A NOTE ON CONNECTEDNESS IN INTUITIONISTIC FUZZY SPECIAL TOPOLOGICAL SPACES

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©Hindawi Publishing Corp.

A NOTE ON CONNECTEDNESS IN INTUITIONISTIC FUZZY SPECIAL TOPOLOGICAL SPACES

SELMA ÖZÇA˘G and DO˘GAN ÇOKER (Received 31 July 1998)

Abstract.We prove some properties of several types of connectedness deﬁned in intuitionistic fuzzy special topological spaces.

Keywords and phrases. Intuitionistic fuzzy special set, intuitionistic fuzzy special topological space, connectedness.

2000 Mathematics Subject Classiﬁcation. 54A99, 03E99.

1. Introduction. After the introduction of the concept of fuzzy sets by Zadeh [12], several researches were conducted on the generalizations of the notion of fuzzy set.

The idea of “intuitionistic fuzzy set” was ﬁrst given by Atanassov [2, 3]. Later this concept is generalized to intuitionistic sets in Çoker [6] and intuitionistic topological spaces in [5, 9, 10]. An introduction to connectedness in these spaces is given in [10].

2. Preliminaries. First we present the fundamental deﬁnitions (see [6]).

Deﬁnition2.1(cf. [5, 9]). LetXbe a nonempty ﬁxed set. An intuitionistic fuzzy special set (IFSS for short)Ais an object having the formA= x,A1,A2, whereA1

andA2are subsets ofXsatisfyingA1∩A2= ∅. The setA1is called the set of members ofA, whileA2is called the set of nonmembers ofA.

The reader may consult [6, 9] to see several types of relations and operations on IFSS’s, and intuitionistic fuzzy special points (IFSP’s for short) and vanishing intuitionistic fuzzy special points (VIFSP’s for short).

Deﬁnition2.2(cf. [5, 7, 8, 9, 10, 11]). An intuitionistic fuzzy special topology (IFST for short) on a nonempty setX is a family τ of IFSS’s inX containing∅

∼, X and closed under ﬁnite inﬁma and arbitrary suprema. In this case the pair(X,τ)is∼

called an intuitionistic fuzzy special topological space (IFSTS for short) and any IFSS inτ is known as an intuitionistic fuzzy special open set (IFSOS for short) inX. The complement ¯Aof an IFSOSAin an IFSTS(X,τ)is called an intuitionistic fuzzy special closed set (IFSCS for short) inX.

Using a similar construction as in [7], one can easily deﬁne the interior and closure operators in IFSTS’s.

3. Types of connectedness in intuitionistic fuzzy special topological spaces.

Throughout this section(X,τ)and(Y ,Φ)will always denote IFSTS’s. Here we deﬁne several types of connectedness in IFSTS’s.

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Notice that two IFSS’sAandBin(X,τ)are said to be weakly separated, if cl(A)⊆B¯ and cl(B)⊆A; and¯ q-separated, if cl(A)∩B= ∅

∼ =A∩cl(B).

Lemma3.1.

A∩B= ∅_∼ ⇒A⊆B;¯ (3.1)

AB¯⇒A∩B≠∅

∼. (3.2)

Deﬁnition3.1(cf. [1, 10, 11]). Let(X,τ)be an IFSTS inX.

(a)X is calledCS-disconnected, if there exist weakly separated nonzero IFSS’sA and B in (X,τ) such thatX_∼ =A∪B. (X,τ) is calledCS-connected, if(X,τ)is not CS-disconnected.

(b)Xis calledCM-disconnected, if there existq-separated nonzero IFSS’sAandBin (X,τ)such thatX_∼=A∪B.Xis calledCM-connected, ifXis notCM-disconnected.

The idea ofCi-connectedness in fuzzy topological spaces and in intuitionistic fuzzy topological spaces (see [1, 11]) can be generalized to the intuitionistic case.

Deﬁnition3.2(cf. [10]). LetNbe an IFSS in(X,τ).

(a) If there exist IFSOS’sMandW inXsatisfying the following properties, thenN is calledCi-disconnected(i=1,2,3,4).

C1: N⊆M∪W,M∩W⊆N,¯ N∩M≠∅_∼,N∩W≠∅_∼, C2: N⊆M∪W,M∩W∩N= ∅

∼,N∩M≠∅

∼,N∩W≠∅

∼, C3: N⊆M∪W,M∩W⊆N,M¯ N,¯ WN,¯

C4: N⊆M∪W,M∩W∩N= ∅_∼,MN,¯ WN.¯

(b) N is said to be Ci-connected (i =1,2,3,4) if N is not Ci-disconnected (i= 1,2,3,4).

Corollary3.1. P, Qare weakly separated if and only if ∃M, W ∈τ such that P⊆M,Q⊆W,P⊆W¯, andQ⊆M.¯

Proof. (⇐)Suppose there existM,W∈τ such thatP⊆M, Q⊆W, P⊆W, and¯ Q⊆M. Then cl(P)¯ ⊆cl(W )¯ =W¯(since ¯Wis an IFSCS) and cl(Q)⊆cl(M)¯ =M¯⇒cl(P)⊆ W¯⊆Q¯⇒cl(P)⊆Q¯and cl(Q)⊆M¯⊆P¯⇒cl(Q)⊆P¯⇒P,Qare weakly separated.

(⇒)Let cl(P)⊆Q, cl(Q)¯ ⊆P. Now take¯ W=cl(P)andM=cl(Q)which are IFSOS’s in (X,τ). Hence ¯W⊆Q¯and ¯M⊆P¯⇒P⊆M,Q⊆W. We also haveW=cl(P)⊆P¯⇒P⊆W¯ andM=cl(Q)⊆Q¯⇒Q⊆M.¯

Here we deﬁneCS-connectedness andCM-connectedness of an IFSS in(X,τ).

Deﬁnition3.3(cf. Ajmal-Kohli [1]). An IFSSNin(X,τ)is said to beCS-disconnected (CM-disconnected) if and only if there are two nonempty weakly separated (q- separated) IFSS’s A and B in (X,τ)such that N=A∪B. N is calledCS-connected (CM-connected) if and only ifNis notCS-disconnected (CM-disconnected).

Theorem3.1. IfNisC3-connected, thenNisCM-connected.

Proof. LetNbeCM-disconnected. Then there exist IFSS’sA,Bsuch thatN=A∪B, A,B≠∅_∼ andA,Bareq-separated. LetP=cl(A)andQ=cl(B). ThenP,Qare IFSOS’s.

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Now

cl(A)∩cl(B)⊆A¯∩B¯=A∪B=N¯ ⇒N

⊆cl(A)∩cl(B)=cl(A)∪cl(B)

=P∪Q ⇒N⊆P∪Q,

(3.3)

P∩Q=cl(A)∩cl(B)=cl(A)∪cl(B)=cl(A∪B)⊆A∪B=N¯ ⇒P∩Q⊂N.¯ (3.4) IfP⊆N, then¯ N⊆cl(A)⇒N∩B= ∅_∼ (since cl(A)∩B= ∅_∼)andN∩B=(A∪B)∩B= B= ∅

∼. This is a contradiction. HencePN¯ follows.QN¯ can be proved similarly.

Theorem3.2. IfNisC1-connected, thenNisCS-connected.

Proof. LetNbeCS-disconnected. Then there exist IFSS’sA,Bsuch thatN=A∪B, A,B≠∅

∼ andA,Bare weakly separated. LetP=cl(A)andQ=cl(B). ThenP,Qare IFSOS’s. We have seen thatN⊆P∪QandP∩Q⊆N. If¯ P∩N= ∅_∼, thenP⊆N¯⇒N⊆ P¯⇒N⊆cl(A)⊆B¯⇒N⊆B. Since¯ N=A∪BandA∪B⊆B, we obtain a contradiction.¯ HenceP∩N≠∅

∼ follows. Similarly, it can be proved thatQ∩N≠∅

∼. Theorem3.3. IfNisCS-connected, thenNisC2-connected.

Proof. Suppose, on the contrary, that N is C2-disconnected. Hence there exist IFSOS’sM, W such thatN⊆M∪W, N∩M∩W = ∅

∼,N∩M ≠∅

∼, N∩W ≠∅

∼. Now, takeP=N∩M andQ=N∩W. SinceN⊆M∪W, we getN=N∩(M∪W )=(N∩ M)∪(N∩W )=P∪Q. We show thatPandQare weakly separated. LetP⊆M,Q⊆W.

Suppose thatPW. Then¯ P∩W ≠∅_∼ ⇒(N∩M)∩W≠∅_∼, a contradiction, in other wordsP⊆W¯ follows. Similarly one can also show thatQ⊆M. Thus¯ P,Qare weakly separated, which is a contradiction. ThereforeNisC2-connected.

Theorem3.4. IfNisCS-connected, thenNisC3-connected.

Proof. Similar to the previous one.

CS-connectedness does not implyC1-connectedness in general:

Counterexample3.1. LetX= {a,b,c}andτ= {∅

∼,X_∼,A1,A2,A3}where A1=

x,{c},{a,b}

, A2=

x,{a},{b,c}

, A3=

x,{a,c},{b}

. (3.5) IfN= x,{a},{b}, thenNisCS-connected, since there exist no two nonempty weakly separated IFSS’sA,B≠∅

∼ such thatN=A∪B. ButNisC1-disconnected.

IfNisC2-connected (C3-connected), thenNmay not beCS-connected.

Counterexample3.2. LetX= {a,b,c,d}andτ= {∅

∼,X

∼,A1,A2,A3,A4}, where A1=

x,{c},{a,b}

, A3=

x,{a},{b}

,

A2=

x,{a,c},{b}

, A4=

x,∅,{a,b}

. (3.6)

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Now takeN= x,{a},{b,c}.NisC2-connected (C3-connected) but notCS-connected, since there exist two nonempty weakly separated IFSS’sA,B≠∅

∼ such thatN=A∪B;

namely

A=

x,∅,{a,b,c}

, B=

x,{a},{b,c}

. (3.7)

C2-connectedness does not implyCM-connectedness in general as shown below.

Counterexample3.3. LetX= {a,b,c}andτ= {∅_∼,X_∼,A1,A2,A3,A4}, where A1=

x,{b},{c}

, A3=

x,{b,c},∅

,

A2=

x,{c},{a}

, A4=

x,∅,{a,c}

. (3.8)

N= x,{c},{a}isC2-connected, but notCM-connected, sinceNcan be expressed as the join of two nonemptyq-separated IFSS’s

A=

x,{c},{a,b}

, B=

x,∅,{a,c}

. (3.9)

Similarly,CM-connectedness does not implyC3- (C4-)connectedness in general:

Counterexample3.4. LetX= {a,b,c}andτ= {∅

∼,X

∼,A1,A2,A3}, where A1=

x,{c},{a,b}

, A2=

x,{a},{b,c}

, A3=

x,{a,c},{b}

. (3.10) LetN= x,{a},{b}.NisCM-connected, since there exist no two nonemptyq-separated IFSS’sA,B≠∅

∼ such thatN=A∪B. ButNisC3-disconnected (C4-disconnected).

IfNisC4-connected, thenNmay not beCM-connected.

∼,X

∼,A1,A2,A3,A4}, where A1=

x,{c},{a,b}

, A3=

x,{a},{b}

,

A2=

x,{a,c},{b}

, A4=

x,∅,{a,b}

. (3.11)

IfN= x,{a},{b,d}, thenNisC4-connected, but notCM-connected. This is because, Ncan be expressed as the join of two nonemptyq-separated IFSS’sAandB, where

A=

x,∅,{a,b,d}

, B=

x,{a},{b,c,d}

. (3.12)

Now, we summarize the relations between several types of connectedness.

C1-connectedness

//

_C_S-connectedness

//

C2-connectedness

CM-connectedness C3-connectedness

oo //

_C₄-connectedness.

(3.13)

None of these implications are reversible, as given here and in [10]. The following example shows that the closure ofC1- (C2-)connected IFSS need not beC1-connected (C2-connected).

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Counterexample3.6. LetX= {a,b,c,d}andτ= {∅_∼,X_∼,A1,A2,A3}, where A1=

x,{a,b},{c,d}

, A2=

x,{d},{a,b}

, A3=

x,{a,b,d},∅

. (3.14) IfN= x,{b},{c,d}, thenNisC1-connected (C2-connected), but cl(N)isC1-disconnected (C2-disconnected).

Theorem3.5. The closure of C3-connected (C4-connected) IFSS is C3-connected (C4-connected)

Proof. LetN beC3-connected, but cl(N) beC3-disconnected. Hence there exist IFSOS’sM,W≠∅_∼ such that cl(N)⊆M∪W,M∩W⊆cl(N),Mcl(N),Wcl(N). We easily deduceN⊆cl(N)⊆M∪W andM∩W⊆cl(N)⊆N. Since¯ NisC3-connected, M⊆N¯ orW⊆N¯follows. IfM⊆N, then¯ N⊆M¯⇒cl(N)⊆cl(M)¯ =int(M)=M, i.e.,¯ cl(N)⊆M¯ orM⊆cl(N). But this is a contradiction to the factMcl(N). Similarly, we obtain a contradiction in caseW⊆N. Therefore cl(N)¯ is alsoC3-connected. The other case can be proved similarly.

Theorem3.6. IfNisC3-connected (C4-connected) IFSS in(X,τ)andN⊆P⊆cl(N), thenP isC3-connected (C4-connected) IFSS in(X,τ), too.

Proof. Assume the contrary and letM,Wbe IFSOS’s inXsuch thatN⊆P⊆M∪W, M∩W⊆P¯⊆N. Since¯ N is C3-connected,M⊆N¯ orW ⊆N¯follows. If M⊆N, then¯ N⊆M¯ ⇒cl(N)⊆cl(M)¯ =int(M)=M¯ ⇒cl(N)⊆M. On the other hand, if¯ N⊆W,¯ then cl(N)⊆cl(W )¯ =int(W )=W¯ ⇒cl(N)⊆W.¯ P ⊆cl(N)⊆M¯ and P⊆cl(N)⊆W.¯ ThereforePisC3-connected.

This theorem fails in the cases ofC1- (C2-)connectedness as shown by the following example.

∼,X

∼,A1,A2,A3}, where A1=

x,{a,b},{c,d}

, A2=

x,{d},{a,b}

, A3=

x,{a,b,d},∅

. (3.15) IfN= x,{a},{c,d}, thenNis C2-connected. If we take the IFSSP = x,{a},{d}, then P satisﬁes the inclusions N⊆ P ⊆cl(N), and P is not C2-connected. On the other hand, if we consider theC1-connected IFSSN= x,{b},{c,d}in(X,τ), then P= x,{b},{d}satisﬁes the inclusionsN⊆P⊆cl(N), but it is notC1-connected.

Theorem3.7. IfN1and N2 are intersectingC1-connected IFSS’s, then N1∪N2 is alsoC1-connected.

Proof. Assume thatN1∪N2is C1-disconnected. Thus there exist IFSOS’sM and Wsuch thatN1∪N2⊆M∪WandM∩W⊆N1∪N2,(N1∪N2)∩M≠∅_∼ and(N1∪N2)∩

W ≠∅_∼. SinceN1and N2 areC1-connected, then (N1∩M = ∅_∼ orN1∩W = ∅_∼)and (N2∩M= ∅

∼ orN2∩W= ∅

∼)follow. SinceN1∩N2≠∅

∼,∃p

≈∈(N1∩N2), there exist four cases:

Case1. LetN1∩M= ∅_∼ andN2∩M= ∅_∼. In this case we get(N1∩M)∪(N2∩M)= (N1∪N2)∩M= ∅_∼, a contradiction.

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Case2. LetN1∩M= ∅_∼ andN2∩W= ∅_∼. Thenp

≈∉M,p

≈∉W. But this is impossible, sincep

≈∈N1∪N2⊆M∪W.

Case3and Case4. N1∩W= ∅_∼ andN2∩M= ∅_∼, orN1∩W= ∅_∼ andN2∩W= ∅_∼. These cases may be treated similarly.

Hence it is seen thatN1∪N2isC1-connected.

Theorem3.8. IfN1and N2 are intersectingC2-connected IFSS’s, then N1∪N2 is alsoC2-connected.

Proof. Assume thatN1∪N2isC2-disconnected. Then there exist IFSOS’sM and W such thatN1∪N2⊆M∪W (N1∪N2)∩M∩W = ∅_∼, (N1∪N2)∩M≠∅_∼ and(N1∪ N2)∩W≠∅_∼. SinceN1∩N2≠∅_∼,∃p

≈∈N1∩N2, and sinceN1andN2areC2-connected, then(N1∩M= ∅

∼ orN1∩W= ∅

∼)and(N2∩M= ∅

∼ orN2∩W= ∅

∼).

Case1. LetN1∩M= ∅_∼andN2∩M= ∅_∼. Then(N1∪N2)∩M=(N1∩M)∪(N2∩M)=

∅∼, a contradiction.

Case2. LetN1∩M= ∅

∼ andN2∩W= ∅

∼. Then we obtainp

≈∉M,p

≈∉Wa contradiction top

≈∈N1∪N2⊆M∪W.

Case3and Case4. They are similar to the ones given above.

HenceN1∪N2isC2-connected.

Deﬁnition 3.4. Two IFSS’s A and B are said to be overlapping, if N1 N2. Conversely,N1andN2are said to be nonoverlapping, ifN1⊆N2.

Notice that

N1N2⇐⇒N₁⁽¹⁾N₂⁽²⁾orN₁⁽²⁾N₂⁽¹⁾

⇐⇒ ∃x

x∈N₁⁽¹⁾,x∉N₂⁽²⁾

or∃y

y∈N₂⁽¹⁾,y∉N₁⁽²⁾

⇐⇒ ∃x

x∼∈N1,x

≈∈N2

or∃y

y∼∈N2,y

≈∈N1

.

(3.16)

Theorem3.9. IfN1andN2are overlappingC3-connected IFSS’s, then so isN1∪N2. Proof. LetN1∪N2beC3-disconnected. Then there exist IFSOS’sM and W such thatN1∪N2⊆M∪W,M∩W⊆N1∪N2,MN1∪N2,WN1∪N2. SinceN1andN2

are overlapping, ∃x(x

∼ ∈N1,x

≈ ∈N2)or∃y(y

∼ ∈N2,y

≈ ∈N1). Since N1 and N2 are C3-connected, then we obtain:(M⊆N1orW⊆N1)and(M⊆N2orW⊆N2).

Case1. LetM⊆N1 andM⊆N2. ThenM⊆N1∩N2=N1∪N2, a contradiction to MN1∪N2⇒

Case2. Let M⊆N1 and W ⊆N2. Now suppose that ∃x(x_∼ ∈N1,x_≈ ∈N2). From M⊆N1 andW ⊆N2, we obtainN1∪N2⊆M∪W⊆N1∪N2=N1∩N2⇒N1∩N2⊆ N1∪N2=N1∩N2. Butx

∼∈N1,x

≈∈N2⇒x

≈∈N1⇒x

≈∈N2⇒x

≈∈N1∩N2⊆N1∩N2⇒ x_≈∈N1,x_≈∈N2means a contradiction. Similarly, if∃y(y

∼∈N2,y

≈∈N1), we arrive at a contradiction again.

Case3and Case4. They are similar to the previous ones.

Hence it follows thatN1∪N2is alsoC3-connected.

Theorem3.10. IfN1andN2are overlappingC4-connected IFSS’s, then so isN1∪N2.

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Using the last two theorems we get the following lemmas immediately:

Lemma3.2. IfN1andN2areC3-connected IFSS’s such that[](N1∩N2)≠∅

∼, then N1∪N2isC3-connected, too.

Proof. For IFSSA, the set[]Awas deﬁned as[]A= x,A1,A2ifA= x,A1,A2. If [](N1∩N2)≠∅

∼, then we see thatN₁⁽¹⁾∩N₂⁽¹⁾≠φ, i.e.,∃x∈N₁⁽¹⁾∩N₂⁽¹⁾⇒x∈N₁⁽¹⁾and x∈N₂⁽¹⁾⇒x

∼∈N1andx∉N₂⁽²⁾⇒x

∼∈N1andx

≈∈N2, i.e.,N1andN2are overlapping.

Hence, the required result follows from a previous theorem.

Lemma3.3. IfN1andN2areC4-connected IFSS’s such that[](N1∩N2)≠∅_∼, then N1∪N2isC4-connected, too.

Now, we give generalized versions of these theorems. Here, a family(Ni)i∈Jof IFSS’s is said to be nonoverlapping if and only if for eachi∈J,Niand∩j≠iNjare nonoverlapping, i.e.,Ni⊆ ∩j≠iNj.

Theorem3.11. Let(Ni)i∈J be a family ofC1-connected IFSS’s such that∩Nj≠∅

∼. Then∪NiisC1-connected, too.

Proof. LetN= ∪NibeC1-disconnected. Then there exist IFSOS’sMandW such thatN⊆M∪W,M∩W⊆N,¯ N∩M≠∅_∼,N∩W≠∅_∼.

Now consider any indexi0∈J. SinceNi₀ isC1-connected, we haveNi₀∩M= ∅_∼ or Ni0∩W= ∅_∼. Hence there exist three cases:

Case1. LetNi∩M= ∅_∼ for eachi∈J. Then, we may write downN∩M=(∪Ni)∩M=

∪(Ni∩M)= ∪∅_∼ = ∅_∼, which is a contradiction.

Case2. LetNi∩W= ∅

∼ for eachi∈J. Then we obtain a similar contradiction.

Case3. Let Ni∩M= ∅

∼ for each i∈J1and Ni∩W = ∅

∼ for eachi∈J2, where J=J1∪J2 and J1≠∅,J2≠∅. Since∩Nj≠∅_∼,∃p

≈ ∈ ∩Nj. in this case we getp

≈∉ M and p

≈ ∉W, which is a contradiction with p

≈ ∈N⊆M∪W. Therefore, N is also C1-connected.

Theorem3.12. Let(Ni)i∈J be a family ofC2-connected IFSS’s such that∩Nj≠∅_∼. Then∪NiisC2-connected, too.

Theorem3.13. Let(Ni)i∈J be an overlapping family ofC3-connected IFSS’s. Then

∪NiisC3-connected, too.

Proof. LetN= ∪NibeC3-disconnected. Then there exist IFSOS’sMandW such thatN⊆M∪W,M∩W⊆N,¯ MN,¯ WN. Now consider any index¯ i∈J. SinceNiis C3-connected, we haveM⊆NiorW⊆Ni. Since(Ni)is an overlapping family, suppose further that∃i0∈Jsuch that

∃x

x∼∈Ni0,x

≈∈ ∩

j≠i₀Nj

or ∃y

y∼∈ ∩

j≠i₀Nj,y

≈∈Ni0

. (3.17)

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Hence there exist three cases:

Case1. LetM⊆Nifor eachi∈J. Then we may write downM⊆ ∩Ni= ∪Ni=N, which is an obvious contradiction.

Case2. LetW⊆Nifor eachi∈J. Then we obtain a similar contradiction.

Case3. LetM⊆Nifor eachi∈J1andW⊆Nifor eachi∈J2, whereJ=J1∪J2and J1≠∅,J2≠∅. Hence

N⊆M∪W⊆

i∈J∩1Ni

∪

i∈J∩2Ni

=

∪Ni i∈J1

∪

∪Ni i∈J2

⇒

i∈J∪1Ni

∩

i∈J∪2Ni

⊆N= ∩

i∈JNi

(3.18)

follows.

Now, let∃x(x_∼∈Ni0,x_≈∈ ∩

j≠i0Nj). Sincex_≈∈Ni0and hencex_≈∈ ∩Ni. We see thatx_≈∈ N⇒x

≈∈Ni0, a contradiction tox

∼∈Ni0. Secondly, let∃y(y

∼∈ ∩

j≠i0Nj,y

≈∈Ni0). From these data we get y

≈ ∈ ∩Nj and hence y

≈ ∈N. Without loss of generality, we may assume that the index setJ\{i0}has cardinality greater than 1; in other words,∃i1∈J such thati1≠i0. Thusy

∼∈Ni1 andy

≈∈Ni1, an obvious contradiction. Therefore,N is alsoC3-connected.

Theorem3.14. Let(Ni)i∈J be an overlapping family ofC4-connected IFSS’s. Then

∪NiisC4-connected, too.

Proof. Similar to the above proof.

Now, we show that intuitionistic points are alwaysCiconnected, unlessXis one- point space(i=1,2,3,4).

Lemma3.4. Let(X,τ)be an IFSTS andp∈X. Then (a) p

∼ isC1-connected.

(b) p

(c) p

(d) p

Proof. (a) Assume the contrary, and letp

∼ beC1-disconnected. Hence there exist IFSOS’s M and W such thatp

∼ ⊆M∪W, M∩W ⊆p

∼ = x,{p}^c, {p}, p

∼∩M ≠∅

∼, p∼∩W≠∅_∼. Sincep

∼∩M≠∅_∼, and p

∼∩W ≠∅_∼, we get p

≈ ∈M and p

≈ ∈W; but from M∩W⊆p

∼, we see thatM1∩W1⊆ {p}^candM2∪W2⊇ {p}, which is impossible. Hence p∼ isC1-connected.

(c) Assume the contrary, and letp

∼ beC3-disconnected. Hence there exist IFSOS’s M andW such thatp

∼⊆M∪W,M∩W⊆p

∼= x,{p}^c,{p},Mp

∼ andWp

∼. Since Mp

∼andWp

∼, we getp

≈∈Mandp

≈∈W; and the same reasoning may be applied in this case, too. Hencep

∼ isC3-connected.

(b) and (d) are similar to the ﬁrst part.

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Lemma3.5. (a)p

≈isC2-connected.

(b)p

≈isC3-connected.

(c)p

≈ isC4-connected.

Proof. (a) Suppose the contrary, i.e., let there exist IFSOS’sM and W such that p≈⊆M∪W,M∩W∩p

≈= ∅

∼, p

≈∩M≠∅

∼, and p

≈∩W ≠∅

∼. Hence,{p} ∩M₂^c∩W₂^c = ∅

∼, p∈M₂^c,p∈W₂^cfollow, which is a contradiction.

(b) Suppose not, i.e., let there exist IFSOS’sMandWsuch thatp

≈⊆M∪W,M∩W⊆p

≈, Mp

≈, andWp

≈. HenceM1∩W1⊆ {p}^c,p∈M1,p∈W1, a contradiction, i.e.,p

≈ is C3-connected.

(c) Similar to (a) and (b).

Notice that IFSSN= x,N1,N2is called proper if and only ifN1∪N2≠X.

Corollary3.2. In discrete intuitionistic fuzzy special topological space(X,I(X)) any nonempty proper IFSS,NisC1- disconnected.

Proof. TakeM:=N,W:=N∈I(X). ThenN⊆N∪N,N∩N⊆N,N∩N=N≠∅ andN∩N≠∅ ∼

∼ hold, since, for example

N∩N= x,N1∩N2,N1∪N2 = x,∅,N1∪N2≠x,∅,X = ∅_∼. (3.19)

Corollary3.3. In discrete intuitionistic fuzzy special topological space(X,I(X)) any proper IFSS N= x,N1,N2, whereN1≠∅, isC2- disconnected.

Proof. Take a pointp∈Xsuch thatp∈N₁^cand p∈N₂^c and letM:=p

∼,W:=p in this IFST. Then we getN⊆M∪W,M∩W∩N= ∅ ∼

∼,N∩M≠∅

∼ andN∩W≠∅

∼, as required.

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Özça˘g: Department of Mathematics, Hacettepe University, Beytepe,06532Ankara, Turkey

E-mail address:[email protected]

Çoker: Department of Mathematics, Akdeniz University,07200Antalya, Turkey E-mail address:[email protected], [email protected]