Intuitionistic fuzzy resolvable and intuitionistic fuzzy irresolvable spaces.

[section] 1. Introduction

The fuzzy concept has invaded almost all branches of mathematics ever since the introduction of fuzzy sets by L. A. Zadeh [14]. The theory of fuzzy topological space was introduced and developed by C. L. Chang [7] and since then various notions in classical topology have been extended to fuzzy topological space. The idea of "intuitionistic fuzzy set" was first published by Atanassov W and many works by the same author and his colleagues appeared in the literature [2-4]. Later, this concept was generalized to "intuitionistic L--fuzzy sets" by Atanassov and Stoeva [5]. In classical topology the class of somewhat continuous functions was introduced by Karl R. Gentry and Hunghes B. Hoyle III in [11]. We have extended this concepts to fuzzy topological space and in this connection, we have introduced the concept of somewhat fuzzy continuous functions and somewhat fuzzy open hereditarily irresolvable by G. Thangaraj and G. Balasubramanian in [12]. The concepts of resolvability and irresolvability in topological spaces was introduced by E. Hewit in [10]. The concept of open hereditarily irresolvable spaces in the classical topology was introduced by A. Geli'kin in [9]. The concept on fuzzy resolvable and fuzzy irresolvable spaces was introduced by G. Thangaraj and G. Balasubramanian in [13]. In this paper the concept of intuitionistic fuzzy resolvable, intuitionistic fuzzy irresolvable, intuitionistic fuzzy open hereditarily irresolvable spaces and maximally intuitionistic fuzzy irresolvable space are introduced. Also we discuss and study several interest properties of the intuitionistic fuzzy open hereditarily irresolvable spaces besides giving characterization of these spaces by means of somewhat intuitionistic fuzzy continuous functions and somewhat intuitionistic fuzzy open functions. Some interesting properties and related examples are given.

[section] 2. Preliminaries

Definition 2.1. [3] Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS for short) A is an object having the form A = {{x, [[mu].sub.A](x), [[delta].sub.A](x)) : x [member of] X} where the function [[mu].sub.A] : X [right arrow] I and [[delta].sub.A] : X [right arrow] I denote the degree of membership (namely [[mu].sub.A](x)) and the degree of nonmembership ([[delta].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) + [[delta].sub.A](x) [less than or equal to] 1 for each x [member of] X.

Definition 2.2. [3] Let X be a nonempty set and the intuitionistic fuzzy sets A and B in the form A = {{x, [[mu].sub.A](x), [[delta].sub.A](x)) : x [member of] X}, B = {{x, [[mu].sub.B](x), [[delta].sub.B](x)) : x [member of] X}. Then

(a) A [intersection] B = {<x, [[mu].sub.A](x) [conjunction] [[mu].sub.B](x), [[delta].sub.A](x) [disjunction] [[delta].sub.B](x)> : x [member of] X};

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

Now we shall define the image and preimage of intuitionistic fuzzy sets. Let X and Y be two nonempty sets and f : X [right arrow] Y be a function.

Definition 2.3. [3] (a) If B = {{y, [[mu].sub.B](y), [[delta].sub.B](y)) : y [member of] Y} is an intuitionistic fuzzy set in Y, then the preimage of B under f, denoted by [f.sup.-1](B), is the intuitionistic fuzzy set in X defined by [f.sup.-1](B) = {{x, [f.sup.-1]([[mu].sub.B])(x), [f.sup.-1]([[delta].sub.B])(x)) : x [member of] X}.

(b) If A = {<x, [[lambda].sub.A](x), [v.sub.A](x)> : x [member of] X} is an intuitionistic fuzzy set in X, then the image of A under f, denoted by f (A), is the intuitionistic fuzzy set in Y defined by f (A) = {<y, f([[lambda].sub.A])(y), (1 - f (1 - [v.sub.A]))(y)> : y [member of] Y}.

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the sake of simplicity, let us use the symbol [f.sub.-]([v.sub.A]) for 1 - f (1 - [v.sub.A]).

Definition 2.4. [8] Let A be an intuitionistic fuzzy set in intuitionistic fuzzy topological space (X,T). Then

I Fint(A) = [universal]{G | G is an intuitionistic fuzzy open in X and G [subset or equal to] A} is called an intuitionistic fuzzy interior of A;

I Fcl(A) = [intersection]|{G | G is an intuitionistic fuzzy closed in X and G [contains or equal to] A} is called an intuitionistic fuzzy closure of A.

Proposition 2.1. [8] Let (X,T) be any intuitionistic fuzzy topological space. Let A be an intuitionistic fuzzy sets in (X, T). Then the intuitionistic fuzzy closure operator satisfy the following properties:

(i) 1 - I Fcl(A) = I Fint(1 - A);

(ii) 1 - I Fint(A) = I Fcl(1 - A).

Definition 2.5. [12] A fuzzy set A in a fuzzy topological space (X, T) is called fuzzy dense if there exists no fuzzy closed set [mu] in (X, T) such that [lambda] < [mu] < 1.

Definition 2.6. [13] Let (X, T) be a fuzzy topological space. (X, T) is called fuzzy resolvable if there exists a fuzzy dense set [lambda] in (X, T) such that cl (1 - [lambda]) = 1. Otherwise (X, T) is called fuzzy irresolvable.

Definition 2.7. [6] A fuzzy topological space (X,T) is called a fuzzy submaximal space if for each fuzzy set A in (X, T) such that cl([lambda]) = 1, then [lambda] [member of] T.

Definition 2.8. [13] Let (X,T) be a fuzzy topological space. (X,T) is called fuzzy open hereditarily irresolvable if intcl([lambda]) [not equal to] 0 then int[lambda] [not equal to] 0 for any fuzzy set [lambda] in (X, T).

Definition 2.9. [12] Let (X, T) and (Y, S) be any two fuzzy topological spaces. A function f : (X,T) [right arrow] (Y, S) is called somewhat fuzzy continuous if [lambda] [member of] S and [f.sup.-]([lambda]) [not equal to] [0.sub.~] there exists [mu] [member of] T such that [mu] [not equal to] 0 and [mu] [less than or equal to] [f.sup.-]([lambda]).

Definition 2.10. [12] Let (X, T) and (Y, S) be any two fuzzy topological spaces. A function f : (X, T) [right arrow] (Y, S) is called somewhat fuzzy open if [lambda] [member of] T and [lambda] [not equal to] [0.sub.~] there exists [mu] [member of] S such that [mu] [not equal to] 0 and [mu] [less than or equal to] f([lambda]).

[section] 3. Intuitionistic fuzzy resolvable and intuitionistic fuzzy irresolvable

Definition 3.1. An intuitionistic fuzzy set A in intuitionistic fuzzy topological space (X, T) is called intuitionistic fuzzy dense if there exists no intuitionistic fuzzy closed set B in (X, T) such that A [subset] B [subset] [1.sub.~].

Definition 3.2. Let (X,T) be an intuitionistic fuzzy topological space. (X,T) is called intuitionistic fuzzy resolvable if there exists a intuitionistic fuzzy dense set A in (X, T) such that IFcl(1 - A) = [1.sub.~]. Otherwise, (X,T) is called intuitionistic fuzzy irresolvable.

Example 3.1. Let X = {a, b, c}. Define the intuitionistic fuzzy sets A, B and C as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly T = {[0.sub.~], [1.sub.~], A} is an intuitionistic fuzzy topology on X. Thus (X,T) is an intuitionistic fuzzy topological space. Now IFint(B) = [0.sub.~], IFint(C) = 0^, IFint(1 - B) = [0.sub.~], IFint(1 - C) = A, IFcl(B) = [1.sub.~], IFcl(C) = [1.sub.~], IFcl(1 - B) = [1.sub.~] and IFcl(1 - C) = 1 - A. Hence there exists a intuitionistic fuzzy dense set B in (X, T), such that IFint(1 - B) = [1.sub.~]. Hence the intuitionistic fuzzy topological space (X, T) is called a intuitionistic fuzzy resolvable.

Definition 3.2. Let X = {a,b,c}. Define the intuitionistic fuzzy sets A, B and C as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly T = {[0.sub.~], [1.sub.~], A} is an intuitionistic fuzzy topology on X. Thus (X,T) is an intuitionistic fuzzy topological space. Now IFint(B) = A, IFint(C) = A, IFcl(B) = [1.sub.~], IFcl(C) = [1.sub.~] and IFcl(B) = [1.sub.~]. Thus B and C are intuitionistic fuzzy dense set in (X, T), such that IFcl(1 - B) = 1 - A and IFcl(1 - C) = 1 - A. Hence the intuitionistic fuzzy topological space (X, T) is called a intuitionistic fuzzy irresolvable.

Proposition 3.1. Let (X, T) be an intuitionistic fuzzy topological space. (X, T) is an intuitionistic fuzzy resolvable space iff (X, T) has a pair of intuitionistic fuzzy dense set [A.sub.1] and [A.sub.2] such that [A.sub.1] [subset or equal to] 1 - [A.sub.2].

Proof. Let (X, T) be an intuitionistic fuzzy topological space and (X, T) is an intuitionistic fuzzy resolvable space. Suppose that for all intuitionistic fuzzy dense sets [A.sub.i] and [A.sub.j], we have [A.sub.i] [not subset or equal to] 1 - [A.sub.j]. Then [A.sub.i] [contains] 1 - [A.sub.j]. Then IFcl([A.sub.i]) [contains] IFcl(1 - [A.sub.j]) which implies that [1.sub.~] [contains] IFcl(1 - [A.sub.j]) then IFcl(1 - [A.sub.j]) [not equal to] [1.sub.~]. Also [A.sub.j] [contains] 1 - [A.sub.i] then IFcl([A.sub.j]) [contains] IFcl(1 - [A.sub.i]) which implies that [0.sub.~] [contains] IFcl(1 - [A.sub.i]). Then IFcl(1 - [A.sub.i]) [not equal to] [1.sub.~]. Hence IFcl([A.sub.i]) = [1.sub.~], but IFcl(1 - [A.sub.i]) [not equal to] [1.sub.~] for all intuitionistic fuzzy set [A.sub.i] in (X,T). Which is a contradiction. Hence (X, T) has a pair of intuitionistic fuzzy dense set [A.sub.1] and [A.sub.2] such that [A.sub.1] [subset or equal to] 1 - [A.sub.2].

Converse, suppose that the intuitionistic fuzzy topological space (X, T) has a pair of intuitionistic fuzzy dense set [A.sub.1] and [A.sub.2], such that [A.sub.1] [subset or equal to] 1 - [A.sub.2]. Suppose that (X, T) is a intuitionisic fuzzy irresolvable space. Then for all intuitionistic fuzzy dense set [A.sub.1] and [A.sub.2] in (X, T), we have IFcl(1 - [A.sub.1]) [not equal to] [1.sub.~]. Then IFcl(1 - [A.sub.2]) [not equal to] [1.sub.~] implies that there exists a intuitionistic fuzzy closed set B in (X, T), such that 1 - [A.sub.2] [subset] B [subset] [1.sub.~]. Then [A.sub.1] [subset or equal to] 1 - [A.sub.2] [subset] B [subset] [1.sub.~] implies that [A.sub.1] [subset] B [subset] [1.sub.~]. Which is a contradiction. Hence (X,T) is a intuitionistic fuzzy resolvable space.

Proposition 3.2. If (X,T) is intuitionistic fuzzy irresolvable iff IFint(A) [not equal to] [0.sub.~] for all intuitionistic dense set A in (X, T).

Proof. Since (X, T) is an intuitionistic fuzzy irresolvable space, for all intuitionistic fuzzy dense set A in (X, T), IFcl(1 - A) [not equal to] [1.sub.~]. Then 1 - IFint(A) [not equal to] [1.sub.~], which implies IFint (A) [not equal to] [0.sub.~].

Conversely IFint(A) = [0.sub.~], for all intuitionistic fuzzy dense set A in (X, T). Suppose that (X, T) is intuitionistic fuzzy resolvable. Then there exists a intuitionistic fuzzy dense set A in (X,T), such that IFcl(1 - A) = [1.sub.~] implies that 1 - IFint (A) = [1.sub.~], implies I Fint (A) = [0.sub.~]. Which is a contradiction. Hence (X, T) is intuitionistic fuzzy irresolvable space.

Definition 3.3. An intuitionistic fuzzy topological space (X,T) is called a intuitionistic fuzzy submaximal space if each intuitionistic fuzzy set A in (X,T) such that IFcl(A) = [1.sub.~], then A [member of] T.

Proposition 3.3. If the intuitionistic fuzzy topological space (X, T) is intuitionistic fuzzy submaximal, then (X, T) is intuitionistic fuzzy irresolvable.

Proof. Let (X, T) be a intuitionistic fuzzy submaximal space. Assume that (X, T) is a intuitionistic fuzzy resolvable space. Let A be a intuitionistic fuzzy dense set in (X, T). Then IFcl (1 - A) = [1.sub.~]. Hence 1 - IFint(A) = [1.sub.~] which implies that IFint (A) = [0.sub.~]. Then A [not member of] T. Which is a contradiction to intuitionistic fuzzy submaximal space of (X, T). Hence (X, T) is intuitionistic fuzzy irresolvable space.

The converse Proposition 3.3 is not true. See Example 3.2.

Definition 3.4. An intuitionistic fuzzy topological space (X,T) is called a maximal intuitionistic fuzzy irresolvable space if (X, T) is intuitionistic fuzzy irresolvable and every intuitionistic fuzzy dense set A of (X, T) is intuitionistic fuzzy open.

Example 3.3. Let X = {a,b,c}. Define the intuitionistic fuzzy sets A, B, A [intersection] B and A [union] B as follows,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly T = {[1.sub.~], [1.sub.~], A, B, A [intersection] B, A [union] B} is an intuitionistic fuzzy topology on X. Thus (X,T) is an intuitionistic fuzzy topological space. Now IFint(1 - A) = [0.sub.~], IFint(1 - B) = V{[0.sub.~], B, A [intersection] B} = B, IFint(1 - A [union] B) = [0.sub.~], IFint(1 - A [intersection] B) = [disjunction]{[0.sub.~], B, A [intersection] B} = B and IFcl(A) = [1.sub.~], IFcl(B) = 1 - B, IFcl(A [union] B) = [1.sub.~], IFcl(A [intersection] B) = 1 - B, IFcl(1 - A [union] B) = [conjunction]{[1.sub.~], 1 - A [union] B, 1 - B, 1 - A [intersection] B} = 1 - A [union] B, IFcl(1 - A) = [conjunction]{[1.sub.~], 1 - A, 1 - A [intersection] B} = 1 - A, IFcl([0.sub.~]) [not equal to] [1.sub.~]. Thus intuitionistic fuzzy dense set in (X,T) are A, A [union] B, [1.sub.~] are intuitionistic fuzzy open in (X, T). Hence (X, T) is an intuitionistic fuzzy irresolvable and every intuitionistic fuzzy dense set of (X, T) is intuitionistic fuzzy open. Therefore (X, T) is a maximally intuitionistic fuzzy irresolvable space.

[section] 4. Intuitionistic fuzzy open hereditarily irresolvable

Definition 4.1. (X, T) is said to be intuitionistic fuzzy open hereditarily irresolvable if IFint(IFcl(A)) [not equal to] [0.sub.~] then IFint (A) [not equal to] [0.sub.~] for any intuitionistic fuzzy set A in (X,T).

Example 4.1. Let X = {a,b,c}. Define the intuitionistic fuzzy sets [A.sub.1], [A.sub.2] and [A.sub.3] as follows,

[A.sub.1] <x, (x,(a/0.4, b/0.4, c/0.4), (a/0.5, b/0.5, c/0.5)>,

[A.sub.2] = <x,(a/0.6, b/0.5, c/0.4), (a/0.4,b/0.5, c/0.4)>,

[A.sub.3] = <x,(a/0.4, b/0.4, c/0.5), (a.0.4, b/0.4, c/0.5)>.

Clearly T = {[0.sub.~], [1.sub.~], [A.sub.1], [A.sub.2]} is an intuitionistic fuzzy topology on X. Thus (X,T) is an intuitionistic fuzzy topological space. Now IFcl([A.sub.1]) = 1-[A.sub.1]; IFcl([A.sub.2]) = [1.sub.~] and IFint([A.sub.3]) [A.sub.1]. Also IFint(IFcl([A.sub.1])) = I Fint (1 - [A.sub.1]) = 1 - [A.sub.1] [not equal to] [0.sub.~] and IFint([A.sub.1]) = A1 [not equal to] [0.sub.~], IFint(IFcl([A.sub.2])) = IFint([1.sub.~]) = [1.sub.~] = [0.sub.~] and IFint([A.sub.2]) = [A.sub.2] = [0.sub.~], IFint(IFcl([A.sub.3])) = IFint(1 - [A.sub.1]) = 1 - [A.sub.1] = [0.sub.~] and IFint([A.sub.3]) = [A.sub.1] = [0.sub.~] and IFint(IFcl(1 - [A.sub.3])) = IFint(1 - [A.sub.1]) = 1 - [A.sub.1] [not equal to] [0.sub.~] and IFint (1 - [A.sub.3]) = [A.sub.1] = [0.sub.~]. Hence if IFint(IFcl(A)) [not equal to] [0.sub.~] then IFint(A) [not equal to] [0.sub.~] for any non zero intuitionistic fuzzy set A in (X,T). Thus (X,T) is a intuitionistic fuzzy open hereditarily irresolvable space.

Proposition 4.1. Let (X, T) be an intuitionistic fuzzy topological space. If (X, T) is intuitionistic fuzzy open hereditarily irresolvable then (X, T) is intuitionistic fuzzy irresolvable.

Proof. Let A be an intuitionistic fuzzy dense set in (X,T). Then IF cl(A) = [1.sub.~], which implies that IFint(IFcl(A)) = [1.sub.~] =[0.sub.~]. Since (X,T) is intuitionistic fuzzy open hereditarily irresolvable, we have IFint(A) [not equal to] [0.sub.~]. Therefore by Proposition 3.2, IFint(A) [not equal to] [0.sub.~] for all intuitionistic fuzzy dense set in (X, T), implies that (X, T) is intuitionistic fuzzy irresolvable. The converse is not true (See Example 4.2).

Example 4.2. Let X = {a, b, c}. Define the intuitionistic fuzzy sets A, B and C as follows,

A = <x (a/0.3, b/0.3, c/0.4), (a/0.5, b/0.5, c/0.5)>,

B = <x,(a/0.4, b/0.5, c/0.4), (a/0.4, b/0.4, c/0.4)>,

and

C = <x, (a/0.4, b/0.4, c/0.4), (a/0.3, b/0.3, c/0.3)>.

Clearly T = {[0.sub.~], [1.sub.~], A, B} is an intuitionistic fuzzy topology on X. Thus (X,T) is an intuitionistic fuzzy topological space. Now C and [1.sub.~] are intuitionistic fuzzy dense sets in (X,T). Then IFint(C) = A [not equal to] [0.sub.~] and IFint([1.sub.~]) = [0.sub.~] Hence (X,T) i an intuitionistic fuzzy irresolvable. But IFint(IFcl(1 - C)) = IFint(1 - A) = A [not equal to] [0.sub.~] and I Fint (1 - C) = [0.sub.~] Therefore (X, T) is not a intuitionistic fuzzy open hereditarily irresolvable space.

Proposition 4.2. Let (X, T) be an intuitionistic fuzzy open hereditarily irresolvable. Then IFint(A) [not subset or equal to] 1 - IFint(B) for any two intuitionistic fuzzy dense sets A and B in (X, T).

Proof. Let A and B be any two intuitionistic fuzzy dense sets in (X, T). Then IFcl(A) = [1.sub.~] and IFcl(B) = [1.sub.~] implies that IFint(IFcl (A)) [not equal to] [0.sub.~] and IFint(IFcl(B)) [not equal to] 0 [??]. Since (X, T) is intuitionistic fuzzy open hereditarily irresolvable, IFint(A) [not equal to] [0.sub.~] and IFint(B) [not equal to] [0.sub.~]. Hence by Proposition 3.1, A [not subset equal to] 1 - B. Therefore IFint(A) [subset or equal to] A [not subset equal to] 1 - B [subset or equal to] 1 - IFint(B). Hence we have IFint(A) [subset or equal to] 1 - IFint(B) for any two intuitionistic fuzzy dense sets A and B in (X, T).

Proposition 4.3. Let (X, T) be an intuitionistic fuzzy topological space. If (X, T) is intuitionistic fuzzy open hereditarily irresolvable then IFint(A) = [0.sub.~] for any nonzero intuitionistic fuzzy dense set A in (X, T) implies that I Fint(IFcl (A)) = [0.sub.~].

Proof. Let A be an intuitionistic fuzzy set in (X, T), such that IFint(A) = [0.sub.~]. We claim that IFint(IFcl(A)) = [0.sub.~]. Suppose that IFint(IFcl(A)) = [0.sub.~]. Since (X,T) is intuitionistic fuzzy open hereditarily irresolvable, we have IFint(A) [not equal to] [0.sub.~]. Which is a contradiction to IFint(A) = [0.sub.~]. Hence I Fint(IFcl(A)) = [0.sub.~].

Proposition 4.4. Let (X, T) be an intuitionistic fuzzy topological space. If (X, T) is intuitionistic fuzzy open hereditarily irresolvable then IFcl(A) = [1.sub.~] for any nonzero intuitionistic fuzzy dense set A in (X, T) implies that IFcl(IFint(A)) = [0.sub.~].

Proof. Let A be an intuitionistic fuzzy set in (X,T), such that IFcl (A) = [1.sub.~]. Then we have 1 - IFcl(A) = [0.sub.~], which implies that IFint(1 - A) = [0.sub.~]. Since (X,T) is intuitionistic fuzzy open hereditarily irresolvable by Proposition 4.3. We have that IFint(IFcl(1 - A)) = [0.sub.~]. Therefore 1 - IFcl(IFint(A)) = [0.sub.~] implies that IFcl (I Fint (A)) = [1.sub.~].

[section] 5. Somewhat intuitionistic fuzzy continuous and somewhat intuitionistic fuzzy open

Definition 5.1. Let (X,T) and (Y, S) be any two intuitionistic fuzzy topological spaces. A function f : (X, T) [right arrow] (Y, S) is called somewhat intuitionistic fuzzy continuous if A [member of] S and [f.sup.-1](A) [not equal to] [0.sub.~], then there exists a B [member of] T, such that B [not equal to] [0.sub.~] and B [subset or equal to] [f.sup.-1](A).

Definition 5.2. Let (X, T) and (Y, S) be any two intuitionistic fuzzy topological spaces. A function f : (X, T) [right arrow] (Y, S) is called somewhat intuitionistic fuzzy open if A [member of] T and A [not equal to] [0.sub.~], then there exists a B [member of] S, such that B [not equal to] [0.sub.~] and B [subset or equal to] f (A).

Proposition 5.1. Let (X, T) and (Y, S) be any two intuitionistic fuzzy topological spaces. If the function f : (X, T) [right arrow] (Y, S) is somewhat intuitionistic fuzzy continuous and 1-1 and if IFint(A) = [0.sub.~] for any nonzero intuitionistic fuzzy set A in (X,T) then IFint(f (A)) = [0.sub.~] in (Y,S).

Proof. Let A be a nonzero intuitionistic fuzzy set in (X, T), such that IFint(A) = [0.sub.~]. To prove that IFint(f(A)) = [0.sub.~]. Suppose that IFint(f (A)) [not equal to] [0.sub.~] in (Y, S). Then there exists an nonzero intuitionistic fuzzy set B in (Y, S), such that B [subset or equal to] f(A). Then [f.sup.-1] (B) [subset or equal to] [f.sup.-1] (f(A)). Since f is somewhat intuitionistic fuzzy continuous, there exists a C [member of] T, such that C [not equal to] [0.sub.~] and C [subset or equal to] [f.sup.-1](B). Hence C [subset or equal to] [f.sup.-1](B) [subset or equal to] A, which implies that IFint(A) [not equal to] [0.sub.~] Which is a contradiction. Hence IFint(f (A)) = [0.sub.~] in (Y, S).

Proposition 5.2. Let (X, T) and (Y, S) be any two intuitionistic fuzzy topological spaces. If the function f : (X, T) [right arrow] (Y, S) is somewhat intuitionistic fuzzy continuous and 1-1 and if IFint(IFcl(A)) = [0.sub.~] for any nonzero intuitionistic fuzzy set A in (X,T) then IFint(IFcl(f (A))) = [0.sub.~] in (Y, S).

Proof. Let A be a nonzero intuitionistic fuzzy set in (X, T), such that IFint(IFcl(A)) = [0.sub.~]. We claim that IFint(IFcl(f (A))) = 0 [right arrow] in (Y, S). Suppose that IFint(IFcl(f (A))) = [0.sub.~] in (Y,S). Then IFcl(f(A)) [not equal to] [0.sub.~]. Then 1 - IFcl(f(A)) [not equal to] [0.sub.~] Now 1 - IFcl(f(A)) = [0.sub.~] [member of] S and since f is somewhat intuitionistic fuzzy continuous, there exists a B [member of] T, such that B [not equal to] [0.sub.~] and B [subset or equal to] [f.sup.-1] (1-IFcl(f(A))). That is B [subset or equal to] 1 - [f.sup.-1] (IFcl(f(A))), which implies that [f.sup.-1] (IFcl(f(A))) [subset or equal to] 1 - B. Since f is 1-1, thus A [subset or equal to] [f.sup.-1] (f(A) [f.sup.-1](IFcl(f(A))) [subset or equal to] 1 - B, implies that A [subset or equal to] 1 - B. Therefore B [subset or equal to] 1 - A implies that IFint(1 - A) [not equal to] [0.sub.~]. Let IFint(1 - A) = C [not equal to] [0.sub.~]. Then we have IFcl(IFint(1 - A)) = IFcl(C) [not equal to] [1.sub.~], implies that IFint(IFcl(A)) [not equal to] [0.sub.~]. Which is a contradiction. Hence IFint(Ifcl(f (A))) = [0.sub.~] in (Y, S).

Proposition 5.3. Let (X, T) and (Y, S) be any two intuitionistic fuzzy topological spaces. If the function f : (X, T) [right arrow] (Y, S) is somewhat intuitionistic fuzzy open and if IFint(A) = [0.sub.~] for any nonzero intuitionistic fuzzy set A in (Y, S) then IFint([f.sup.-1](A)) [not equal to] [0.sub.~] in (X, T).

Proof. Let A be a nonzero intuitionistic fuzzy set in (Y, S), such that IFint(A) = [0.sub.~]. We claim that IFint([f.sup.-1](A)) = [0.sub.~] in (X,T). Suppose that IFint([f.sup.-1](A)) [not equal to] [0.sub.~] in (X,T). Then there exists a nonzero intuitionistic fuzzy open set B in (X, T), such that B [subset or equal to][f.sup.-1](A). Then we have f(B)[subset or equal to] f([f.sup.-1](A))[subset or equal to] A. Which implies that f(B) [subset or equal to] A. Since f is somewhat intuitionistic fuzzy open, there exists a C [member of] S, such that C [not equal to] [0.sub.~] and C [subset or equal to] f(B). Hence C [subset or equal to] f(B) [subset or equal to] A, which implies that C [subset or equal to] A. Hence IFint(A) [not equal to] [0.sub.~]. Which is a contradiction. Hence IFint([f.sup.-1](A)) = [0.sub.~] in (X,T).

Proposition 5.4. Let (X, T) and (Y, S) be any two intuitionistic fuzzy topological spaces. Let (X, T) be an intuitionistic fuzzy open hereditarily irresolvable space. If f : (X, T) [right arrow] (Y, S) is somewhat intuitionistic fuzzy open and somewhat intuitionistic fuzzy continuous, 1-1 and onto function then (Y, S) is intuitionistic fuzzy open hereditarily space.

Proof. Let A be a nonzero intuitionistic fuzzy set in (Y, S), such that IFint(A) = [0.sub.~]. Now IFint(A) = [0.sub.~] and f is somewhat intuitionistic fuzzy open implies that by Proposition 5.3, IFint([f.sup.-1](A)) = [0.sub.~] in (X,T). Since (X,T) is intuitionistic fuzzy open hereditarily irresolvable space, we have IFint(IFcl([f.sup.-1](A))) = [0.sub.~] in (X,T), by Proposition 4.3. Since IFint(IFcl([f.sup.-1](A))) = [0.sub.~] and f is somewhat intuitionistic fuzzy continuous by Proposition 5.2, we have IFint(IFcl(f ([f.sup.-1] (A)))) = [0.sub.~]. Since f is onto, thus IFintIFcl(A) = [0.sub.~]. Hence by Proposition 4.3. (Y, S) is an intuitionistic fuzzy open hereditarily irresolvable space.

References

[1] K. Atanassov, Intuitionistic fuzzy sets, in: V. Sgurev. Ed., VII ITKR's Session, Sofia Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984.

[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96.

[3] K. Atanassov, Review and new results on intuitionistic fuzzy sets, Preprint IM-MFAIS1-88, Sofia, 1988.

[4] K. Atanassov and S. Stoeva, Intuitionistic fuzzy sets, in: Polish Syrup. on Interval & Fuzzy Mathematics, Poznan, 8(1983), 23-26.

[5] K. Atanassov and S. Stoeva, Intuitionistic L-fuzzy sets, in: R. Trappl, Ed., Cybernetics and System Research, Elsevier, Amsterdam, 1984, 539-540.

[6] G. Balasubramanian, Maximal Fuzzy Topologies, Kybernetika, 31(1995), No. 5, 459-464.

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

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

[9] A. Gel'kin, Ultrafilters and undecomposable spaces, Vestnik. Mosk. Univ. Mat, 5(1969), No. 24, 51.

[10] E. Hewit, A Problem in set Theoretic Topology, Duke. Math. Jour, 10(1943), 309-333.

[11] Karl R. Gentry and Hughes B. Hoyle III, Somewhat Continuous Function, Czech. Math. Journal, 96(1971), No. 21, 5-12.

[12] G. Thangaraj and G. Balasubramanian, On Somewhat Fuzzy Continuous Functions, J. Fuzzy. Math, 11(2003), No. 2, 725-736.

[13] G. Thangaraj and G. Balasubramanian, On Fuzzy resolvable and Fuzzy Irresolvable spaces, Fuzzy Sets Rough Sets and Multivalued Operations and Applications, 2(2009), No. 1, 173-180.

[14] L. A. Zadeh, Fuzzy sets, Inform and Control, 8(1965), 338-353.

R. Dhavaseelan ([dagger]), E. Roja ([double dagger]) and M. K. Uma (#)

([dagger]) Department of Mathematics, Sona College of Techology, Salem, 636005, Tamil Nadu, India.

([double dagger]) (#) Department of Mathematics, Sri Saradha College for Women, Salem, 16, Tamil Nadu, India.

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