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Int Jr. of Mathematics Sciences & Applications
Vol. 2, No. 2, May 2012
Copyright Mind Reader Publications
ISSN No: 2230-9888
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Intuitionistic Bifuzzy Generalized Structure Space
R.Dhavaseelan1 , E.Roja2 and M.K.Uma2
1
2
Department of Mathematics, Sona College of Technology, Salem-636005,
Tamil Nadu ,India.
e-mail : [email protected]
Department of Mathematics, Sri Saradha College for Women, Salem - 16,
Tamil Nadu , India.
Abstract
In this paper,the concept of intuitionistic bifuzzy generalized structure space is introduced.The concept of compactness is studied.In this connection,good extension is also established.
Keywords: intuitionistic bifuzzy GX space; intuitionistic fuzzy G∗X α compact space; intuitionistic fuzzy
GX P α compact space.
2000 Mathematics subject classification:54A40,03E72.
1
Introduction
The fuzzy concept has invaded almost all branches of mathematics ever since the introduction of fuzzy sets
by L.A.Zadeh [18]. These Fuzzy sets have applications in many fields such as information [15] and control
[16].The theory of fuzzy topological space was introduced and developed by C.L.Chang [7] and since then
various notions in classical topology has extended the fuzzy topological space.
The idea of ”intuitionistic fuzzy set” was first published by Atanassov [1] and many works by the same author and his colleagues appeared in the literature [2,3,4].Different types of compactness in bifuzzy topological
spaces was introduced by A.S.M.Abu Safiya,A.A.Fora and M.W.Warner[6].
In this paper,the concept of intuitionistic bifuzzy generalized space is introduced.The concept of compactness is studied.In this connection,good extension is also established.
Throughout the paper if A ⊆ X, let χA denote the characteristic map of the subset A,that is χA (x) = 1
if x ∈ A and χA (x) = 0 if x 6∈ A.
2
Preliminary
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 = {hx, µA (x), δA (x)i : x ∈ X} where the function µA : X → I and δA : X → I denote the
degree of membership ( namely µA (x) )and the degree of nonmembership ( δA (x) ) of each element x ∈ X
to the set A, respectively, and 0 ≤ µA (x) + δA (x) ≤ 1 for each x ∈ X.
Definition: 2.2. [3] 0∼ = {hx, 0, 1i : x ∈ X} and 1∼ = {hx, 1, 0i : x ∈ X}.
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Definition: 2.3. [9] An intuitionistic fuzzy topology(IF T for short) on a nonempty set X is a family T of
intuitionistic fuzzy sets in X satisfying the following axioms:
(1) 0∼ ,1∼ ∈ T ,
(2) G1 ∩ G2 ∈ T for any G1 , G2 ∈ T ,
(3) ∪Gi ∈ T for any arbitary family {Gi : i ∈ J} ⊆ T .
In this case the pair (X, T ) is called an intuitionistic fuzzy topological space (IF T S for short) and any IFS
in T is known as an intuitionistic fuzzy open set (IF OS for short) in X.
The complement of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set.
Definition: 2.4. (cf.[10]) Let X be a nonempty fixed set.An intuitionistic set[IS for short] A is an object
having the form A = hx, A1 , A2 i where A1 and A2 are subsets of X satisfying A1 ∩ A2 = ∅.The set A1 is
called the set of members of A,while A2 is called the set of nonmembers of A.
Definition: 2.5. [9] An intuitionistic topology[IT for short] on a nonempty set X is a family τ of IS 0 s in
X satisfying the following axioms:
(i) ∅∼ , X∼ ∈ τ ,
(ii) G1 ∩ G2 ∈ τ ,for any G1 , G2 ∈ τ ,
S
(iii) Gi ∈ τ ,for any arbitrary family {Gi : i ∈ J} ⊆ τ .
In this case the pair (X, τ ) is called an intuitionistic topological space[IT S for short] and any IS in τ is
known as an intuitionistic open set [IOS for short] in X.
Definition: 2.6. Let (X, τ ) be an intuitionistic topological space.An intuitionistic set A in (X, τ ) is said to
be generalized intuitionistic closed (in shortly GI-closed) if cl(A) ⊆ G whenever A ⊆ G and G is intuitionistic
open.The complement of a GI closed set GI open.
Definition: 2.7. [8] Let (X, T ) be an intuitionistic fuzzy topological space. An intuitionistic fuzzy set A in
(X, T ) is said to be generalized intuitionistic fuzzy closed (in shortly GIF-closed) if IF cl (A) ⊆ G whenever
A ⊆ G and G is intuitionistic fuzzy open. The complement of a GIF-closed set is GIF-open.
Definition: 2.8. [8]Let (X, T ) be an intuitionistic fuzzy topological space and A be an intuitionistic fuzzy
set in X.Then intuitionistic fuzzy generalized closure and intuitionistic fuzzy generalized interior of A are
defined by
T
(a) IF Gcl(A) = {G: G is a GIF closed set in X and A ⊆ G }.
S
(b) IF Gint(A) = {G: G is a GIF open set in X and A ⊇ G}.
Definition: 2.9. [6] Let (X, τ1 , τ2 ) be a bifuzzy topological space.A fuzzy set µ is called τ1 τ2 -open,τ1 τ2 -closed
provided µ ∈ τ1 ∪ τ2 , µc ∈ τ1 ∪ τ2 respectively.Whenever we call a collection U ⊆ I X S-open(P -open) we
mean that U ⊆ τ1 ∪ τ2 (U ⊆ τ1 ∪ τ2 and U contains a non-zero τ1 -open set and a non-zero τ2 -open set).A
collection C is called S-closed (P -closed) iff U = {λc : λ ∈ C} is S-open (P -open).
Definition: 2.10. [12] A collection U ⊆ I X is called an α-shading of X if for each x ∈ X,there exists µ ∈ U
with µ(x) > α where 0 ≤ α < 1.A subcollection V of U that is also an α-shading is called an α-subshading.
Definition: 2.11. [5] A collection U ⊆ τ1 ∪τ2 is called an S-α-shading(P -α-shading) iff for each x ∈ X,there
exists µ ∈ U such that µ(x) > α where 0 ≤ α < 1 and U is S-open(P -open).
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Intuitionistic Bifuzzy Generalized Structure Space
Definition: 2.12. [14] A bifuzzy topological space (X, τ1 , τ2 ) is called S-α-compact (P -α-compact) iff if
every S-α-shading (P -α-shading) of X has a finite α-subshading.
Definition: 2.13. [12] Let X be a non-empty set and α ∈ I.A collection P of fuzzy sets in X is said to
be α-centered if for all finite collections µi ∈ P, i = 1, 2, ..., n,there exists x ∈ X with µk (x) ≥ 1 − α for all
k ∈ {1, 2, ..., n}.
Definition: 2.14. [13] A fuzzy set f in a fuzzy topological space(X, τ ) is said to be compact if for every
family G ⊆ τ such that sup{g : g ∈ G} ≥ f and for every > 0,there exists a finite subfamily G such that
sup{g : g ∈ G } ≥ f − .
3
Intuitionistic Bifuzzy Generalized Structure and Intuitionistic
Bifuzzy Generalized Compactness
Definition: 3.1. A family G of generalized intuitionistic fuzzy open sets in an intuitionistic fuzzy topological
space (X, T ) satisfying the following axioms.
(1) 0∼ ,1∼ ∈ G,
(2) A1 ∩ A2 ∈ G,for any A1 , A2 ∈ G,
S
(3) Ai ∈ G,for any arbitrary family of GIF open sets {Ai : i ∈ J} ⊆ G.
Then G is called intuitionistic fuzzy generalized structure.The pair (X, G) is called intuitionistic fuzzy generalized structure(in shortly intuitionistic fuzzy GX )space.The member of (X, G) are called intuitionistic fuzzy
GX open.
The complement of an intuitionistic fuzzy GX open set is an intuitionistic fuzzy GX closed set.
Definition: 3.2. Let (X, G1 , G2 ) be an intuitionistic bifuzzy GX space.An intuitionistic fuzzy set A =
hx, µA (x), γA (x)i is called intuitionistic fuzzy generalized G1 G2 open, intuitionistic fuzzy generalized G1 G2
closed provided A in G1 ∪ G2 ,Ain G1 ∪ G2 respectively.
Definition: 3.3. Let (X, G1 , G2 ) be an intuitionistic bifuzzy GX space.A collection of intuitionistic fuzzy
sets U = {Ui : i ∈ J} where Ui = hx, µUi (x), γUi (x)i is intuitionistic fuzzy G∗X open(intuitionistic fuzzy GX P
open), we mean that U ⊆ G1 ∪ G2 (U ⊆ G1 ∪ G2 and U contains a non-zero intuitionistic fuzzy GX open
sets in G1 and non-zero intuitionistic fuzzy GX open sets in G2 ).
A collection of intuitionistic fuzzy sets C = {Ci : i ∈ J} where Ci = hx, µCi (x), γCi (x)i is called intuitionistic
fuzzy G∗X closed(intuitionistic fuzzy GX P closed) iff U = Ci : Ci ∈ C, i ∈ J} is intuitionistic fuzzy G∗X open
(intuitionistic fuzzy GX P open)
Definition: 3.4. Let (X, G1 , G2 ) be an intuitionistic bifuzzy GX space.A collection of intuitionistic fuzzy
sets Ci = {hx, µCi (x), γCi (x)i : i ∈
fuzzy G∗X (intuitionistic
S J} with Ci ⊆ G1 ∪G2 is said to be an intuitionistic
∗
fuzzy GX P ) open cover for X iff i∈J Ci = 1∼ and Ci is an intuitionistic fuzzy GX (intuitionistic fuzzy GX P )
open.
Definition: 3.5. An intuitionistic bifuzzy GX space (X, G1 , G2 ) is said to be an intuitionistic fuzzy G∗X (intuitionistic
fuzzy GX P )compact iff every intuitionistic fuzzy G∗X (intuitionistic fuzzy GX P ) open cover of X has a finite
subcover.
Definition: 3.6. A collection of intuitionistic fuzzy sets U = {Ui : i ∈ J},where Ui = hx, µUi (x), γUi (x)i
is called an intuitionistic fuzzy α shading if for each x ∈ X,there exists an intuitionistic fuzzy set A =
hx, µA (x), γA (x)i in U with µA (x) > α and γA (x) < 1 − α where 0 ≤ α < 1.
A subcollection of intuitionistic fuzzy sets V of U that is also an intuitionistic fuzzy α shading is called
an intuitionistic fuzzy α subshading.
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Definition: 3.7. A collection of intuitionistic fuzzy sets U = {Ui : i ∈ J},where Ui = {hx, µUi (x), γUi (x)i :
x ∈ X} with U ⊆ G1 ∪ G2 is called intuitionistic fuzzy G∗X α shading (intuitionistic fuzzy GX P α shading)iff
for each x ∈ X,there exists an intuitionistic fuzzy set A = hx, µA (x), γA (x)i in U such that µA (x) > α and
γA (x) < 1 − α,where 0 ≤ α < 1 and U is intuitionistic fuzzy G∗X (intuitionistic fuzzy GX P )open.
Definition: 3.8. An intuitionistic bifuzzy GX space (X, G1 , G2 ) is called intuitionistic fuzzy G∗X α compact
(intuitionistic fuzzy GX P α compact)iff every intuitionistic fuzzy G∗X α shading(intuitionistic fuzzy GX P α
shading) has a finite intuitionistic fuzzy GX α subshading.
Definition: 3.9. Let X be a non empty set and α ∈ [0, 1].A collection of intuitionistic fuzzy sets P = {Ai :
i ∈ Λ} where Ai = {hx, µAi (x), γAi (x)i : x ∈ X} is said to be intuitionistic fuzzy GX α centered if for all
finite collections Ai ∈ P, i = 1, 2, ..., n, there exists x ∈ X with µAi (x) ≥ 1 − α and γAi (x) ≤ α for all
i ∈ {1, 2, ..., n}.
Definition: 3.10. Let (X, G1 ) and (Y, G2 ) be two intuitionistic fuzzy GX spaces.A function f : X → Y is
intuitionistic fuzzy GX continuous if inverse image of every intuitionistic fuzzy GX open set in (Y, G2 ) is
intuitionistic fuzzy GX open set in (X, G1 ).
Proposition: 3.1. An intuitionistic bifuzzy GX space (X, G1 , G2 ) is intuitionistic fuzzy G∗X α compact
(intuitionistic fuzzy GX P α compact)iff for every intuitionistic fuzzy GX α centered system P of intuitionistic
fuzzy G∗X closed(intuitionistic fuzzy GX P closed) sets in X,there exists x ∈ X and intuitionistic fuzzy set
A,such that µA (x) ≥ 1 − α and γA (x) ≤ α for all A ∈ P .
Proof.
=⇒ Let P be an intuitionistic fuzzy GX α centered collection of intuitionistic fuzzy G∗X closed sets in
X.Assume that for each x ∈ X, there is an intuitionistic fuzzy set A = {hx, µA (x), γA (x)i in P ,such that
µA (x) ≤ 1 − α and γA (x) ≥ α.Then U = {Ai : i ∈ Λ} where Ai = {hx, γA (x), µA (x)i : x ∈ X} is an
i
i
intuitionistic fuzzy G∗X α shading of X.Consequently there is a finite intuitionistic fuzzy GX α subshading
{Ai : i = 1, 2, ..., n} of X.Now {Ai : i = 1, 2, ..., n} is finite so there exists x0 ∈ X such that µAi (x0 ) ≥ 1 − α
and γAi (x0 ) ≤ α for all i.But this contradicts that {Ai : i = 1, 2, ..., n} is an intuitionistic fuzzy GX α
subshading,that is for each x ∈ X there is Ai such that γAi (x) > α and µAi (x) < 1 − α and so γA (x) > α
and µA (x) < 1 − α.Which is a contradiction.
⇐= Let U = {Ai : i ∈ Λ} where Ai = {hx, µAi (x), γAi (x)i : x ∈ X} be an intuitionistic fuzzy G∗X α
shading of X,such that no finite subcollection of U is an intuitionistic fuzzy GX α subshading of X.Hence
for every finite subcollection V of U ,there exists x0 ∈ X such that γA (x) ≤ α and µA (x) ≥ 1 − α for all
i
i
{Ai : i = 1, 2, ..., n} in V .Then {Ai : Ai ∈ U } is intuitionistic fuzzy G∗X closed α centered.Therefore there
exists x0 ∈ X such that γA (x) ≥ 1 − α and µA (x) ≤ α for all {Ai : i ∈ Λ} in U ,we have µAi (x) < α and
i
i
γAi (x) > 1 − α and this contradicts that U is intuitionistic fuzzy G∗X α shading.
Proposition: 3.2. An intuitionistic bifuzzy GX space (X, G1 , G2 ) is intuitionistic fuzzy G∗X α compact iff
(X, G1 , G2 ) is intuitionistic fuzzy GX P α compact and intuitionistic fuzzy GX α compact.
Proof.=⇒ Let U = {Ai : i ∈ Λ} where Ai = {hx, µAi (x), γAi (x)i : x ∈ X} be an intuitionistic fuzzy GX P α
shading of X.Then U is an intuitionistic fuzzy G∗X α shading of (X, G1 , G2 ) and so it has finite intuitionistic
fuzzy GX α subshading.
⇐= Let U = {Ai : i ∈ Λ} where Ai = {hx, µAi (x), γAi (x)i : x ∈ X} be an intuitionistic fuzzy GX P α
shading of X.Then either U ⊆ G1 or U ⊆ G2 or U is an intuitionistic fuzzy GX P α shading of X.Since
(X, G1 , G2 ) is an intuitionistic fuzzy GX P α compact and intuitionistic fuzzy GX α compact,so U has a
finite intuitionitic fuzzy GX α subshading.
Definition: 3.11. An intuitionistic fuzzy set A = hx, µA (x), γA (x)i in an intuitionistic fuzzy GX space
(X, G) is said to be intuitionistic fuzzy GX compact iff for every U = {Ci : i ∈ Λ} ⊆ G where Ci =
{hx, µCi (x), γCi (x)i : x ∈ X} such that A ⊆ sup{Ci : Ci ∈ U } and for every > 0,there exists a finite
subfamily U ⊆ U such that A − ⊆ sup{Ci : Ci ∈ U , i = 1, 2, ..., n}.
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Intuitionistic Bifuzzy Generalized Structure Space
Definition: 3.12. An intuitionistic fuzzy set A = hx, µA (x), γA (x)i in an intuitionistic bifuzzy GX space
(X, G1 , G2 ) is said to be intuitionistic fuzzy G∗X compact(intuitionistic fuzzy GX P compact) iff for every
family U = {Ui : i ∈ Λ} where Ui = {hx, µUi (x), γUi (x)i : x ∈ X} with U ⊆ G1 ∪ G2 of intuitionistic fuzzy
G∗X open(intuitionistic fuzzy GX P open) sets,such that A ⊆ sup{Ui : Ui ∈ U } and for every > 0,there
exists a finite subfamily U ⊆ U such that A − ⊆ sup{Ui : Ui ∈ U , i = 1, 2, ..., n}.
Proposition: 3.3. Let (X, G1 , G2 ) and (Y, L1 , L2 ) be two intuitionistic bifuzzy GX spaces and f : X → Y
be an intuitionistic fuzzy GX continuous surjection
(i) If X is intuitionistic fuzzy GX P α compact then Y is intuitionistic fuzzy GX P -α compact.
(ii) If X is intuitionistic fuzzy G∗X α compact then Y is intuitionistic fuzzy G∗X -α compact.
Proof.
(i) Let U = {Ai : i ∈ Λ} where Ai = {hy, µAi (y), γAi (y)i : y ∈ Y } be an intuitionistic fuzzy GX P α
shading of Y .Then f −1 (U ) = {f −1 (Ai ) : Ai ∈ U } is an intuitionistic fuzzy GX P -α shading of X because
if x ∈ X then f (x) ∈ Y so there exists A ∈ U such that µA (f (x)) > α and γA (f (x)) < 1 − α,that
is µf −1 (A) (x) > α and γf −1 (A) (x) < 1 − α.Hence {f −1 (A) : A ∈ U } has a finite intuitionistic fuzzy GX α
subshading {f −1 (Ai ) : i = 1, 2, ..., n}.Now {Ai : i = 1, 2, ..., n} is a finite intuitionistic fuzzy GX α subshading
of U because if y ∈ Y then y = f (x) for some x ∈ X thus there exists j,such that µf −1 (A ) (x) > α and
j
γf −1 (A ) (x) < 1 − α.This implies that Ai (f (x)) = Aj (f (x)) ⊃ α,that is µAi (f (x)) = µAj (f (x)) > α and
j
γAi (f (x)) = γAj (f (x)) < 1 − α.
(ii)Similar to the Proof of Proposition 3.3(i).
Proposition: 3.4. Let (X, G1 , G2 ) and (Y, L1 , L2 ) be two intuitionistic bifuzzy GX spaces and f : X → Y
be an intuitionistic fuzzy GX continuous surjection
(i) If X is intuitionistic fuzzy G∗X compact then Y is intuitionistic fuzzy G∗X compact.
(ii) If X is intuitionistic fuzzy GX P compact then Y is intuitionistic fuzzy GX P compact.
Proof.
(i) Let A = hy, µA (y), γA (y)i be an intuitionistic fuzzy set in Y and U = {Ui : i ∈ Λ} where Ui =
{hy, µUi (y), γUi (y) : y ∈ Y i} with U ⊆ G1 ∪ G2 be a family of intuitionistic fuzzy G∗X open sets in Y ,such
that A ⊆ sup{Ui : Ui ∈ U }.We have to show that for > 0 there exists a finite subfamily U ⊆ U such
that A − ⊆ sup{Ui : Ui ∈ U , i = 1, 2, ..., n}.Since f is intuitionistic fuzzy GX continuous,then f −1 (U ) =
{f −1 (Ui ) : Ui ∈ U } is a family of intuitionistic fuzzy G∗X open sets in X such that f −1 (A) ⊆ sup{f −1 (Ui ) :
Ui ∈ U } because f −1 (A)(x) = A(f (x)).Since X is an intuitionistic fuzzy G∗X compact,there exists a finite
subfamily {f −1 (Ui ) : Ui ∈ U , i = 1, 2, ..., n} such that f −1 (A) − ⊆ sup{f −1 (Ui ) : Ui ∈ U , i = 1, 2, ..., n}
which implies A − ⊆ sup{Ui : Ui ∈ U , i = 1, 2, ..., n}.Thus Y is intuitionistic fuzzy G∗X compact.
(ii) Similar to the Proof of Proposition 3.4(i).
Definition: 3.13. An intuitionistic bifuzzy GX space (X, G1 , G2 ) is intuitionistic fuzzy G∗X constant compact(intuitionistic fuzzy GX P constant compact) provided that each intuitionistic fuzzy constant map from
X into I is intuitionistic fuzzy G∗X compact(intuitionistic fuzzy GX P compact).
Proposition: 3.5. Let (X, G1 , G2 ) and (Y, L1 , L2 ) be two intuitionistic bifuzzy GX spaces and f : X → Y
be an intuitionistic fuzzy GX continuous surjection
(i) If X is intuitionistic fuzzy GX P constant compact then Y is intuitionistic fuzzy GX P constant compact.
(ii) If X is intuitionistic fuzzy G∗X constant compact then Y is intuitionistic fuzzy G∗X constant compact.
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Proof.
(i) Let c be an intuitionistic fuzzy constant map in Y and U = {Ui : i ∈ Λ} where Ui = hy, µUi (y), γUi (y)i
with U ⊆ G1 ∪ G2 be a family of intuitionistic fuzzy GX P open sets in Y ,such that c ⊆ sup{Ui : Ui ∈ U }.We
have to show that for > 0,there exists a finite subfamily U ⊆ U such that c − ⊆ sup{Ui : Ui ∈
U , i = 1, 2, ..., n}.Since f is an intuitionistic fuzzy GX continuous,then f −1 (U ) = {f −1 (Ui ) : Ui ∈ U }
is a family of intuitionistic fuzzy GX P open sets in X such that c ⊆ sup{f −1 (Ui ) : Ui ∈ U } because
f −1 (U )(x) = U (f (x)).Since X is intuitionistic fuzzy GX P constant compact,there exists a finite subfamily
{f −1 (Ui ) : Ui ∈ U , i = 1, 2, ..., n},such that c − ⊆ sup{f −1 (Ui ) : Ui ∈ U , i = 1, 2..., n} which implies
c − ⊆ sup{Ui : Ui ∈ U , i = 1, 2, ..., n}.Thus Y is intuitionistic fuzzy GX P constant compact.
(ii) Similar to the Proof of Proposition 3.5(i).
4
Good Extension
Definition: 4.1. A family G of generalized intuitionistic open sets in an intuitionistic topological space
(X, τ ) satisfying the following axioms:
(i) ∅∼ , X∼ ∈ G,
(ii) G1 ∩ G2 ∈ G ,for any G1 , G2 ∈ G,
S
(iii) Gi ∈ G,for any arbitrary family of generalized intuitionistic open sets {Gi : i ∈ J} ⊆ G.
Then G is called an intuitionistic generalized structure(in shortly intuitionistic GX ).The pair (X, G) is called
an intuitionistic generalized structure (in shortly intuitionistic GX ) space.The member of (X, G) are called
intuitionistic GX open.
The complement of an intuitionistic GX open set is an intuitionistic GX closed set.
Definition: 4.2. An intuitionistic biGX space (X, G1 , G2 ) is called an intuitionistic G∗X compact (intuitionistic GX P compact) iff every intuitionistic G∗X open(intuitionistic GX P open) cover of X has a finite
subcover.
Definition: 4.3. A property PA of an intuitionistic bifuzzy GX space is said to be a good extension of the
property P in classical intuitionistic biGX structure iff whenever the intuitionistic bifuzzy GX generated say
by (X, G1 , G2 ) then (X, ω(G1 ), ω(G2 )) has property PA iff (X, G1 , G2 ) has the property P .
Notation:A−1 (x) = hx, µA−1 (x), γA−1 (x)i
Note:If (X, G)be any classical intuitionistic generalized structure space then there are two intuitionistic
fuzzy generalized structure corresponding to G,namely X/G = {χA : A ∈ G} and ω(G)= the set of all lower
semicontinuous functions from X into the closed interval [0, 1].It is clear that X/G ⊆ ω(G).
Proposition: 4.1. Let (X, G1 , G2 ) be an intuitionistic biGX space.Then the following are equivalent:
(i) (X, G1 , G2 ) is intuitionistic G∗X compact
(ii) (X, ω(G1 ), ω(G2 )) is an intuitionistic fuzzy G∗X α compact,for each 0 ≤ α < 1 .
(iii) (X, ω(G1 ), ω(G2 )) is an intuitionistic fuzzy G∗X compact,for some 0 ≤ α < 1 .
Proof.
(i) ⇒ (ii) Let U = {Ai : i ∈ Λ} where Ai = hx, µAi (x), γAi (x)i be an intuitionistic fuzzy G∗X α shading
of (X, ω(G1 ), ω(G2 )).Then G = {A−1 (α, 1] : A ∈ U } is GS ∗ open cover for (X, G1 , G2 ) because for each
x ∈ X there exists A ∈ U such that µA (x) > α and γA (x) < 1 − α (that is A(x) ⊃ α) which implies
x ∈ A−1 (α, 1].Since (X, G1 , G2 ) is GS ∗ compact,G has a finite subcover {A−1
i (α, 1] : i = 1, 2, ..., n}.Now the
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Intuitionistic Bifuzzy Generalized Structure Space
set {Ai : i = 1, 2, ..., n} is an intuitionitic fuzzy GX α subshading for (X, ω(G1 ), ω(G2 )) because if x ∈ X
then there exists i ∈ {1, 2, ..., n} such that x ∈ A−1
i (α, 1] which implies µAi (x) > α and γAi (x) < 1 − α.
(ii) ⇒ (iii) Trivial.
(iii) ⇒ (i) Let V be an intuitionistic G∗X open cover of (X, G1 , G2 ),where (X, ω(G1 ), ω(G2 )) is an
intuitionistic fuzzy G∗X α compact for some α ∈ [0, 1).Then U = {χvi : vi ∈ V } is an intuitionistic fuzzy
G∗X α shading for (X, ω(G1 ), ω(G2 )) because if x ∈ X then there exists vi ∈ V such that µχv (x) > α and
i
γχv (x) < 1 − α and so there exists a finite intuitionistic fuzzy GX α subshading,say {χvi : vi ∈ V, i =
i
1, 2, ..., n}.Now the collection {vi : i = 1, 2, ..., n} is a finite subcover of V because if x ∈ X then there exists
i ∈ {1, 2, ..., n} such that µχv (x) > α and γχv (x) < 1 − α which implies that x ∈ vi .
i
i
Proposition: 4.2. Let (X, G1 , G2 ) be an intuitionistic biGX space.Then the following are equivalent:
(i) (X, G1 , G2 ) is an intuitionistic G∗X compact
(ii) There exists c : 0 < c ≤ 1 and c is an intuitionistic fuzzy G∗X compact in (X, ω(G1 ), ω(G2 )).
(iii) (X, ω(G1 ), ω(G2 )) is an intuitionistic fuzzy G∗X constant compact.
Proof.
(i) ⇒ (ii) Let b be an intuitionistic fuzzy constant set in X and U = {Ai : i ∈ Λ} where Ai =
hx, µAi (x), γAi (x)i be a family of intuitionistic fuzzy G∗X open sets of X in (X, ω(G1 ), ω(G2 )) such that
b ⊆ sup{A : A ∈ U } and let > 0.Then the family U = {A−1 (b − , 1] : A ∈ U } is an intuitionistic G∗X
open cover of X because if x ∈ X then there exists A ∈ U such that b − ⊆ A that is A(x) ∈ (b − , 1] or
x ∈ A−1 (b − , 1].Since X is intuitionistic G∗X compact,so U has a finite subcover V = {A−1
i (b − , 1] : i =
1, 2, ..., n}.Now b − sup{Ai : i = 1, 2, ..., n},which shows that b is intuitionistic fuzzy G∗X compact.
(ii) ⇒ (iii) Let b be an intuitionistic fuzzy constant set in X and U = {Ai : i ∈ Λ} where Ai =
hx, µAi (x), γAi (x)i be a family of intuitionistic fuzzy G∗X open sets in (X, ω(G1 ), ω(G2 )) such that b ⊆
sup{A : A ∈ U }.Let > 0 and c be an intuitionistic fuzzy G∗X compact in X.Since b ⊆ sup{A : A ∈ U } then
c ⊂ sup{A − b + c : A ∈ U }.Now {A − b + c : A ∈ U } is an intuitionistic fuzzy G∗X open and c is intuitionistic
fuzzy G∗X compact.Therefore there exists a finite subcover {Ai − b + c : i = 1, 2, ..., n} of c − .That is
c− ⊆ sup{Ai −b+c : i = 1, 2, ..., n}.This implies b− ⊆ c−+b−c ⊆ sup{Ai −b+c+b−c : i = 1, 2, ..., n}that
is b − ⊆ sup{Ai : i = 1, 2, ..., n}.consequently b is intuitionistic fuzzy G∗X compact.Hence (X, ω(G1 ), ω(G2 ))
is an intuitionistic fuzzy G∗X constant compact.
(iii) ⇒ (i) Let U be an intuitionistic G∗X open cover of (X, G1 , G2 ).Then G = {χv : v ∈ U } is a
family of intuitionistic fuzzy G∗X open sets such that hc, 0.5, 0.5i ⊆ sup{χv : v ∈ U }.Since hc, 0.5, 0.5i is an
intuitionistic fuzzy GX compact,there exists a finite subfamily {χvi : vi ∈ U, i = 1, 2, ..., n} ⊆ G such that
hx, 0.4, 0.4i ⊆ sup{χvi : vi ∈ U, i = 1, 2, ..., n}.Now {vi : i = 1, 2, ..., n} is a finite subcover of U for X because
if x ∈ X then there exists v ∈ U such that χvi ∈ G,that is χvi (x) = 1∼ ⊇ hc, 0.5, 0.5i for some i,that is
x ∈ χ−1
(0.4, 1].
v
i
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Acknowledgement
The authors are thankful to the referee for valuable suggestions in rewriting the paper in the present form.
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