Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -Continuity

S.S.Thakur & Jyoti Pandey Bajpai
International Journal of Contemporary Advanced Mathematics (IJCM), Volume (1): Issue (1) 1
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic
Fuzzy W -Continuity
S. S. Thakur samajh_singh@rediffmail.com
Department of Applied Mathematics
Jabalpur Engineering College
Jabalpur (M.P.) 482011
Jyoti Pandey Bajpai yk1305@ gmail.com
Department of Applied Mathematics
Jabalpur Engineering College
Jabalpur (M.P.) 482001
Abstract
The aim of this paper is to introduce and study the concepts of intuitionistic fuzzy w-
closed sets, intuitionistic fuzzy w-continuity and inttuitionistic fuzzy w-open &
intuitionistic fuzzy w-closed mappings in intuitionistic fuzzy topological spaces.
Key words: Intuitionistic fuzzy w-closed sets, Intuitionistic fuzzy w-open sets, Intuitionistic fuzzy w-
connectedness, Intuitionistic fuzzy w-compactness, intuitionistic fuzzy w-continuous mappings.
2000, Mathematics Subject Classification: 54A
1. INTRODUCTION
After the introduction of fuzzy sets by Zadeh [23] in 1965 and fuzzy topology by Chang [4] in 1967,
several researches were conducted on the generalizations of the notions of fuzzy sets and fuzzy
topology. The concept of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of
fuzzy sets. In the last 25 years various concepts of fuzzy mathematics have been extended for
intuitionistic fuzzy sets. In 1997 Coker [5] introduced the concept of intuitionistic fuzzy topological spaces.
Recently many fuzzy topological concepts such as fuzzy compactness [7], fuzzy connectedness [21],
fuzzy separation axioms [3], fuzzy continuity [8], fuzzy g-closed sets [15] and fuzzy g-continuity [16] have
been generalized for intuitionistic fuzzy topological spaces. In the present paper we introduce the
concepts of intuitionistic fuzzy w-closed sets; intuitionistic fuzzy w-open sets, intuitionistic fuzzy w-
connectedness, intuitionistic fuzzy w-compactness and intuitionistic fuzzy w-continuity obtain some of
their characterization and properties.

2. PRELIMINARIES
Let X be a nonempty fixed set. An intuitionistic fuzzy set A[1] in X is an object having the form A =
{<x, µA(x), γA(x)> : x ∈ X }, where the functions µA :X→[0,1] and ϒA:X→[0,1] denotes the degree of
membership µA(x) and the degree of non membership γA(x) of each element x∈X to the set A respectively
and 0 ≤ µA(x)+ γA(x) ≤ 1 for each x∈X. The intutionistic fuzzy sets 0 = {< x, 0, 1 > : x ∈ X} and 1 = {<x, 1,
0> : x ∈ X } are respectively called empty and whole intuitionistic fuzzy set on X. An intuitionistic fuzzy set
A = {<x, µA(x), γA(x)> : x ∈ X} is called a subset of an intuitionistic fuzzy set B = {<x, µB(x), γB(x)> : x ∈ X}
(for short A ⊆ B) if µA(x) ≤ µB(x) and γA(x) ≥ γB(x) for each x ∈ X. The complement of an intuitionistic fuzzy
set A = {<x, µA(x), γA(x)> : x ∈ X } is the intuitionistic fuzzy set A
c
= { <x,γA(x), µA(x) >: x ∈ X}. The
intersection (resp. union) of any arbitrary family of intuitionistic fuzzy sets Ai = {< x, µAi(x) , γAi(x) > : x ∈ X ,
( i∈∧∧∧∧) } of X be the intuitionistic fuzzy set ∩Ai ={<x , ∧ µAi(x) , ∨ γAi(x) > : x ∈ X } (resp. ∪Ai ={ <x, ∨ µAi(x)
, ∧ γAi(x) >: x ∈ X }). Two intuitionistic fuzzy sets A = {<x, µA(x), γA(x)> : x ∈ X } and B = {<x, µB(x), γB(x)> : x
∈ X} are said be q-coincident (AqB for short) if and only if ∃ an element x∈ X such that µA(x) > γB(x) or
γA(x)< µB(x). A family ℑ of intuitionistic fuzzy sets on a non empty set X is called an intuitionistic fuzzy
topology [5] on X if the intuitionistic fuzzy sets 0, 1∈ ℑ, and ℑ is closed under arbitrary union and finite
intersection. The ordered pair (X,ℑ) is called an intuitionistic fuzzy topological space and each
intuitionistic fuzzy set in ℑ is called an intuitionistic fuzzy open set. The compliment of an intuitionistic
fuzzy open set in X is known as intuitionistic fuzzy closed set .The intersection of all intuitionistic fuzzy
closed sets which contains A is called the closure of A. It denoted cl(A). The union of all intuitionistic
fuzzy open subsets of A is called the interior of A. It is denoted int(A) [5].
Lemma 2.1 [5]: Let A and B be any two intuitionistic fuzzy sets of an intuitionistic fuzzy
topological space (Χ, ℑ). Then:
(a). (AqB) ⇔ A ⊆ B
c
.
(b). A is an intuitionistic fuzzy closed set in X ⇔ cl (A) = A.
(c). A is an intuitionistic fuzzy open set in X ⇔ int (A) = A.
(d). cl (A
c
) = (int (A))
c
.
(e). int (A
c
) = (cl (A))
c
.
(f). A ⊆ B ⇒ int (A) ⊆ int (B).
(g). A ⊆ B ⇒ cl (A) ⊆ cl (B).
(h). cl (A ∪ B) = cl (A) ∪ cl(B).
(i). int(A ∩ B) = int (A) ∩ int(B)
Definition 2.1 [6]: Let X is a nonempty set and c∈X a fixed element in X. If α∈(0, 1] and β∈[0, 1) are two
real numbers such that α+β≤1 then:
(a) c(α,β) = < x,cα, c1-β > is called an intuitionistic fuzzy point in X, where α denotes the degree of
membership of c(α,β), and β denotes the degree of non membership of c(α,β).
(b) c(β) = < x,0, 1-c1-β > is called a vanishing intuitionistic fuzzy point in X, where β denotes the
degree of non membership of c(β).
Definition 2.2[7] : A family { Gi : i∈∧} of intuitionistic fuzzy sets in X is called an intuitionistic fuzzy open
cover of X if ∪{ Gi : i∈∧} =1 and a finite subfamily of an intuitionistic fuzzy open cover { Gi: i∈∧}of X which
also an intuitionistic fuzzy open cover of X is called a finite sub cover of { Gi: i∈∧}.
Definition 2.3[7]: An intuitionistic fuzzy topological space (X,ℑ) is called fuzzy compact if every
intuitionistic fuzzy open cover of X has a finite sub cover.

Definition 2.4[8]: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,ℑ) is called
intuitionistic fuzzy semi open (resp. intuitionistic fuzzy semi closed) if there exists a intuitionistic fuzzy
open (resp. intuitionistic fuzzy closed) U such that U ⊆ A ⊆ cl(A) (resp.int(U) ⊆ A ⊆ U)
Definition 2.5 [21]: An intuitionistic fuzzy topological space X is called intuitionistic fuzzy connected if
there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy open and intuitionistic
fuzzy closed .
Definition 2.6[15]: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,ℑ) is called:
(a) Intuitionistic fuzzy g-closed if cl (A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open.
(b) Intuitionistic fuzzy g-open if its complement A
c
is intuitionistic fuzzy g-closed.
Remark 2.1[15]: Every intuitionistic fuzzy closed set is intuitionistic fuzzy g-closed but its converse may
not be true.
(a) Intuitionistic fuzzy sg-closed if scl (A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy semi open.
(b) Intuitionistic fuzzy sg -open if its complement A
c
is intuitionistic fuzzy sg-closed.
Remark 2.2[18]: Every intuitionistic fuzzy semi-closed (resp. Intuitionistic fuzzy semi-open) set is
intuitionistic fuzzy sg-closed (intuitionistic fuzzy sg-open) but its converse may not be true.
(a) Intuitionistic fuzzy gs-closed if scl (A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open.
(b) Intuitionistic fuzzy gs -open if its complement A
c
is intuitionistic fuzzy gs-closed.
Remark 2.3[12]: Every intuitionistic fuzzy sg-closed (resp. Intuitionistic fuzzy sg-open) set is intuitionistic
fuzzy gs-closed (intuitionistic fuzzy gs-open) but its converse may not be true.
Definition 2.9: [5] Let X and Y are two nonempty sets and f: X → Y is a function. :
(a) If B = {<y, µB(y), γB(y)> : y ∈ Y}is an intuitionistic fuzzy set in Y, then the pre image of B under f
denoted by f
-1
(B), is the intuitionistic fuzzy set in X defined by
f
-1
(B) = <x, f
-1
(µB) (x), f
-1
(γB) (x)>: x ∈ X}.
(b) If A = {<x, λA(x), νA(x)> : x ∈ 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 (λA) (y), f(νA) (y)>: y ∈ Y}
Where f (νA) = 1 – f (1- νA).
Definition 2.10[8]: Let (X,ℑ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f: X→Y be a
function. Then f is said to be
(a) Intuitionistic fuzzy continuous if the pre image of each intuitionistic fuzzy open set of Y is an
intuitionistic fuzzy open set in X.
(b) Intuitionistic fuzzy semi continuous if the pre image of each intuitionistic fuzzy open set of Y is an
intuitionistic fuzzy semi open set in X.
(c) Intuitionistic fuzzy closed if the image of each intuitionisic fuzzy closed set in X is an intuitionistic
fuzzy closed set in Y.
(d) Intuitionistic fuzzy open if the image of each intuitionisic fuzzy open set in X is an intuitionistic
fuzzy open set in Y.

Definition 2.6[12, 16,17 19]: Let (X,ℑ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let
f: X→Y be a function. Then f is said to be
(a) Intuitionistic fuzzy g-continuous [16] if the pre image of every intuitionistic fuzzy closed set in Y is
intuitionistic fuzzy g –closed in X.
(b) Intuitionistic fuzzy gc-irresolute[17]if the pre image of every intuitionistic fuzzy g-closed in Y is
intutionistic fuzzy g-closed in X
(c) Intuitionistic fuzzy sg-continuous [19] if the pre image of every intuitionistic fuzzy closed set in Y is
intuitionistic fuzzy sg –closed in X.
(d) Intutionistic fuzzy gs-continuous [12] if the pre image of every intuitionistic fuzzy closed set in Y is
intuitionistic fuzzy gs –closed in X.
Remark 2.4[12, 16, 19]:
(a) Every intuitionistic fuzzy continuous mapping is intuitionistic fuzzy g-continuous, but the
converse may not be true [16].
(b) Every intuitionistic fuzzy semi continuous mapping is intuitionistic fuzzy sg-continuous, but the
(c) Every intuitionistic fuzzy sg- continuous mapping is intuitionistic fuzzy gs-continuous, but the
(d) Every intuitionistic fuzzy g- continuous mapping is intuitionistic fuzzy gs-continuous, but the
3. INTUITIONISTIC FUZZY W-CLOSED SET
Definition 3.1: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,ℑ) is called an
intuitionistic fuzzy w-closed if cl (A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy semi open.
Remark 3.1: Every intuitionistic fuzzy closed set is intuitionistic fuzzy w-closed but its converse may not
be true.
Example 3.1: Let X = {a, b} and ℑ = {0, 1, U} be an intuitionistic fuzzy topology on X, where U=
{< a,0.5,0.5>,< b, 0.4, 0.6 > }.Then the intuitionistic fuzzy set A = {<a,0.5,0.5>,<b,0.5,0.5>}
is intuitionistic fuzzy w -closed but it is not intuitionistic fuzzy closed.
Remark 3.2: Every intuitionistic fuzzy w-closed set is intuitionistic fuzzy g-closed but its converse may not
be true.
{< a,0.7,0.3>,< b, 0.6, 0.4 >}.Then the intuitionistic fuzzy set A = {<a,0.6,0.4>,<b,0.7,0.3>}
is intuitionistic fuzzy g -closed but it is not intuitionistic fuzzy w-closed.
Remark 3.3: Every intuitionistic fuzzy w-closed set is intuitionistic fuzzy sg-closed but its converse may
not be true.
{< a,0.5,0.5>,< b, 0.4, 0.6 >}.Then the intuitionistic fuzzy set A ={<a,0.5,0.5>,<b,0.3,0.7>}
is intuitionistic fuzzy sg -closed but it is not intuitionistic fuzzy w-closed.

Remark 3.4: Remarks 2.1, 2.2, 2.3, 3.1, 3.2, 3.3 reveals the following diagram of implication.
Intuitionistic fuzzy Intuitionistic fuzzy Intuitionistic fuzzy
Closed w-closed g-closed
Semi closed sg-closed gs-closed
Theorem 3.1: Let (X,ℑ) be an intuitionistic fuzzy topological space and A is an intuitionistic fuzzy set of
X. Then A is intuitionistic fuzzy w-closed if and only if (AqF) ⇒ (cl (A)qF) for every intuitionistic fuzzy
semi closed set F of X.
Proof: Necessity: Let F be an intuitionistic fuzzy semi closed set of X and  (AqF). Then by Lemma
2.1(a), A ⊆ F
c
and F
c
intuitionistic fuzzy semi open in X. Therefore cl(A) ⊆ F
c
by Def 3.1 because A is
intuitionistic fuzzy w-closed. Hence by lemma 2.1(a),  (cl (A)qF).
Sufficiency: Let O be an intuitionistic fuzzy semi open set of X such that A ⊆ O i.e. A ⊆ (O)
c
)
c
Then by
Lemma 2.1(a), (AqO
c
) and O
c
is an intuitionistic fuzzy semi closed set in X. Hence by hypothesis
 (cl (A)qO
c
). Therefore by Lemma 2.1(a), cl (A) ⊆((O)
c
)
c
i .e. cl (A) ⊆ O Hence A is intuitionistic fuzzy w-
closed in X.
Theorem 3.2: Let A be an intuitionistic fuzzy w-closed set in an intuitionistic fuzzy topological space (X,ℑ)
and c(α,β) be an intuitionistic fuzzy point of X such that c(α,β)qcl (A) then cl(c(α,β))qA.
Proof: If cl(c(α,β))qA then by Lemma 2.1(a),cl(c(α,β) ⊆ A
c
which implies that A ⊆ (cl(c(α,β)))
c
and so cl(A)
⊆ (cl(c(α,β)))
c
⊆ (c(α,β))
c
, because A is intuitionistic fuzzy w-closed in X. Hence by Lemma 2.1(a),
 (c(α,β)q (cl (A))), a contradiction.
Theorem 3.3: Let A and B are two intuitionistic fuzzy w-closed sets in an intuitionistic fuzzy topological
space (X,ℑ), then A∪B is intuitionistic fuzzy w-closed.
Proof: Let O be an intuitionistic fuzzy semi open set in X, such that A∪B ⊆ O. Then A ⊆ O and B ⊆ O.
So, cl (A) ⊆ O and cl (B) ⊆ O. Therefore cl (A) ∪ cl (B) = cl (A∪B) ⊆ O. Hence A∪B is intuitionistic fuzzy
w-closed.
Remark 3.2: The intersection of two intuitionistic fuzzy w-closed sets in an intuitionistic fuzzy topological
space (X,ℑ) may not be intuitionistic fuzzy w-closed. For,
Example 3.2: Let X = {a, b, c} and U, A and B be the intuitionistic fuzzy sets of X defined as follows:
U = {<a, 1, 0>, <b, 0, 1 >, < c, 0, 1>}
A = {<a, 1, 0 >, < b, 1, 0 >, < c, 0, 1>}
B = {<a, 1, 0 >, < b, 0, 1>, < c, 1, 0>}

Let ℑ = {0, 1, U} be intuitionistic fuzzy topology on X. Then A and B are intuitionistic fuzzy w-closed in
(X,ℑ) but A ∩ B is not intuitionistic fuzzy w-closed.
and A ⊆ B ⊆ cl (A). Then B is intuitionistic fuzzy w-closed in X.
Proof: Let O be an intuitionistic fuzzy semi open set such that B ⊆ O. Then A ⊆ O and since A is
intuitionistic fuzzy w-closed, cl (A) ⊆ O. Now B ⊆ cl (A) ⇒ cl (B) ⊆ cl (A) ⊆ O. Consequently B is
intuitionistic fuzzy w-closed.
Definition 3.2: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,ℑ) is called
intuitionistic fuzzy w-open if and only if its complement A
c
is intuitionistic fuzzy w-closed.
Remark 3.5 Every intuitionistic fuzzy open set is intuitionistic fuzzy w-open. But the converse may not be
true. For
{<a, 0.5, 0.5>, <b, 0.4, 0.6>}. Then intuitionistic fuzzy set B defined by B={ <a,0.5,0.5>, <b,0.5,0.5>}is an
intuitionistic fuzzy w-open in intuitionistic fuzzy topological space (X, ℑ) but it is not intuitionistic fuzzy
open in (X, ℑ).
Remark 3.6: Every intuitionistic fuzzy w-open set is intuitionistic fuzzy g-open but its converse may not be
true.
{<a,0.5,0.5>,<b,0.4,0.6>}.Then the intuitionistic fuzzy set A={<a,0.4,0.6>,<b,0.3,0.7>}
is intuitionistic fuzzy g-open in (X, ℑ ) but it is not intuitionistic fuzzy w-open in (X, ℑ).
Theorem 3.5: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X,ℑ) is intuitionistic
fuzzy w-open if F⊆⊆⊆⊆ int (A) whenever F is intuitionistic fuzzy semi closed and F ⊆⊆⊆⊆ A.
Proof: Follows from definition 3.1 and Lemma 2.1
Remark 3.4: The union of two intuitionistic fuzzy w-open sets in an intuitionistic fuzzy topological space
(X,ℑ) may not be intuitionistic fuzzy w-open. For the intuitionistic fuzzy set C ={ <a,0.4,0.6> ,<b,0.7,0.3> }
and D = {< a,0.2,0.8> ,<b,0.5,0.5>} in the intuitionistic fuzzy topological space (X,ℑ) in Example 3.2 are
intuitionistic fuzzy w-open but their union is not intuitionistic fuzzy w-open.
Theorem 3.6: Let A be an intuitionistic fuzzy w-open set of an intuitionistic fuzzy topological space (X,ℑ)
and int (A) ⊆ B ⊆ A. Then B is intuitionistic fuzzy w-open.
Proof: Suppose A is an intuitionistic fuzzy w-open in X and int(A) ⊆ B ⊆A. ⇒A
c
⊆ B
c
⊆ (int(A))
c
⇒ A
c
⊆
B
c
⊆ cl(A
c
) by Lemma 2.1(d) and A
c
is intuitionistic fuzzy w-closed it follows from theorem 3.4 that B
c
is
intuitionistic fuzzy w-closed .Hence B is intuitionistic fuzzy w-open.
Definition 3.3: An intuitionistic fuzzy topological space (X, ℑ) is called intuitionistic fuzzy semi normal if
for every pair of two intuitionistic fuzzy semi closed sets F1 and F2 such that (F1qF2), there exists two
intuitionistic fuzzy semi open sets U1 and U2 in X such that F1⊆ U1, F2⊆ U2 and (U1qU2).

Theorem 3.7: If F is intuitionistic fuzzy semi closed and A is intuitionistic fuzzy w--closed set of an
intuitionistic fuzzy semi normal space (X,ℑ) and (AqF). Then there exists intuitionistic fuzzy semi open
sets U and V in X such that cl (A) ⊂U, F⊂V and (UqV).
Proof: Since A is intuitionistic fuzzy w-closed set and (AqF), by Theorem (3.1), (cl (A)qF) and (X,ℑ) is
intuitionistic fuzzy semi normal. Therefore by Definition 3.3 there exists intuitionistic fuzzy semi open sets
U and V in X such that cl (A) ⊂ U, F ⊂ V and (UqV).
and f: (X,ℑ) → (Y,ℑ
*
) is an intuitionistic fuzzy irresolute and intuitionistic fuzzy closed mapping then f (A)
is an intuitionistic w-closed set in Y.
Proof: Let A be an intuitionistic fuzzy w-closed set in X and f: (X,ℑ) → (Y,ℑ
*
) is an intuitionistic fuzzy
continuous and intuitionistic fuzzy closed mapping. Let f(A) ⊆ G where G is intuitionistic fuzzy semi open
in Y then A ⊆ f
–1
(G) and f
–1
(G) is intutionistic fuzzy semi open in X because f is intuitionistic fuzzy
irresolute .Now A be an intuitionistic fuzzy w-closed set in X , by definition 3.1 cl(A) ⊆ f
–1
(G). Thus
f(cl(A)) ⊆ G and f(cl(A)) is an intuitionistic fuzzy closed set in Y( since cl(A) is intuitionistic fuzzy closed
in X and f is intuitionistic fuzzy closed mapping). It follows that cl (f (A) ⊆ cl (f (cl (A))) = f (cl (A)) ⊆ G.
Hence cl (f (A)) ⊆ G whenever f (A) ⊆ G and G is intuitionistic fuzzy semi open in Y. Hence f (A) is
intuitionistic fuzzy w-closed set in Y.
Theorem 3.9: Let(X,ℑ) be an intuitionistic fuzzy topological space and IFSO(X) (resp.IFC(X)) be the
family of all intuitionistic fuzzy semi open (resp. intuitionistic fuzzy closed) sets of X. Then IFSO(X) =
IFC(X) if and only if every intuitionisic fuzzy set of X is intuitionistic fuzzy w -closed.
Proof :Necessity : Suppose that IFSO(X) = IFC(X) and let A is any intuitionistic fuzzy set of X such that
A⊆ U ∈ IFSO(X) i.e. U is intuitionistic fuzzy semi open. Then cl (A) ⊆cl (U) = U because U∈IFSO(X)
=IFC(X). Hence cl (A) ⊆ U whenever A ⊆ U and U is intuitionistic fuzzy semi open. Hence A is w- closed
set.
Sufficiency: Suppose that every intuitionistic fuzzy set of X is intuitionistic fuzzy w- closed. Let U ∈
IFSO(X) then since U ⊆ U and U is intuitionistic fuzzy w- closed, cl (U) ⊆ U then U ∈ IFC(X).Thus
IFSO(X) ⊆ IFC(X). If T ∈ IFC(X) then T
c
∈ IFO(X) ⊆IFSO ⊆ IFC(X) hence T∈ IFO(X) ⊆ IFSO(X).
Consequently IFC(X) ⊆ IFSO(X) and IFSO(X) =IFC(X).
4: INTUITIONISTIC FUZZY W-CONNECTEDNESS AND INTUITIONISTIC FUZZY W-
COMPACTNESS
Definition 4.1: An intuitionistic fuzzy topological space (X ℑ )is called intuitionistic fuzzy w – connected
if there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy w- open and
intuitionistic fuzzy w- closed .
Theorem 4.1: Every intuitionistic fuzzy w-connected space is intuitionistic fuzzy connected.
Proof: Let (X, ℑ) be an intuitionisic fuzzy w –connected space and suppose that (X, ℑ) is not intuitionistic
fuzzy connected .Then there exists a proper intuitionistic fuzzy set A( A≠ 0, A≠ 1 ) such that A is both

intuitionistic fuzzy open and intuitionistic fuzzy closed. Since every intuitionistic fuzzy open set (resp.
intuitionistic fuzzy closed set) is intuitionistic w-open ((resp. intuitionistic fuzzy w-closed), X is not
intuitionistic fuzzy w-connected, a contradiction.
Remark 4.1: Converse of theorem 4.1 may not be true for ,
Example 4.1: Let X = {a, b} and ℑ = {0, 1, U} be an intuitionistic fuzzy topology on X, where U =
{< a,0.5,0.5>,< b, 0.4, 0.6 > }.Then intuitionistic fuzzy topological space (X, ℑ) is intuitionistic fuzzy
connected but not intuitionistic fuzzy w-connected because there exists a proper intuitionistic fuzzy set
A={<a,0.5,0.5>,<b,0.5,0.5>} which is both intuitionistic fuzzy w -closed and intuitionistic w-open in X.
Theorem 4.2: An intuitionistic fuzzy topological (X,ℑ) is intuitionistic fuzzy w-connected if
and only if there exists no non zero intuitionistic fuzzy w-open sets A and B in X such that A=B
c
.
Proof: Necessity: Suppose that A and B are intuitionistic fuzzy w-open sets such that A≠ 0≠ B and A =
B
c
. Since A=B
c
, B is an intuitionistic fuzzy w-open set which implies that B
c
= A is intuitionistic fuzzy w-
closed set and B≠ 0 this implies that B
c
≠ 1  i.e. A≠ 1 Hence there exists a proper intuitionistic fuzzy
set A( A≠ 0, A≠ 1 ) such that A is both intuitionistic fuzzy w- open and intuitionistic fuzzy w-closed. But
this is contradiction to the fact that X is intuitionistic fuzzy w- connected.
Sufficiency: Let (X,ℑ) is an intuitionistic fuzzy topological space and A is both intuitionistic fuzzy w-open
set and intuitionistic fuzzy w-closed set in X such that 0 ≠ A ≠ 1. Now take B = A
c
.In this case B is an
intuitionistic fuzzy w-open set and A≠ 1.This implies that B =A
c
≠ 0 which is a contradiction. Hence there
is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy w- open and intuitionistic
fuzzy w- closed. Therefore intuitionistic fuzzy topological (X,ℑ) is intuitionistic fuzzy w-connected
Definition 4.2: Let (X,ℑ) be an intuitionistic fuzzy topological space and Abe an intuitionistic fuzzy set X.
Then w-interior and w-closure of A are defined as follows.
wcl (A) = ∩ {K: K is an intuitionistic fuzzy w-closed set in X and A⊆ K}
wint (A) = ∪ {G: G is an intuitionistic fuzzy w-open set in X and G⊆ A}
Theorem 4.3: An intuitionistic fuzzy topological space (X, ℑ) is intuitionistic fuzzy w-connected if and only
if there exists no non zero intuitionistic fuzzy w-open sets A and B in X such that B= A
c
, B ={wcl(A))
c
, A=
(wcl(B))
c
.
Proof: Necessity : Assume that there exists intuitionistic fuzzy sets A and B such that A≠ 0 ≠ B in X
such that B=A
c
,B =(wcl(A))
c
,A=(wcl(B))
c
. Since (wcl (A))
c
and (wcl (B))
c
are intuitionistic fuzzy w-open
sets in X, which is a contradiction.
Sufficiency: Let A is both an intuitionistic fuzzy w-open set and intuitionistic fuzzy w-closed set such that
0 ≠ A ≠ 1. Taking B= A
c
, we obtain a contradiction.
Definition 4.3: An intuitionistic fuzzy topological space (X,ℑ) is said to be intuitionistic fuzzy
w- T1/2 if every intuitionistic fuzzy w-closed set in X is intuitionistic fuzzy closed in X.
Theorem 4.4: Let (X,ℑ) be an intuitionistic fuzzy w- T1/2 space, then the following conditions are
equivalent:
(a) X is intuitionistic fuzzy w-connected.

(b) X is intuitionistic fuzzy connected.
Proof: (a) ⇒(b) follows from Theorem 4.1
(b) ⇒(a): Assume that X is intuitionistic fuzzy w- T1/2 and intuitionistic fuzzy w-connected space. If
possible, let X be not intuitionistic fuzzy w-connected, then there exists a proper intuitionistic fuzzy set A
such that A is both intuitionistic fuzzy w-open and w-closed. Since X is intuitionistic fuzzy w-T1/2 , A is
intuitionistic fuzzy open and intuitionistic fuzzy closed which implies that X is not intuitionistic fuzzy
connected, a contradiction.
Definition 4.4 : A collection { Ai : i∈ Λ} of intuitionistic fuzzy w- open sets in intuitionistic fuzzy topological
space (X,ℑ) is called intuitionistic fuzzy w- open cover of intuitionistic fuzzy set B of X if B ⊆ ∪{ Ai : i∈
Λ}
Definition 4.5: An intuitionistic fuzzy topological space (X,ℑ) is said to be intuitionistic fuzzy w-compact if
every intuitionistic fuzzy w- open cover of X has a finite sub cover.
Definition 4.6 : An intuitionistic fuzzy set B of intuitionistic fuzzy topological space (X,ℑ) is said to be
intuitionistic fuzzy w- compact relative to X, if for every collection { Ai : i∈ Λ} of intuitionistic fuzzy w- open
subset of X such that B ⊆ ∪{ Ai : i∈ Λ} there exists finite subset Λo of Λ such that B ⊆ ∪{ Ai : i∈ Λo}
.
Definition 4.7: A crisp subset B of intuitionistic fuzzy topological space (X,ℑ) is said to be intuitionistic
fuzzy w- compact if B is intuitionistic fuzzy w- compact as intuitionistic fuzzy subspace of X .
Theorem 4.5: A intuitionistic fuzzy w-closed crisp subset of intuitionistic fuzzy w- compact space is
intuitionistic fuzzy w- compact relative to X.
Proof: Let A be an intuitionistic fuzzy w- closed crisp subset of intuitionistic fuzzy w- compact space(
X,ℑ). Then A
c
is intuitionistic fuzzy w- open in X. Let M be a cover of A by intuitionistic fuzzy w- open sets
in X. Then the family {M, A
c
} is intuitionistic fuzzy w- open cover of X. Since X is intuitionistic fuzzy w-
compact, it has a finite sub cover say {G1, G2, G3 ......., Gn}. If this sub cover contains A
c
, we discard it.
Otherwise leave the sub cover as it is. Thus we obtained a finite intuitionistic fuzzy w – open sub cover of
A. Therefore A is intuitionistic fuzzy w – compact relative to X.
5: INTUTIONISTIC FUZZY W- CONTINUOUS MAPPINGS
Definition 5.1:A mapping f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w- continuous if inverse image of every
intuitionistic fuzzy closed set of Y is intuitionistic fuzzy w-closed set in X.
Theorem 5.1: A mapping f : (X,ℑ). →(Y,σ) is intuitionistic fuzzy w- continuous if and only if the inverse
image of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy w- open in X.
Proof: It is obvious because f
-1
(U
c
) = (f
-1
(U))
c
for every intuitionistic fuzzy set U of Y.
Remark5.1 Every intuitionistic fuzzy continuous mapping is intuitionistic fuzzy w-continuous, but converse
may not be true. For,
Example 5.1 Let X = {a, b}, Y ={x, y } and intuitionistic fuzzy sets U and V are defined as follows :
U= {< a, 0.5, 0.5>, < b, 0.4, 0.6>}
V= {<x, 0.5, 0.5>, <y, 0.5, 0.5>}

Let ℑℑℑℑ = {0, 1, U} and σ = {0, 1, V} be intuitionistic fuzzy topologies on X and Y respectively. Then the
mapping f: (X,ℑ). →(Y, σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy w- continuous but not
intuitionistic fuzzy continuous.
Remark5.2 Every intuitionistic fuzzy w-continuous mapping is intuitionistic fuzzy g-continuous, but
converse may not be true. For,
Example 5.2: Let X = {a, b}, Y ={x, y} and intuitionistic fuzzy sets U and V are defined as follows:
U= {< a, 0.7, 0.3>, < b, 0.6, 0.4>}
V= {<x, 0.6, 0.4>, <y, 0.7, 0.3>}
Let ℑℑℑℑ = { 0, 1 , U } and σ ={ 0, 1 , V } be intuitionistic fuzzy topologies on X and Y respectively. Then the
mapping f: (X,ℑ). →(Y, σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy g- continuous but not
intuitionistic fuzzy w- continuous.
Remark5.3 Every intuitionistic fuzzy w-continuous mapping is intuitionistic fuzzy sg-continuous, but
converse may not be true. For,
Example 5.1 Let X = {a, b}, Y ={x, y} and intuitionistic fuzzy sets U and V are defined as follows:
U= {< a, 0.5, 0.5>, < b, 0.4, 0.6>}
V= {<x, 0.5, 0.5>, <y, 0.3, 0.7>}
Let ℑℑℑℑ = { 0, 1 , U } and σ ={ 0, 1 , V } be intuitionistic fuzzy topologies on X and Y respectively. Then the
mapping f: (X,ℑ). →(Y, σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy sg- continuous but not
intuitionistic fuzzy w- continuous.
Remark 5.4: Remarks 2.4, ,5.1, 5.2, 5.3 reveals the following diagram of implication:
Continuous w-continuous g-continuous
Semi continuous sg-continuous gs-continuous
Theorem 5.2: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy w- continuous then for each intuitionistic fuzzy point
c(α,β) of X and each intuitionistic fuzzy open set V of Y such that f(c(α,β)) ⊆ V there exists a intuitionistic
fuzzy w- open set U of X such that c(α,β) ⊆ U and f(U) ⊆ V.
Proof : Let c(α,β) be intuitionistic fuzzy point of X and V be a intuitionistic fuzzy open set of Y such that
f(c(α,β) ) ⊆ V. Put U = f
-1
(V). Then by hypothesis U is intuitionistic fuzzy w- open set of X such that c(α,β)
⊆ U and f (U) = f (f
-1
(V)) ⊆ V.
Theorem 5.3: Let f: (X,ℑ). →(Y,σ) is intuitionistic fuzzy w- continuous then for each intuitionistic fuzzy
point c(α,β) of X and each intuitionistic fuzzy open set V of Y such that f(c(α,β))qV, there exists a
intuitionistic fuzzy w- open set U of X such that c(α,β)qU and f(U) ⊆ V.
Proof: Let c(α,β) be intuitionistic fuzzy point of X and V be a intuitionistic fuzzy open set of Y such that
f(c(α,β))q V. Put U = f
-1
(V). Then by hypothesis U is intuitionistic fuzzy w- open set of X such that
c(α,β)q U and f(U)= f(f
-1
(V) ) ⊆ V.

Theorem 5.4: If f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous, then f(wcl(A) ⊆ cl(f(A)) for every
intuitionistic fuzzy set A of X.
Proof: Let A be an intuitionistic fuzzy set of X. Then cl(f(A)) is an intuitionistic fuzzy closed set of Y. Since
f is intuitionistic fuzzy w –continuous, f
-1
(cl(f(A))) is intuitionistic fuzzy w-closed in X. Clearly A ⊆ f
-1
(cl
((A)). Therefore wcl (A)⊆ wcl (f
-1
(cl(f(A)))) = f
-1
(cl(f(A))). Hence f (wcl (A) ⊆ cl (f (A)) for every intuitionistic
fuzzy set A of X.
.
Theorem 5.5: A mapping f from an intuitionistic fuzzy w-T1/2 space (X,ℑ) to a intuitionistic fuzzy
topological space (Y, σ) is intuitionistic fuzzy semi continuous if and only if it is intuitionistic fuzzy w –
continuous.
Proof: Obvious
Remark 5.5: The composition of two intuitionistic fuzzy w – continuous mapping may not be
Intuitionistic fuzzy w – continuous. For
Example 5-5: Let X = {a, b}, Y= {x, y} and Z= {p, q} and intuitionstic fuzzy sets U,V and W defined as
follows :
U = {< a, 0.5, 0.5>, < b, 0.4, 0.6>}
V = {<x, 0.5, 0.5>, <y, 0.3, 0.7>}
W = {< p, 0.6, 0.4>, < q, 0.4, 0.6>}
Let ℑℑℑℑ = { 0, 1 , U } , σ ={ 0, 1 , V } and µ={ 0, 1 , W } be intuitionistic fuzzy topologies on X , Y and Z
respectively. Let the mapping f: (X,ℑ). →(Y, σ) defined by f(a) = x and f(b) = y and g : (Y,σ) →(Z,µ)
defined by g(x) = p and g(y) = q. Then the mappings f and g are intuitionistic fuzzy w-continuous but the
mapping gof: (X,ℑ) →(Z, µ ) is not intuitionistic fuzzy w-continuous.
Theorem 5.6: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous and g :( Y, σ) →(Z, µ) is intuitionistic
fuzzy continuous. Then gof : (X,ℑ) →(Z,µ) is intuitionistic fuzzy w-continuous.
Proof: Let A is an intuitionistic fuzzy closed set in Z. then g
-1
(A) is intuitionstic fuzzy closed in Y because
g is intuitionistic fuzzy continuous. Therefore (gof )
-1
(A) =f
-1
(g
-1
(A)) is intuitionistic fuzzy w – closed in X.
Hence gof is intuitionistic fuzzy w –continuous.
Theorem 5.7 : If f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous and g : (Y,σ) . →(Z,µ) is intuitionistic
fuzzy g-continuous and (Y,σ) is intuitionistic fuzzy (T1/2) then gof : (X,ℑ) →(Z,µ) is intuitionistic fuzzy w-
continuous.
Proof: Let A is an intuitionistic fuzzy closed set in Z, then g
-1
(A) is intuitionstic fuzzy g-closed in Y. Since
Y is (T1/2), then g
-1
(A) is intuitionstic fuzzy closed in Y. Hence (gof )
-1
(A) =f
-1
(g
-1
(A)) is intuitionistic fuzzy
w – closed in X. Hence gof is intuitionistic fuzzy w – continuous.
Theorem 5.8: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy gc-irresolute and g :( Y, σ) →(Z, µ) is intuitionistic
fuzzy w-continuous. Then gof : (X,ℑ) →(Z,µ) is intuitionistic fuzzy g-continuous.
Proof: Let A is an intuitionistic fuzzy closed set in Z, then g
-1
(A) is intuitionstic fuzzy w-closed in Y,
because g is intuitionistic fuzzy w-continuous. Since every intuitionistic fuzzy w-closed set is intuitionistic
fuzzy g-closed set, therefore g
-1
(A) is intuitionstic fuzzy g-closed in Y .Then (gof )
-1
(A) =f
-1
(g
-1
(A)) is
intuitionistic fuzzy g-closed in X ,because f is intuitionistic fuzzy gc- irresolute. Hence gof : (X,ℑ) →(Z,µ)
is intuitionistic fuzzy g-continuous.
Theorem 5.9: An intuitionistic fuzzy w – continuous image of a intuitionistic fuzzy w-compact space is
intuitionistic fuzzy compact.

Proof: Let. f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous map from a intuitionistic fuzzy w-compact
space (X,ℑ) onto a intuitionistic fuzzy topological space (Y, σ). Let {Ai: i∈ Λ} be an intuitionistic fuzzy
open cover of Y then {f
-1
(Ai) : i∈ Λ} is a intuitionistic fuzzy w –open cover of X. Since X is intuitionistic
fuzzy w- compact it has finite intuitionistic fuzzy sub cover say { f
-1
(A1) , f
-1
(A2) ,----f
-1
(An) } . Since f is
onto {A1, A2, --------------An} is an intuitionistic fuzzy open cover of Y and so (Y, σ) is intuitionistic fuzzy
compact.
Theorem 5.10: If f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous surjection and X is intuitionistic
fuzzy w-connected then Y is intuitionistic fuzzy connected.
Proof: Suppose Y is not intuitionistic fuzzy connected. Then there exists a proper intuitionistic fuzzy set
G of Y which is both intuitionistic fuzzy open and intuitionistic fuzzy closed. Therefore f
-1
( G) is a proper
intuitionistic fuzzy set of X, which is both intuitionistic fuzzy w- open and intuitionistic fuzzy w – closed ,
because f is intuitionistic fuzzy w– continuous surjection. Hence X is not intuitionistic fuzzy w –
connected, which is a contradiction.
6. INTUITIONISTIC FUZZY W-OPEN MAPPINGS
Definition 6.1: A mapping f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-open if the image of every intuitionistic
fuzzy open set of X is intuitionistic fuzzy w-open set in Y.
Remark 6.1 : Every intuitionistic fuzzy open map is intuitionistic fuzzy w-open but converse may not be
true. For,
Example 6.1: Let X = {a, b} , Y = {x , y} and the intuitionistic fuzzy set U and V are defined as follows :
U = {< a, 0.5. 0.5 > , < b ,0.4,0.6>}
V = {< x, 0.5, 0.5 >, <y, 0.3, 0.7 >}
Then ℑ = {0 ,U, 1 } and σ = { 0 ,V, 1 } be intuitionistic fuzzy topologies on X and Y respectively . Then
the mapping f : (X,ℑ). →(Y, σ) defined by f(a) = x and f(b) =y is intuitionistic fuzzy w-open but it is not
intuitionistic fuzzy open.
Theorem 6.1: A mapping f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-open if and only if for every
intuitionisic fuzzy set U of X f(int(U)) ⊆ wint(f(U)).
Proof: Necessity Let f be an intuitionistic fuzzy w-open mapping and U is an intuitionistic fuzzy open set
in X. Now int(U) ⊆ U which implies that f (int(U) ⊆ f(U). Since f is an intuitionistic fuzzy w-open mapping,
f(Int(U) is intuitionistic fuzzy w-open set in Y such that f(Int(U) ⊆ f(U) therefore f(Int(U) ⊆ wint f(U).
Sufficiency: For the converse suppose that U is an intuitionistic fuzzy open set of X. Then
f (U) = f (Int (U) ⊆ wint f(U). But wint (f (U)) ⊆ f (U). Consequently f (U) = wint (U) which implies that f(U)
is an intuitionistic fuzzy w-open set of Y and hence f is an intuitionistic fuzzy w-open .
Theorem 6.2: If f : (X,ℑ). →(Y, σ) is an intuitionistic fuzzy w-open map then int (f
-1
(G) ⊆ f
-1
(wint (G) for
every intuitionistic fuzzy set G of Y.
Proof: Let G is an intuitionistic fuzzy set of Y. Then int f
-1
(G) is an intuitionistic fuzzy open set in X. Since
f is intuitionistic fuzzy w-open f(int f
-1
(G) ) is intuitionistic fuzzy w-open in Y and hence f(Int f
-1
(G) ) ⊆
wint(f( f
-1
(G)) ⊆ wint(G). Thus int f
-1
(G) ⊆ f
-1
(wint (G).

Theorem 6.3: A mapping f : (X,ℑ). →(Y,σ) is intuitionistic fuzzy w-open if and only if for each
intuitionistic fuzzy set S of Y and for each intuitionistic fuzzy closed set U of X containing f
-1
(S) there is
a intuitionistic fuzzy w-closed V of Y such that S ⊆ V and f
-1
(V) ⊆ U .
Proof: Necessity: Suppose that f is an intuitionistic fuzzy w- open map. Let S be the intuitionistic fuzzy
closed set of Y and U is an intuitionistic fuzzy closed set of X such that f
-1
(S) ⊆ U . Then V = ( f
-1
(U
c
))
c
is intuitionistic fuzzy w- closed set of Y such that f
-1
(V) ⊆ U .
Sufficiency: For the converse suppose that F is an intuitionistic fuzzy open set of X. Then
f
-1
((f(F))
c
⊆ F
c
and F
c
is intuitionistic fuzzy closed set in X. By hypothesis there is an intuitionistic fuzzy
w-closed set V of Y such that ( f (F))
c
⊆ V and f
-1
(V) ⊆ F
c
.Therefore F ⊆ ( f
-1
(V))
c
. Hence V
c
⊆ f (F)
⊆ f(( f
-1
(V))
c
) ⊆V
c
which implies f(F) = V
c
. Since V
c
is intuitionistic fuzzy w-open set of Y. Hence f (F) is
intuitionistic fuzzy w-open in Y and thus f is intuitionistic fuzzy w-open map.
Theorem 6.4: A mapping f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-open if and only if
f
-1
(wcl (B) ⊆ cl f
-1
(B) for every intuitionistic fuzzy set B of Y.
Proof: Necessity: Suppose that f is an intuitionistic fuzzy w- open map. For any intuitionistic fuzzy set B
of Y f
-1
(B) ⊆cl( f
-1
(B)) Therefore by theorem 6.3 there exists an intuitionistic fuzzy w-closed set F in Y
such that B ⊆ F and f
-1
(F) ⊆cl(f
-1
(B) ). Therefore we obtain that f
-1
(wcl(B)) ⊆ f
-1
(F) ⊆ cl f
-1
((B)).
Sufficiency: For the converse suppose that B is an intuitionistic fuzzy set of Y. and F is an intuitionistic
fuzzy closed set of X containing f
-1
( B ). Put V= cl (B) , then we have B ⊆ V and Vis w-closed and f
-
1
(V) ⊆ cl ( f
-1
(B)) ⊆ F. Then by theorem 6.3 f is intuitionistic fuzzy w-open.
Theorem 6.5: If f: (X,ℑ). →(Y, σ) and g :( Y, σ) →(Z, µ) be two intuitionistic fuzzy map and gof : (X,ℑ)
→(Z,µ) is intuitionistic fuzzy w-open. If g :( Y, σ) →(Z, µ) is intuitionistic fuzzy w-irresolute then f: (X,ℑ).
→(Y, σ) is intuitionistic fuzzy w-open map.
Proof: Let H be an intuitionistic fuzzy open set of intuitionistic fuzzy topological space(X,ℑ). Then (go f)
(H) is intuitionistic fuzzy w-open set of Z because gof is intuitionistic fuzzy w- open map. Now since g :(
Y, σ) →(Z, µ) is intuitionistic fuzzy w-irresolute and (gof) (H) is intuitionistic fuzzy w-open set of Z
therefore g
-1
(gof (H)) = f(H) is intuitionistic fuzzy w-open set in intuitionistic fuzzy topological space Y.
Hence f is intuitionistic fuzzy w-open map.
7. INTUITIONISTIC FUZZY W-CLOSED MAPPINGS
Definition 7.1: A mapping f : (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-closed if image of every intuitionistic
fuzzy closed set of X is intuitionistic fuzzy w-closed set in Y.
Remark 7.1 Every intuitionistic fuzzy closed map is intuitionistic fuzzy w-closed but converse may not be
true. For,
Example 7.1: Let X = {a, b}, Y = {x, y}
Then the mapping f : (X,ℑ). →(Y, σ) defined in Example 6.1 is intuitionistic fuzzy w- closed but it is not
intuitionistic fuzzy closed.
Theorem 7.1: A mapping f : (X,ℑ). →(Y,σ) is intuitionistic fuzzy w-closed if and only if for each
intuitionistic fuzzy set S of Y and for each intuitionistic fuzzy open set U of X containing f
-1
(S) there is a
intuitionistic fuzzy w-open set V of Y such that S ⊆ V and f
-1
(V) ⊆ U .

Proof: Necessity: Suppose that f is an intuitionistic fuzzy w- closed map. Let S be the intuitionistic fuzzy
closed set of Y and U is an intuitionistic fuzzy open set of X such that f
-1
(S) ⊆ U . Then V =Y - f
-1
(U
c
) is
intuitionistic fuzzy w- open set of Y such that f
-1
(V) ⊆ U .
Sufficiency: For the converse suppose that F is an intuitionistic fuzzy closed set of X. Then (f(F))
c
is an
intuitionistic fuzzy set of Y and F
c
is intuitionistic fuzzy open set in X such that f
-1
((f(F))
c
) ⊆ F
c
. By
hypothesis there is an intuitionistic fuzzy w-open set V of Y such that ( f(F))
c
⊆ V and f
-1
(V) ⊆ F
c
.Therefore F ⊆ ( f
-1
(V))
c
. Hence V
c
⊆ f (F) ⊆ f( ( f
-1
(V))
c
) ⊆ V
c
which implies f(F) = V
c
. Since V
c
is
intuitionistic fuzzy w-closed set of Y. Hence f (F) is intuitionistic fuzzy w-closed in Y and thus f is
intuitionistic fuzzy w-closed map.
Theorem 7.2: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy semi continuous and intuitionistic fuzzy w-closed
map and A is an intuitionistic fuzzy w-closed set of X ,then f (A) intuitionistic fuzzy w-closed.
Proof: Let f(A) ⊆ O where O is an intuitionistic fuzzy semi open set of Y. Since f is intuitionistic fuzzy
semi continuous therefore f
-1
(O) is an intuitionistic fuzzy semi open set of X such that A ⊆ f
-1
(O). Since
A is intuitionistic fuzzy w-closed of X which implies that cl(A) ⊆ (f
-1
(O) ) and hence f( cl (A) ⊆ O which
implies that cl ( f ( cl(A) ) ⊆ O therefore cl ( f ((A) ) ⊆ O whenever f(A) ⊆ O where O is an intuitionistic
fuzzy semi open set of Y. Hence f(A) is an intuitionistic fuzzy w-closed set of Y.
Corollary 7.1: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy w-continuous and intuitionistic fuzzy closed map
and A is an intuitionistic fuzzy w-closed set of X ,then f (A) intuitionistic fuzzy w-closed.
Theorem 7.3: If f: (X,ℑ). →(Y, σ) is intuitionistic fuzzy closed and g :( Y, σ) →(Z, µ) is intuitionistic fuzzy
w-closed. Then gof : (X,ℑ) →(Z,µ) is intuitionistic fuzzy w-closed.
Proof: Let H be an intuitionistic fuzzy closed set of intuitionistic fuzzy topological space(X,ℑ). Then f (H)
is intuitionistic fuzzy closed set of (Y, σ) because f is inuituionistic fuzzy closed map. Now( gof) (H) =
g(f(H)) is intuitionistic fuzzy w-closed set in intuitionistic fuzzy topological space Z because g is
intuitionistic fuzzy w-closed map. Thus gof : (X,ℑ) →(Z,µ) is intuitionistic fuzzy w-closed.
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