Download On Intuitionistic Fuzzy Semi-Generalized Closed Sets and its

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 34, 1677 - 1688
On Intuitionistic Fuzzy Semi-Generalized Closed
Sets and its Applications
R. Santhi
Department of Mathematics, Nallamuthu Gounder
Mahalingam College, Pollachi - 642 001, India
[email protected]
K. Arun Prakash
Department of Mathematics, Kongu Engineering College
Perundurai - 638 052, India
Abstract. The purpose of this paper is to introduce and study the concepts of
intuitionistic fuzzy semi-generalized closed set and intuitionistic fuzzy semigeneralized open sets in intuitionistic fuzzy topological space. Also we study the
application of intuitionistic fuzzy semi-generalized closed sets namely
intuitionistic fuzzy semi-T1/2 space.
Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy semi-generalized
closed set, Intuitionistic fuzzy semi-generalized open set and Intuitionistic fuzzy
semi-T1/2 space.
Mathematics Subject Classification: 54A40, 03F55
1. Introduction
Fuzzy set (FS), proposed by Zadeh [13] in 1965, as a framework to
encounter uncertainty, vagueness and partial truth, represents a degree of
membership for each member of the universe of discourse to a subset of it. By
adding the degree of non-membership to FS, Atanassov proposed intuitionistic
fuzzy set (IFS) in 1983 [1] which looks more accurately to uncertainty
quantification and provides the opportunity to precisely model the problem based
on the existing knowledge and observations. Later on fuzzy topology was
introduced by Chang in 1967.After this, there have been several generalizations
of notions of fuzzy sets and fuzzy topology. In last few years various concepts in
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R. Santhi and K. Arun Prakash
fuzzy were extended to intuitionistic fuzzy sets. In 1997, Coker introduced the
concept of intuitionistic fuzzy topological space. In this paper, we introduce one
of the concept namely semi-generalized closed sets which were introduced
initially by P.Bhattacharyya and B.K.Lahiri in General Topology in 1987 [2].
After this many researchers worked on this set and developed many interesting
properties and applications. We have studied some of the basic properties
regarding it. We also
introduced application of intuitionistic fuzzy semigeneralized closed sets namely intuitionistic fuzzy semiT1/2 space and obtained
some characterizations and several preservation theorems of such spaces.
2. Preliminaries
Definition 2.1 [1] An intuitionistic fuzzy set (IFS) A in X is an object
having the form
A= {〈 x , μA (x) , γA (x) 〉 / x ∈ X }
where the functions μA : X Æ [0,1] and γA: X Æ [0,1] denote the degree of the
membership (namely μA(x)) and the degree of non- membership (namely γA(x) )
of each element x∈X to the set A respectively, 0 ≤ μA (x) + γA (x) ≤ 1 for each
x ∈ X.
Definition 2.2 [1] Let A and B be IFS’s of the forms
A = {〈 x , μA (x) , γA (x) 〉 / x ∈ X} and B= {〈 x , μB (x) , γB (x) 〉 / x ∈ X}.
Then
(a) A ⊆ B if and only if μA (x) ≤ μB (x) and γA (x) ≥ γB (x) for all x ∈ X.
(b) A = B if and only if A ⊆ B and B ⊆ A.
(c) Ā = {〈 x, γA (x), μA (x) 〉 / x ∈ X}.
(d) A ∩ B = {〈 x, μA (x) ∧ μB (x) , γA (x) ∨ γB (x)〉 / x ∈ X}.
(e) A ∪ B = {〈 x, μA (x) ∨ μB (x) , γA (x) ∧ γB (x)〉 / x ∈ X}.
(f) 0 = { x,0,1 : x ∈ X } and 1 = { x,1,0 : x ∈ X }.
~
~
(g) 1~ = 0 and 0~ = 1 .
~
~
Definition 2.3 [5] An intuitionistic fuzzy point (IFP), written as p(α,β) , is defined
to be an IFS of X given by
⎧(α , β ), x = p
p(α , β ) = ⎨
⎩(0,1), otherwise
Definition 2.4 [4] An intuitionistic fuzzy topology (IFT for short) on a nonempty
X is a family τ of IFSs in X satisfying the following axioms:
(i) 0~, 1~ ∈ τ,
(ii) G1 ∩ G2 ∈ τ for any G1, G2 ∈ τ,
(iii) ∪ Gi ∈ τ, for any arbitrary family {Gi | i ∈ J} ⊆ τ.
Intuitionistic fuzzy semi-generalized closed sets
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In this case the pair (X,τ) is called an intuitionistic fuzzy topological space
(IFTS for short) and any IFS in τ is known as an intuitionistic fuzzy open set
(IFOS for short) in X . The complement of an IFOS A in IFTS (X,τ) is called
an IFCS ( IFCS for short ) in X.
Definition 2.5 [4] Let (X,τ) be an IFTS and A = 〈 x ,μA ,γA〉 be an IFS in X. Then
the intuitionistic fuzzy interior and intuitionistic fuzzy closure of A are defined
by
int(A) = ∪ {G | G is an IFOS in X and G ⊆ A},
cl(A) = ∩ {K | K is an IFCS in X and A ⊆ K}.
Note that, for any IFS A in (X,τ), we have
cl (A ) = int ( A) and int (A ) = cl ( A)
Definition 2.6 An IFS A = {〈 x, μA (x), γA (x) 〉 / x ∈ X} in an IFTS (X,τ) is called
(i)
an intuitionistic fuzzy semi open set (IFSOS) if A ⊆ cl (int( A)) . [6]
(ii)
an intuitionistic fuzzy α-open set (IFαOS) if A ⊆ int (cl (int ( A))) . [6]
(iii) an intuitionistic fuzzy pre-open set (IFαOS) if A ⊆ int(cl ( A)) . [6]
(iv)
an intuitionistic fuzzy regular open set (IFROS) if int(cl ( A)) = A . [6]
(v)
an intuitionistic fuzzy semi-pre open set (IFSPOS) if there exists
B ∈ IFPO (X), such that B ⊆ A ⊆ cl (B ) . [11]
(vi)
an intuitionistic fuzzy generalized open set (IFGOS), if U ⊆ int( A) ,
whenever U ⊆ A and U is an IFCS [9]
(vii) an intuitionistic fuzzy regular generalized open set (IFRGOS), if
U ⊆ int( A) , whenever U ⊆ A and U is an IFRCS.[9]
An IFS A is called intuitionistic fuzzy semiclosed set , intuitionistic fuzzy α-closed set , intuitionistic fuzzy pre-closed set ,
intuitionistic fuzzy regular closed set, intuitionistic fuzzy semi-pre-closed set,
intuitionistic fuzzy generalized-closed set and intuitionistic fuzzy regular
generalized-closed set (IFCS , IFαCS, IFPCS, IFRCS, IFSPCS, IFGCS and
IFRGCS resp.) , if the complement of is an IFSOS , IFαOS, IFPOS , IFROS,
IFSPOS, IFGOS and IFRGOS respectively.
The family of all intuitionistic fuzzy semi-open ( resp. intuitionistic
fuzzy α-open , intuitionistic fuzzy pre-open , intuitionistic fuzzy regular-open,
intuitionistic fuzzy semi-preopen, intuitionistic fuzzy generalized-open and
intuitionistic fuzzy regular generalized-open ) sets of an IFTS (X,τ ) is denoted
by IFSO(X) ( resp. IFαO(X) , IFPO(X) , IFRO(X), IFSPO(X), IFGO(X) and
IFRGO(X) ).
Definition.2.7 [5] An IFTS(X,τ) is said to be intuitionistic fuzzy T1/2 space if
every intuitionistic fuzzy g-closed set in X is intuitionistic fuzzy closed in X.
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R. Santhi and K. Arun Prakash
Definition.2.8 [9] An IFTS(X,τ) is said to be intuitionistic fuzzy regular-T1/2
space if every intuitionistic fuzzy rg-closed set in X is intuitionistic fuzzy regular
closed in X.
Definition.2.9 [10] Two IFSs A and B in an IFTS (X,τ) are called q-separated if
0~
.
3. Intuitionistic fuzzy semi-generalized closed sets
In this section, we introduce intuitionistic fuzzy semigeneralized closed sets and studied some of their basic properties.
Definition 3.1 An IFS A of an IFTS (X, τ) is called an intuitionistic fuzzy
semi-generalized closed (IFSGCS for short) set if sc1(A) ⊆ U, whenever A ⊆ U
and U is an IFSOS.
The complement of an IFSGCS is called an intuitionistic fuzzy semigeneralized open set (IFSGOS for short).The family of all IFSGCS (resp,
IFSGOS) sets of an IFTS (X,τ) will be denoted by IFSGC(X) (resp. IFSGO(X))
Example 3.2 Let X= {a, b, c}
Let
,
,
.
,
,
.
Then τ = {0~, 1~, G1, G2} is an IFT on X.
,
, ,
0 ~ , 1~ ,
,
,
0,0.35 ,
,
,
, ,
0 ~ , 1~ ,
0.4,1 ,
α ,β
;
1 1
where H a, b,
c
,
,
,
,
,
;
, α2 ,β2 , α3 ,β3
=
.
,
.
.
0.4,1 ,
,
,
,
0,0.35 ,
0,0.4 ,
1,2,3
a
,
b
,
c
α1 α2 α3
,
0,0.3 ,
1,2,3
0,0.3 ,
1,
,
.
,
.
0.1,1 ,
1,
, ,
,
.
,
.
0.3,1 ,
,
,
,
.
,
.
0,0.4 ,
,
, ,
where
,
,
,
.
a
,
b
,
c
β1 β2 β3
Let A
,
, ,
,
, ,
be any IFS in X.
.
.
.
.
.
.
Then sc1 (A) = A .Hence A is an IFSGCS in X.
Theorem 3.3 Every IFCS in (X,τ) is an IFSGCS, but not conversely.
Proof: Obvious.
,
,
,
Example 3.4 Let X = {a, b} and let A
x,
.
.
.
.
0.3,1 ,
0.1,1 ,
Then τ = {0~, 1~, A} is an IFT on X.
Intuitionistic fuzzy semi-generalized closed sets
0 ~ , 1~ ,
,
,
0.4,1 ,
where
,
,
,
0 ~ , 1~ ,
where
,
,
,
;
0.3,1 ,
0,0.2 ,
,
,
,
,
0,0.5 ,
,
,
,
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1,
1,2
,
,
,
;
0,0.5 ,
0.3,1 ,
,
,
0,0.2 ,
0.4,1 ,
,
1,
1,2
,
Let
,
,
,
,
be any IFS in X.
.
.
.
.
Then
. Clearly
, whenever
X.Therefore B is an IFSGCS, but not an IFCS in X.
, for all IFSOS U in
Theorem 3.5 Every IFSCS in (X,τ) is an IFSGCS.
Proof: Obvious.
The converse of the Theorem.3.5 is not true as seen from the
following example.
Example 3.6 Let X = {a, b} and let A
Then τ = {0~, 1~, A} is an IFT on X.
,
, ,
0 ~ , 1~ ,
,
1,
where
,
,
,
0 ~ , 1~ ,
,
Let
,
,
,
,
,
.
,
.
.
,
.
,
.
0.4,0.6 , ,
,
,
,
,
,
,
,
.
0.4,0.6 ,
,
,
,
0.4,0.6 ,
where
;
,
.
1,2
,
,
x,
;
,
0.4,0.6 ,
,
1,
1,2
,
be any IFS in X.
Then
,
,
,
,
. Clearly
, whenever
.
.
.
for all IFSOS U in X. Therefore B is an IFSGCS in X, but B IFSCS(X).
,
Theorem 3.7 Every IFαCS in an (X,τ) is an IFSGCS, but not conversely.
Proof: In [4], it has been proved that every IFαCS is an IFSCS. By Theorem 3.5,
every IFSCS is an IFSGCS. Hence every IFαCS is an IFSGCS in X.
Example 3.8 Let X = {a, b} and let A
Then τ = {0~, 1~, A} is an IFT on X.
x,
.
,
.
,
.
,
.
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R. Santhi and K. Arun Prakash
0 ~ , 1~ ,
,
,
,
1,
where
,
,
,
0 ~ , 1~ ,
,
,
Let
,
Clearly
IFSGCS in X.
,
.
,
,
,
.
, whenever
0.4,0.6 , ,
0.4,0.6 ,
;
,
0.4,0.6 ,
,
1,
1,2
,
be any IFS in X.
, for all IFSOS U in X. Therefore B is an
.
, int cl B
,
,
,
int A
Clearly cl int cl B
,
,
,
,
,
,
,
0.4,0.6 ,
where
;
1,2
,
,
,
A
A , cl int cl B
cl A
A.
B. Hence B is not an IFαCS in X.
Theorem 3.9 Every IFSGCS in an IFT (X,τ) is an IFSPCS, but not conversely.
Proof: Follows from definitions.
Example 3.10 Let X = {a, b} and let A
Then τ = {0~, 1~, A} is an IFT on X.
,
, ,
0 ~ , 1~ ,
,
1,
Where
,
,
,
0 ~ , 1~ ,
,
,
,
0.3,0.5 ,
Where
Let
,
,
,
.
,
0~
,
.
But scl B
,
,
.
0.3,0.5 ,
,
0.1,0.4 ,
,
1,
1,2
,
,
.
. Hence B is an IFSPCS in X.
,
.
,
0.1,0.4 , ,
;
,
1~ , B
.
be any IFS in X.
.
0~
scl B
,
.
,
,
,
,
,
.
,
,
,
,
,
;
.
1,2
,
,
x,
,
.
.
,
,
.
.
,
.
.
,
.
.
. Therefore B is not an IFSGCS in X.
Theorem 3.11 Every IFRCS in an IFT (X,τ) is an IFSGCS, but not conversely.
Proof: In [4], it has been proved that every IFRCS is an IFCS. By Theorem 3.3,
every IFCS is an IFSGCS. Hence every IFRCS is an IFSGCS.
Intuitionistic fuzzy semi-generalized closed sets
1683
Example 3.12 In Example 3.6, the IFS B is an IFSGCS but not an IFRCS in X,
since
,
,
.
Remark 3.13 The concepts of IFGCS and IFSGCS are independent of each other
as seen from the following two examples.
Example 3.14 In Example 3.8, the IFS B is not an IFSGCS but an IFGCS in X,
since
1~
,
B U , U is an IFOS, cl B
U. Hence
B is an IFGCS in X.
Example 3.15 In Example 3.10, the IFS B is an IFSGCS but not an IFRCS in X,
since
,
B A , cl B
A. Hence B is an not IFGCS in X.
Remark 3.16 The concepts of IFPCS and IFSGCS are independent of each other
as seen from the following two examples.
Example 3.17 Let X = {a, b} and let A
Then τ = {0~, 1~, A} is an IFT on X.
,
, ,
0 ~ , 1~ ,
,
1,
Where
,
,
,
0 ~ , 1~ ,
,
,
,
0.3,0.6 ,
Where
Let
,
,
,
,
,
,
.
.
.
1~ only, also
0~
,
.
,
;
.
,
.
0.3,0.6 ,
,
.
.
,
1,
1,2
,
be any IFS in X.
1~ . Therefore B is an IFSGCS in X.
1~
.
. Therefore B is not an IFPCS in X.
.
,
,
0.2,0.4 ,
,
0~ ,
0~
Therefore C is an IFPCS in X.
1~
,
,
,
.
.
0.2,0.4 , ,
,
Example 3.18 In Example 3.17, consider an IFS
,
,
,
,
,
Clearly
but
.
,
,
,
,
;
,
.
1,2
,
,
x,
.
0~
.
,
.
.
.
,
.
,
.
,
.
.
,
. Hence C is not an IFSGCS in X.
Remark 3.19 The concepts of IFRGCS AND IFSGCS are independent of each
other as seen from the following two examples.
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R. Santhi and K. Arun Prakash
,
Example 3.20 In Example 3.17, consider an IFS B
cl
1~
,
1~
Therefore B is an IFRGCS in X.
,
,
1~
1~ .
.
,
but
IFSGCS in X.
,
.
.
,
.
,
.
,
,
.
,
,
1,
where
,
,
;
.
,
.
,
.
.
,
.
x,
0.5,1 ,
.
,
,
.
0.3,1 ,
,
.
.
0,0.2 ,
0,0.4 ,
1,2
,
,
,
,
,
0 ~ , 1~ ,
where
,
. Hence B is an IFRGCS but B is not an
.
Example 3.21 Let X = {a, b} and let A
Then τ = {0~, 1~, A} is an IFT on X.
0 ~ , 1~ ,
.
,
,
,
,
,
,
,
,
,
0,0.4 ,
0.5,1 ,
,
,
,
;
0,0.2 ,
0.3,1 ,
1,
1,2
,
,
,
,
be any IFS in X.
Let
,
.
.
.
Then
. Clearly
, whenever
, for all
IFSOS U in X.Therefore B is an IFSGCS in X.
cl
,
,
, where A is an IFROS.
Therefore B is not an IFRGCS in X.
Theorem 3.22 If A is an IFSGCS and A⊆B⊆scl A , then B is an IFSGCS.
Proof: Let U be an IFSOS such that B ⊆ U. Since A ⊆ B, A ⊆ U. But A is an
IFSGCS, so sc1 (A) ⊆ U. By hypothesis, B⊆ sc1 (A), then sc1 (B) ⊆ sc1 (A).
This implies sc(B) ⊆ U. Hence is an IFSGCS.
Theorem 3.23 If sint(A) ⊆ B ⊆ A and A is an IFSGOS in an IFTS (X,τ).Then B
is an IFSGOS.
Proof: If
, then
. Since is an
IFSGCS, then by Theorem 3.22, B is also an IFSGCS. Therefore B is an IFSGOS.
The following diagram shows the relationships between
IFSGCSs and some other sets. The reverse implications are not true in the
following diagram.
Intuitionistic fuzzy semi-generalized closed sets
IFGCS
IFRCS
IFCS
IFSPCS
IFRGCS
IFαCS
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IFSGCS
IFSCS
IFPCS
Theorem 3.24 An IFS A of an IFTS(X,τ) is an IFSGOS if and only if B ⊆
sint(A), whenever B is an IFSCS and B ⊆ A.
Proof: Necessity: Assume that A is an IFSGOS in X. Also let B be an IFSCS in
X, such that B ⊆ A. Then is an IFSOS in X, such that A
B . Since is an
. But
IFSGCS,
. Hence
. This
implies
.
Sufficiency: Assume that
, whenever B is an IFSCS and B ⊆ A.
, whenever
is an IFSOS and
. Therefore
Then
is an IFSGCS.This implies A is an IFSGOS.
Theorem 3.25 The concepts of IFSCS and intuitionistic fuzzy semi-open
coincide if and only if every intuitionistic fuzzy subset of X is an IFSGCS.
Proof: Sufficiency: Let A be an intuitionistic fuzzy subset of X, such that A ⊆ U,
where U is an IFSOS. Then U is IFSCS such that sc1 (A) ⊆ sc1 (U) = U. Hence
sc1 (A) ⊆ U. Therefore A is an IFSGCS.
Necessity: Let U be an IFSOS. By hypothesis every intuitionistic fuzzy subset of
X is an IFSGCS, therefore U is an IFSGCS. This shows that scl (U) ⊆ U. Thus U
is an IFCS.
Next let us assume that B is an IFCS. Then is an IFSOS. By hypothesis
is an IFSGCS, hence
. This implies is an IFSCS. Therefore
B is an IFSOS in X.
Theorem 3.26 Let A be an IFSGCS subset of (X,τ). Then scl (A)-A does not
contain any non-empty IFSCS.
Proof: Assume that A is IFSGCS. Let F be an non-empty IFSCS, such that
.
(i.e)
.
Therefore A
. Since is an IFSOS,
.
But we have,
. So
(i.e)
φ.Therefore F is empty.
Theorem 3.27 Let A be an IFSGCS in an IFTS (X,τ). Then scl(A) - A contains
no nonempty IFCS.
Proof: Follows from the Theorem 3.26.
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R. Santhi and K. Arun Prakash
Corollary 3.28 An IFSGCS A is an IFSCS if and if only if scl(A) - A is an
IFSCS.
Proof: Let A be an IFSGCS. If A is an IFSCS, then by Theorem 3.26,
scl (A) - A=φ. Therefore scl (A) - A is an IFSCS.
Assume that scl (A) - A is IFSCS. But A is an IFSGCS. By Theorem 3.21,
scl (A) - A=φ. Hence scl (A) = A. A is an IFSCS.
Theorem 3.29 If A is an IFSOS and IFSGCS in (X,τ), then A is an IFSCS in X.
Proof: Since A is an IFSOS and IFSGCS in X, scl (A) ⊆ A. But we have A ⊆
scl(A). This implies scl(A) =A. Therefore A is an IFSCS in X.
4. Applications of Intuitionistic fuzzy semi-generalized closed sets
In this section, we introduce intuitionistic fuzzy T1/2 space, which
utilizes intuitionistic fuzzy semi-generalized closed sets and its characterizations
are proved.
Definition 4.1 An IFTS (X, τ) is called an intuitionistic fuzzy semi-T1/2 space, if
every IFSGCS is an IFSCS.
Theorem 4.2 An IFTS (X, τ) is an intuitionistic fuzzy semi-T1/2 space if and only
if
IFSOS(X) = IFSGOS(X).
Proof: Let A be an IFSGOS in X, then is an IFSGCS in X. By hypothesis is
an IFSCS of X, and therefore A is an IFSOS of X. Hence IFSOS(X) =
IFSGOS(X).
Converse: Let A be an IFSGCS in X, then
is an IFSGOS of X. By our
assumption is an IFSOS in X, which in turn implies A is an IFSCS in X. Hence
(X, τ) is an intuitionistic fuzzy semi-T1/2 space.
Theorem 4.3 For an IFTS(X,τ) the following conditions are equivalent:
(i)
(X,τ) is an intuitionistic fuzzy semi-T1/2 space.
(ii)
Every singleton set of X is either an IFSCS or IFSOS.
Proof: (i) => (ii). Assume that (X,τ) is an intuitionistic fuzzy semi-T1/2 space.
Suppose that {x} is not an IFSCS for some x∈X. Then X-{x} is not IFSOS and
hence X is the only IFSOS containing X-{x}.Therefore, X-{x} is IFSGCS in
(X,τ).Since (X,τ) is an intuitionistic fuzzy semi-T1/2 space, then X-{x} is IFSCS
or equivalently {x} is an IFSOS.
(ii) => (i). Assume that every singleton set of X is either IFSCS or IFSOS. Let A
be an IFSGCS of (X,τ). Let x∈X. We show that x∈A in two cases.
Case(i): Assume that {x} is IFSCS. If x ∉ A, then x ∈ scl (A)-A. Now scl (A)A contains a non - empty IFSCS. Since A is IFSGCS, by Theorem 3.21, we
arrived to a contradiction. Hence x∈A.
Intuitionistic fuzzy semi-generalized closed sets
1687
Case (ii): Assume that {x}is IFSOS. Since x∈scl(A), then {x}∩ A ≠ φ. So x ∈A.
Thus in any case x∈A. So scl(A)⊆ A. Therefore A = scl (A) or equivalently A is
an IFSCS. Thus every IFSGCS is IFCS. Hence (X,τ) is intuitionistic fuzzy semiT1/2 space.
Theorem 4.4 Every intuitionistic fuzzy T1/2 space is an intuitionistic fuzzy semiT1/2 space.
The following example supports that an intuitionistic fuzzy semi-T1/2
space need not be an intuitionistic fuzzy T1/2 space.
Example 4.5 Let X ={a, b}
Let A
x,
,
,
,
.
.
.
.
Then τ ={0~,1~,A} be an IFT on X. Clearly (X,τ) is an intuitionistic fuzzy semiT1/2 space, but not an intuitionistic fuzzy T1/2 space.
Theorem.4.6. Every intuitionistic fuzzy regular T1/2 space is an intuitionistic
fuzzy semi-T1/2 space.
Proof: In ([5].Remark.5.3), it has been proved that every intuitionistic fuzzy
regular T1/2 space is an intuitionistic fuzzy T1/2 space. ByTheorem.4.3, every
intuitionistic fuzzy T1/2 space is an intuitionistic fuzzy semi-T1/2 space. Hence
every intuitionistic fuzzy regular T1/2 space is an intuitionistic fuzzy semi-T1/2
space
Example 4.7 Let X ={a ,b}and let A
x,
,
,
,
.
.
.
.
Then τ ={0~,1~,A} be an IFT on X. Clearly (X,τ) is an intuitionistic fuzzy semiT1/2 space, but not an intuitionistic fuzzy regular T1/2 space.
Theorem 4.9 Let (X, τ) be an IFTS and X is an intuitionistic fuzzy semi-T1/2
space, then the following conditions are equivalent
(i) A IFSGO X (ii) A ⊆ cl intA (iii) A IFRC X .
Proof: (i) ⇒ (ii): Let A be an IFSGOS in X. Since X is an intuitionistic fuzzy
semi-T1/2 space, A is an IFSOS in X. Hence by definition of IFSOS, A ⊆ cl intA .
(ii) ⇒ (iii): Assume that A ⊆ cl intA , then A cl intA . Hence A is an IFRCS
and A
IFRC X .
(iii)⇒ (i): Assume that
, then A cl intA . Since A cl intA ,
A is an IFSOS and hence A is an IFSGOS, therefore
.
Theorem 4.10 Let A and B be q-separated IFSGOS’s in (X, τ), then A B is also
an IFSGOS in X is an intuitionistic fuzzy semi T1/2 space.
Proof: Let F be an IFSCS such that F ⊆ A B. Then F cl A ⊆ A, since
B cl A
0~ . Since A is an IFSOS in X, F cl A ⊆ sint A . Similarly
F cl B ⊆ sint B . Now F F
A B ⊆ F cl A
F cl B ⊆ sint A
1688
sint B ⊆ sint A
is an IFSGOS.
R. Santhi and K. Arun Prakash
B . Therefore F ⊆ sint A
B . Hence by theorem.3.19, A
B
Theorem 4.11 Let A and B be two IFSGCS of an ITFS (X, τ) and suppose that
and are q-separated. Then
is an IFSGCS if X is an intuitionstic fuzzy
semi-T1/2 space.
Proof: Assume that A and B are IFSGCS, then
and
are q-separated
IFSGOS. By theorem 4.10,
is an IFSGOS. Hence A B is an IFSGOS,
which in turn implies that A B is an IFSGOS in X.
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Received: January, 2010