subset or equal to] [union]{[A.sub.i] : i [member of] I}. Since f is IFWG irresolute, B [subset or equal to] [union]{[f.sup.-1] ([A.sub.i]) : i [member of] I} where [f.sup.-1] ([A.sub.i]) is intuitionistic fuzzy weakly generalized open cover in (X, [tau]) for each i. Since B is intuitionistic fuzzy weakly generalized compact relative to (X, [tau]), there exists a finite subset [I.sub.0] of I such that B [subset or equal to] [union] {[f.sup.-1] ([A.sub.i]) : i [member of] [I.sub.0]}. Therefore f (B) [subset or equal to] [union]{[A.sub.i] : I [member of] [I.sub.0]}. Hence f (B) is intuitionistic fuzzy weakly generalized compact relative to (Y, [sigma]).

Theorem 3.5. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IFWG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy nearly weakly generalized compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy weakly generalized closed cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = [f.sup.-1] ([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1] ([A.sub.i]), so {[f.sup.-1] ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized closed cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]([A.sub.1]), [f.sup.-1]([A.sub.2]),... [f.sup.-1]([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=1][f.sup.-1] ([A.sub.i]). Hence

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That is, {[A.sub.1],[A.sub.2],... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy nearly weakly generalized compact.

Corollary 3.3. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IFWG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). Then the following statements hold.

(i) If (X, [tau]) is intuitionistic fuzzy weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

(ii) If (X, [tau]) is intuitionistic fuzzy countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy countable weakly generalized compact.

(iii) If (X, [tau]) is intuitionistic fuzzy weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized Lindelof.

(iv) If (X, [tau]) is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy nearly countable weakly generalized compact.

(v) If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy nearly weakly generalized Lindelof.

Proof. Obvious.

Theorem 3.6. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = f-1 ([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1] ([A.sub.i]), so {[f.sup.-1] ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized closed cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1] ([A.sub.1]), [f.sup.-1] ([A.sub.2]), .. [f.sup.-1] ([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=1][f.sup.-1] ([A.sub.i]). Hence

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That is, {[A.sub.1], [A.sub.2], ... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy compact.

Corollary 3.4. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy Lindelof.

Proof. Obvious.

Theorem 3.7. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy strongly S-closed.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy closed cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = [f.sup.-1] ([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1] ([A.sub.i]), so [f.sup.-1] ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized open cover of (X, [tau]). Since(X, [tau]) is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]([A.sub.1]), [f.sup.-1]([A.sub.2]) ,... [f.sup.-1]([A.sub.n])}. Therefore [1.sub.~] = [U.sup.n.sub.i=1][f.sup.-1]([A.sub.i]). Hence

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That is, {[A.sub.1], [A.sub.2], ... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy strongly S-closed.

Corollary 3.5. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). Then the following statements hold.

(i) If (X, [tau]) is intuitionistic fuzzy weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy strongly S-Lindelof.

(ii) If (X, [tau]) is intuitionistic fuzzy countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy countable strongly S-Closed.

Proof. Obvious.

Theorem 3.8. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy weakly generalized compact (respectively, intuitionistic fuzzy weakly generalized Lindelof, intuitionistic fuzzy countable weakly generalized compact), then (Y, [sigma]) is intuitionistic fuzzy S-closed (respectively, intuitionistic fuzzy S-Lindelof, intuitionistic fuzzy countable S-closed).

Proof. It follows from the statement that each IFRCS is an IFCS.

Theorem 3.9. Let f : (X, [tau]) - (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy almost compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. It follows that [1.sub.~] = [[union].sub.I[member of]I]cl([A.sub.i]). From the relation, [1.sub.~] = [f.sup.-1]( [[union].sub.I[member of]I]cl([A.sub.i])) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1]cl([A.sub.i]), so {[f.sup.-1]cl([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized closed cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1] cl([A.sub.1]),[f.sup.-1] cl([A.sub.2]),...[f.sup.- 1]cl([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=l] [f.sup.-1]cl([A.sub.i]). Hence

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Hence (Y, [sigma]) is intuitionistic fuzzy almost compact.

Corollary 3.6. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG continuous mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). Then the following statements hold.

(i) If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy almost Lindelof.

(ii) If (X, [tau]) is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy countable almost compact.

Proof. Obvious.

Theorem 3.10. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IFWG * open bijective mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact, then (X, [tau]) is intuitionistic fuzzy weakly generalized compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy weakly generalized open cover of (X, [tau]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = f ([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I]f ([A.sub.i]), so {f ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized open cover of (Y, [sigma]). Since (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]([A.sub.1]), [f.sup.- 1] ([A.sub.2]), ... [f.sup.-1] ([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=1][f.sup.-1] ([A.sub.i]). Hence

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That is, {[A.sub.1], [A.sub.2], ... [A.sub.n]} is a finite sub cover of (X, [tau]). Hence (X, [tau]) is intuitionistic fuzzy compact.

Theorem 3.11. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy weakly generalized open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. From the relation, [1.sub.~] = [f.sup.-1]([[union].sub.I[member of]I][A.sub.i]) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1]([A.sub.i]), so {[f.sup.-1]([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized closed cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]([A.sub.1]),[f.sup.-1]([A.sub.2]), ... [f.sup.-1]([A.sub.n])}. Therefore [1.sub.~] = ([[union].sup.n.sub.i=1] [f.sup.-1]([A.sub.i]). Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, {[A.sub.1] ,[A.sub.2],... [A.sub.n]} is a finite sub cover of (Y, [sigma]). Hence (Y, [sigma]) is intuitionistic fuzzy weakly generalized compact.

Corollary 3.7. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). Then the following statements hold.

(i) If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy weakly generalized Lindelof.

(ii) If (X, [tau]) is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy countable weakly generalized compact.

(iii) If (X, [tau]) is intuitionistic fuzzy weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy nearly weakly generalized compact.

(iv) If (X, [tau]) is intuitionistic fuzzy weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy nearly weakly generalized Lindelof.

(v) If (X, [tau]) is intuitionistic fuzzy countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy nearly countable weakly generalized compact.

Proof. Obvious.

Theorem 3.12. Let f : (X, [tau]) [right arrow] (Y, [sigma]) be an IF contra WG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy almost weakly generalized compact.

Proof. Let {[A.sub.i] : i [member of] I} be an intuitionistic fuzzy weakly generalized open cover of (Y, [sigma]). Then [1.sub.~] = [[union].sub.I[member of]I][A.sub.i]. It follows that [1.sub.~] = [[union].sub.I[member of]I]cl([A.sub.i]). From the relation, [1.sub.~] = [f.sup.-1] ([[union].sub.I[member of]I]cl([A.sub.i])) follows that [1.sub.~] = [[union].sub.I[member of]I][f.sup.-1]cl ([A.sub.i]), so {[f.sup.-1]cl ([A.sub.i]) : i [member of] I} is an intuitionistic fuzzy weakly generalized closed cover of (X, [tau]). Since (X, [tau]) is intuitionistic fuzzy nearly weakly generalized compact, there exists a finite sub cover say, {[f.sup.-1]cl ([A.sub.1]), [f.sup.-1]cl ([A.sub.2]),... [f.sup.-1]cl ([A.sub.n])}. Therefore [1.sub.~] = [[union].sup.n.sub.i=1] [f.sup.-1]cl ([A.sub.i]). Hence

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Hence (Y, [sigma]) is intuitionistic fuzzy almost weakly generalized compact.

Corollary 3.8. Let f : (X, [tau]) - (Y, [sigma]) be an IF contra WG irresolute mapping from an IFTS (X, [tau]) onto an IFTS (Y, [sigma]). Then the following statements hold.

(i) If (X, [tau]) is intuitionistic fuzzy nearly weakly generalized Lindelof, then (Y, [sigma]) is intuitionistic fuzzy almost weakly generalized Lindelof.

(ii) If (X, [tau]) is intuitionistic fuzzy nearly countable weakly generalized compact, then (Y, [sigma]) is intuitionistic fuzzy almost countable weakly generalized compact.

Proof. Obvious.

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[3] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl, 24(1968), 182-190.

[4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88(1997), 81-89.

[5] P. Rajarajeswari and R. Krishna Moorthy, On intuitionistic fuzzy weakly generalized closed set and its applications, International Journal of Computer Applications, 27(2011), 9-13.

[6] P. Ra jarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized irresolute mappings, Ultrascientist of Physical Sciences, 24(2012), 204-212.

[7] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized continuous mappings, Far East Journal of Mathematical Sciences, 66(2012), 153-170.

[8] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy quasi weakly generalized continuous, Scientia Magna, 3(2012), 16-24.

[9] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy weakly generalized closed mappings, Journal of Advanced Studies in Topology, 4(2012), 20-27.

[10] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy perfectly weakly generalized continuous mappings, Notes on Intuitionistic Fuzzy Sets, 18(2012), 64-76.

[11] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy contra weakly generalized continuous mappings, Annals of Fuzzy Mathematics and Informatics, 13(2013), 361-369.

[12] P. Rajarajeswari and R. Krishna Moorthy, Intuitionistic fuzzy contra weakly generalized irresolute mappings (submitted).

[13] L. A. Zadeh, Fuzzy Sets, Information and control, 8(1965), 338-353.

P. Rajarajeswari ([dagger]) and R. Krishna Moorthy ([double dagger])

([dagger]) Department of Mathematics, Chikkanna Government Arts College, Tirupur, 641602, Tamil Nadu, India

([double dagger]) Department of Mathematics, Kumaraguru College of Technology, Coimbatore, 641049, Tamil Nadu, India E-mail: p.rajarajeswari29@gmail.com krishnamoorthykct@gmail.com
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Author:Rajarajeswari, P.; Moorthy, R. Krishna
Publication:Scientia Magna
Date:Dec 1, 2012
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