AMS WORDS

Internat. J. Math. & Math. Sci.

VOL. 21 NO. (1998) 33-40 33

ON

CONNECTEDNESS

IN

INTUITIONISTIC FUZZY SPECIAL TOPOLOGICAL SPACES

SELMA

(Z(A(

Departmentof Mathematics

Hacettepe

University, Beytepe 06532-Ankara TURKEY

DO(AN (OKER

Department

ofMathematics Education

Hacettepe

University,

Beytepe

06532-Ankara TURKEY

(Received May i, 1996 and in revised form July 7, 1996)

ABSTRACT.Theaim ofthispaperis to construct the basic concepts relatedtoconnectednessin intuitionisticfuzzy special topological spaces. Hereweintroducethe concepts ofC5-connectedness, connectedness,Cs-connectedness, CM-connectedness,strong connectedness, superconnectedness,

Ci-

connectedness (i=1,2,3,4), and, obtain several preservation properties and some characterizations concerningconnectednessinthesespaces.

KEY WORDS AND PHRASES. Intuitionisticfuzzy specialset;intuitionisticfuzzyspecial topology, intuitionistic fuzzy special topological space, continuity; C5-connectedness; connectedness; CS- connectedness; CM-connectedness; strong connectedness; super cormectedness;

Ci-connectedness 0=1,2,3,4).

1992AMS SUBJECT CLASSI1CICATION

CODE.

04A99.

1. INTRODUCTION

ARer the imroduction of the _concept of fuzzy sets by Zadeh [1] several researches were conductedonthe generalizations of thenotionof fuzzy set. Theideaof intuitionistic fuzzysetwas first published byKrassimir

Atanassov [2]

and many worksbythe same author appeared in the literature

(see Atanassov [2,3])

Laterthis concept is usedtodefine intuitionisticfuzzyspecialsetsby Coker

[4]

and intuitionisticfuzzytopologicalspacesareintroducedby

oker

[5], Coker-Es

[6].In

this direction some preliminary concepts are also defined by

Colkun-oker[7].Here

we shall give the classicalversion of thiskindof fuzzy topological space in the framework ofcormectedness;

especially,weshall makeuseof severaltypes offuzzyconnectedness in intuitionisticfuzzytopological spacesin Turanli-Coker[8].

2. PRELIMINARIES

First weshall presentthefundamental definitions. The followingone is obviously inspired byK.

Atanassov [2,3]

DEFINITION 2.1.(see

Coker

^{[4] Let X be a} nonempty fixed set. An intuitionisticfuzzy specialset

(IFSS

^forshort) Ais anobject having theformA ^<x,A1,

A2 ^>,

^where

A

^and

A2

^are

subsets of X satisfying

A1

^c

A2 =o

The set

AI

^{iscalledthe set}of members of

A,

while

A2

^is

calledthe setof"nonmembersof"A.

ObviouslyeverysetAon anonemptysetXis obviously an IFSShavingtheform<x,A, A% One candefineseveralrelations andoperations betweenIFSS’sasfollows

DEFINITION2.2. (see

Coker

[4,5]) Let Xbe anonemptyset,and theIFSS’s A and B be in the form A=<x,

A A2 >,

^B=<x,

B1 B2 >,

respectively. Furthermore, let Ai ieJ} be an

C2)>

arbitrary family ofIFSS’sin

X,

^whereA

=<x, AI1),A,

^Then

(a) A___BiffA_c_B and A2_B2 (c)

X=<x,

A2,

A

^>

(e) <>A=<x, A2,

A2 >,

(g) ^(’Ai

<x, f"IAI’),kjAI2)>

(b) A=B iffA_BandB_A- (d)[]A

=<x, A, A>,

(f) Ai=<x,

(h)

= ^<x,

^{,X> and X}

^{=<x,X, >.}

We shalldefinethe imageandpreimage ofIFSS’s. Let XandYbetwononemptysetsand fX Ya function.

DEFINITION2.3.(see(oker [4,5]) (a)IfB=<y,B ,B2> is anIFSSinY,then thepreimageorb underf,denoted by

f-

(B),istheIFSSinXdefinedby

ft(B) <x, f

(B),

ft

(B2)>

(b)If

A=<x,A

,A2 >is anISin

X,

then the image ofAunder f,denotedby

f(A),

^istheIFSSinY definedby f(A)=

<y, ffA

^),f_(A2 )>, where

f_(A2)=(f(A))

COROLLARY2.1.LetA,

A.

(ieJ)^beIFSS’sin

X,

B, Bj(jK)IFSS’sinYandf:X->Ya function.Then

(a) A,_A2 :::>f(A1 )_f(A2 Co)

B

^{c_B2 =>}

f’ (B)_f’

(B2)

(c)Ac_f

"

^{(f(A)) and iffisinjective,thenA--f}

" (f(A)).

(d)

f(f’

(B))__B,andiffissurjective, then

f(f’ (B))=B

(e)

f

(wBj)=

f’

(Bj)

(g) f(wAi)=w’(A, (i)

f"

(Y)= X

(k) f(

X )= Y if fissurjective.

(f)

f-I (’Bj)= f"

_(Bj)

(h) f(cA.)cL=_f(A, ),and iffisinjective,thenf(A, )=cf(A. ).

0)

f"

()=

(l) f(ee

(m) ^Iffissurjective, then

f(A)_f(X);

and if,furthermore,fisinjective,wehave(f(A)) f(X) (n)

f()= f-

(B)

CONNECTEDNESS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES 35

DEFINITION2.4(seeCoker[5,9], Coker-Es[6]) Anintuitionisticfuzzy special topology (IFST forshort)on anonemptysetXis afamily z ofIFSS’sinXcontaining @, X, and closedunder finite infima and arbitrarysuprema. Inthis casethe pair(X,z)iscalled an intuitionisticfuzzyspecial topological space(IFSTSforshort)and any IFSS inzisknownas anintuitionisticfuzzy special open set(IFSOS for short)inX.

Anytopologicalspacecanbeobviouslytreated as anIFSTSin ausualmanner.

PROPOSmON2.1.Let(X,z)beanIFSTSonX. Then, wecanalsoconstructseveral IFSTS’s onXin thefollowingway.

(a) 0,

={[]G:G,},

(b)^x0.2={<>G:Gx}.

REMARK2.1 Let (X, z)^beanIFSTS

x

^={

G

G=<x,G ,G

>

is atopological spaceonX

x ^G:

^G=<x,

^GI

^{,G2>1; isthe}family of allclosed sets of the topological space(2

G

G=<x,G,G2

>

onX.

Thecomplement

X

ofanIFSOS A in anIFSTS (X, z)is called an intuitionistic fuzzy special

c,

^losedset(IFSCS for short) in

X,

and theinteriorand closure

f

anIFSS A aredefined by

cl(A)={K

^KisanIFSCSinXandA_cK},

int(Aw{G Gis anIFSOSinXandC,_A}

DEFINITION 2.5. Let (X, ) be an IFSTS on X. If A=int(cl(A)), thenAiscalledan intuitionisticfuzzy specialregularopen setinX

DEFINmON2.6. Let (X, ) and (Y, W) ^{be two}IFSTS’sand let f:X-Y beafunction. Then fis saidto becontinuous iffthepreimage ofeachIFSSinWis anIFSSin

Hereweobtain some characterizationsofcontinuity.

PROPOSITION2.2 Thefollowingare equivalent to eachother:

(a)f.(X, z)-+(Y, W)is continuous.

(b)Thepreimage of eachIFSCSinYis anIFSCSinX (c)

f

(int(B))_int(f

" (B))

^{for each}IFSS B inY.

(d)cl(f

"

^(B))c_

^f’

^{(cl(B)) foreachIFSS B inY}

3. TYPES OF CONNECTEDNESS IN INTUITIONISTIC FUZZY SPECIAL TOPOLOGICAL SPACES

Throughoutthis section(X, )and(Y, W) willalways denote IFSTS’s We shall define several types of connectedness in IFSTS’s

DEFINITION 3.1. (seeChaudhuri-Das

[I0],

Turanli-Coker

[8])

(a)X is called C-disconnected,ifthere exists an IFSS A which isboth inmitionisticfuzzy specialopen and inmitionisticfuzzyspecialclosed, such that .A. X

Co) X

^iscalled

C

-connected,ifX isnot

C

-disconnected.

(c)X

^iscalleddisconnected,if thereexistIFSOS’s A. andB3 such thatAB X and AraB=

OZA

(d) Xiscalled connected,ifXisnotdisconnected.

PROPOSITION 3.1.

C-connectedness

implies connectedness.

PROOF.

Suppose

thatthereexistnonemptyIFSOS’s AandBsuch thatAB= X,

AB=

9, from which we get A,B1 =X, A2B2=,

ABm=,

A2Be=X, in other words,

A=.

Hence A is intuitionisticfuzzy special clopen,i.e.(X,)is

C

-disconnected.

COUNTEREXAMPLE3.1. ConsidertheI:FTSz onX={a,b,c,d},where={ O, X,A,A,A3,A},

A =<x,{a},{b,c}>,A2=<x,{b,c},{a}>, A3 =<x, @,{a,b,c}>, A4

=<x,{a,b,c},O> (X,x) is connected, butnot

C

^-connected(namely,

A

^isintuitionisticfuzzyspecialclopeninX).

PROPOSITION3.2.Let f: (X,z)-->(Y, W)^{be a} continuous surjection. IfXisconnected,then so isY.

PROOF. AssumethatY isdisconnected ThusthereexistIFSOS’s C;e, D- in Y such that CD=-Y,^Cr’)-.--O. Nowwe seethatA---f

"

^(C),

^B=f(D)

^{areIFSOS’sinX, sincef}is continuous From ^C;eO, we get A--f

"

(C);e O (If

f

(C) O, then ^C---f[71 (C)--’-f( O

O,

which is a contradiction.) Similarly,we obtainB:O. Now

CD=Y=:f(C)f(D f(y)=

X =>AB= X,

CD=

^::>

^fl(c)

^c’

^f’l(’D)= fl()=

^:

Ac_xB="

^Butthisisa contradictiontoourhypothesis, thusYisconnected.

PROPOSmON

3.3.

If(X,x)

isdisconnected, then so are theIFSTS’s (X,

z0,)

^and (X,_z0.2 PROOF. LetthereexistIFSOS’s A O andB O such thatAraB= X, AB=O.

In

thiscase we

obtain X X (AB)=( A)( B)^:::, A)w( B) X

=[]

=[](Ac)=(

[]A)([]B)=:>

([]A)([]B)=,

whichis a contradiction.

PROPOSITION3.4.(X,’0is

C-connected

^{iffthereexist no}nonemptyIFSOS’s AandBinXsuch that

A=.

PROOF. (= :)

Suppose

thatAand

B

areIFSOS’sinXsuch thatA;e@;eBand

A=B

Since

A=,

Bis anIFSCS,and A;eO =:,B;eX. Butthis is a contradictiontothefact thatXis

C5

^-connected

:) Let Abe both anIFSOSandIFSCSsuch that ^O;eA;eX. Nowtake

B=X.

InthiscaseBisan IFSOSandA;e

X

=>

B=X

^;e

O,

whichis acontradiction.

PROPOSITION

3.5. (X,x)^is

C

-connected iff thereexistno nonemptyIFSS’s

A

and

B

in

X

such that

B=X,

B=eI(A),

A=el-’i

PROOF.

(::>:) Assume

that there exist IFSS’s A and

B

such that

A;e;eB, B=X, _B=cI(A),

A=cI(B). Sinceel(A)and el(B)areIFSOS’sin

X, A

and

B

are

IFSOS’s

in

X,

whichis a contradiction

CONNECTEDNESS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES 37

:)Let A be both an IFSOS and IFSCS in X such that ;A*X. Taking

B=X

^we obtain a contradiction.

Here we generalize the concepts of

Cs-connectedness

and

Ct-connectedness

given by Chaudhuri-Das [10] ^{to the} intuitionistic case:

LEMMA3.1.(a)

AB=

==>

A_c,

(b)

A

^AB;e

DEFINITION3.2. LetAand Bbenonzero IFSS’sin(X,z). A and B are saidto be weakly separated,if

cl(A)<:_

^and cl(B)c_X andq-separated,ifel(A)cB @ =Ac

cl(B).

DEFINITION3.3. (see Turanli-(oker [8]) (a) An IFSTS (X,z) is saidto beCs-disconnected, if there existweakly separatednonzeroIFSS’s AandBin(X,z)such that X =AwB

(b) (X,x)iscalledCs-connected,if(X,z)isnot

Cs-disconnected.

(c) Xis said tobe CM-disconnected, ifthere exist q-separated nonzeroIFSS’s AandBinX such that X =AB.

(d) Xiscalled CM-connected,ifXisnotCM-disconnected.

Letusgive theconnectionbetween thesetwotypes of connectednessinIFSTS’s:

COROLLARY3.1.If theIFSTSXis

Cs

-connected,thenXisalso

CM

-connected.

DEFINITION3.4.(see Turanli-(oker

[8])

An IFSTS (X,z) is said to be strongly connected, if thereexitnononemptyIFSCS’s AandBinXsuchthatAr=

.

PROPOSITION3.6.Xisstronglyconnected iffthere exist noIFSOS’s A andB in Xsuchthat A;*X^;BandAB X.

PROOF. (==>:) Let AandBbeIFSOS’sinXsuch that^A;

X

^;Band

AB=X.

Ifwetake

C=X

and

D=,

thenCandDbecomeIFSCS’sinXandC*

,D,

^Ccq)=

,

a contradiction.

.

^{Useasimilartechniqueasabove}

PROPOSITION 3.7. Let f (X,x) --> (Y, ) be a cominuous surjection. If X is strongly connected,thenso isY

PROOF.

Suppose

that

Y

isnotstronglyconnected.

In

this casethereexistIFSCS’s Cand

D

in

Y

such that C,*D, Cc-d)=. Since f is continuous,

fl(C)

and

fl(D)

are IFSCS’s in X, and

f(C)f(D)=, f(C);e@, f(D)

@. (Iff

(C)=,

then

f(f (C))=C f()=C

==> =C, a contradiction.) But this is acontradiction, henceYisstronglyconnected,too.

Strong

connectedness doesnotimply

C5

-connectedness, and the sameis treeforIFSTSconverse, i.e.

C5

connectedness does not imply strong connectedness. For this purpose see the following counterexamples:

COUNTEREXAMPLES 3.2. Let

X={a,b,c,d} (a)

^If

z={,X,A,A,A,A},

where

A =<x,{b,c},{d}>, A =<x,{d},{b,c}>, A=<x,,{b,c,d}>, A =<x,{b,c,d},>,

^then the IFSTS (X,z) ^is strongly connected, but not

C-connected.

(b) Ifz={

,

X,A,Aa,A,A4,A}, ^where

A =<x,{b,c},{d}>, A=<x,{a},{c} >, A

=<x,{a,d},{c}>,

A4=<x,{a,b,c},>, A=<x,,{c,d}>,

^{then the} IFSTS (X,z) is C-connected, but not strongly connected

DEFINITION3.5.(see

Turanli-Goker

^[$])(a)Ifthere exists an intuitionisticfuzzy special regular opensetAinXsuch that *A*X, thenXiscalled superdisconnected

(b) Xiscalledsuper connected, ifXis notsuperdisconnected.

Nowwegivesome characterizationsof super connectedness:

PROPOSITION 3.$. Thefollowing assertions areequivalent:

(a) X issuper connected. (b) ForeachIFSOSA* inXwehavecl(A) X (c) ForeachIFSCS^A;X inXwehaveint(A)=

(d)Thereexist noIFSOS’s AandBinsuch thatA

B, A _ _.

(e)There exist noIFSOS’s AandBinXsuchthatA *B,B=cl(A), A=cI(B) (f)Thereexist noIFSCS’s AandBinXsuch that A*

B,

B=mt(A), A=tnt(B)

PROOF. (a)=>(b) Assume that there exists an IFSOS At suchthat cl(A), X. Now take B=int(cl(A)). ^ThenB is a proper intuitionistic fuzzy special regular open set in

X,

and this is in contradiction with thesuper connectedness of X.

(b)=>(c) Let A X ^beanIFSCSinX.Ifwetake

B=X,

then Bis anIFSOSinXandB;c Hence cl(B)=X cl(B)=O int(g)= int(A)= follows.

(c)=>(d) Let AandBbeIFSOS’sinXsuch that ^A;e

B

^and

A_.

Since isanIFCSinX andB* =::>

*

^X,weobtainint(g)= But, from

A_g,

we see that A=int(A)

_

^int()=O,

which is acomradiction.

(d)

(a)

^Let

eA.

X be an intuitionistic fuzzy special regular open set in X. Ifwe take

B=cI(A----,

we get B. (Because, otherwise we have

B=

cI(A)=I ::> cl(A)=X :=>

A--int(cl(A))-nt(X )=X,but the lastresultcontradictsthefact A*

X

.) WealsohaveA

_ ^B,

^andthis

isacomradiction,too.

(a)

=>

(e)

^{Let AandBbeIFSOS’sinXsuch thatA;,B}and B=cI(A), A=cI(B) Nowwehave int(cl(A))nt(g)=cl(B)=A and A;eO, A*X. (Ifnot, i.e. ifA=X, then X=cl(Bj

O=cl(B)

B .) Butthis isacontradiction.

(e) (a)

Let

A

bean

IFSOS

in

X

such that A---int(cl(A)), ^,A,X.

Now

take

B=cI(A--. In

this caseweget B, and

B

isanIFSOS in

X

andB=cI(A) andcl(B)=cl(cl(A)) =int(cl(A))--int(cl(A))=A, which is acontradiction.

CONNECTEDNESS IN INTUITIONISTIC FUZZY TOPOLOGICAL SPACES 39

(e)

=

^{(f) Let AandB IFSCS’sinX suchthatA X eB,} B=int(A), A=int(B) Taking C=A and D=B, C and D become IFSOS’s in X and C;D, cl(C)=cl(A)--int(A)=int(A)=B=D, and similarly

cI(D)=C.

^Butthis is an obviouscontradiction.

(f) (e) Onecanuseasimilartechniqueas in(e) (f).

PROPOSITION 3.9.Superconnectedness implies

C-connectedness.

PROOF.Obvious.

ButthereverseimplicationtoProposition3.9doesnotholdingeneral

COUNTEREXAMPLE3.3.Let X={a,b,c,d}andtheIFST ’={

,X,AI

,A2 ,A3 ,A4 onX, where

A

=<x,{a},{c,d}>,A2=<x,{d},{a,c}>,

A3

=<x,{a,d},{c}>,

A =<x,@,{a,c,d}>.

ThentheIFSTS (X,z)is

C

-connected,butnotsuperconnected

PROPOSITION3.10.Letf:(X,)-->(Y, W)bea continuoussurjection. IfXissuper connected, then so isY.

PROOF. Suppose^thatYissuper disconnected.InthiscasethereexistIFSOS’s CandDinYsuch that

C;;D, C_c

Since f is continuous,

f(C)

^and

f(D)

are IFSOS’s in

X,

and Cc_

=>f(C)c_f()=f-I

(D),

f(C) ; fq(),

which meansthatXissuper disconnected

Nowweshallsummarizethe interrelations betweenseveral types of connectednessinIFSTS’s.

super connectedness

Cs

-connectedness C -connectedness CM-connectedness connectedness

Here we generalize the idea of fuzzy

Ci-connectedness

ⁱⁿ fuzzy topological spaces and in intuitionistic fuzzytopological spaces(seeAjmal-Kohli [11], Chaudhuri-Das 10] and

Turanli-oker

[$]totheintuitionistic case:

DEFINITION3.6.Let Nbe anIFSSin(X,x)

(a)

IfthereexistIFSOS’s MandWin

X

satisfying the following properties, thenNiscalledC,- disconnected(i= 1,2,3,4)

C:N _c.MW,

MrW

c_N,Nr’tMc:,NW=, C2:N

_cMW,NMW= @,NM;

,NW; ,

C3:N

_cMwW,MW

c_N,MN,WN, C4:N

_cMwW,NrxMcW=

,M N,W .

(b) Nis said tobe C,-connected (i=1,2,3,4),ifN is not C,-disconnected(i=1,2,3,4)

Obviously,one can obtainthefollowing implications betweenseveraltypes ofC,-connectedness (i=1,2,3,4)

C

-connectedness -->

C2

-connectedness

C3-connectedness

^-->

C4

-connectedness

Noneof these implicationsarereversible, as the following counterexamplesstate

COUNTEREXAMPLES

3.4.Consider theIFSTzonX={a,b },where

z={

,

X,A,A2,A3,A4,As,A6,A7},

A =<x,{a},}>, A2 =<x,{b},>, A3 =<x,,{a}>, A4 =<x,,{b}>, A

=<x,{a},{b}>, A,

=<x,{b},{a}>, A7 =<x,,>

and take theIFSS

N=<x,,{a}>

^inX.

(a) NisC_-connected, butnot

C-connected.

^[Namely,

A2

^and

A3

do satisfy the propertiesin(C) (b)Nis

C3

-connected,butnot

C-connected

COUNTEREXAMPLE3.5.Consider theIFST onX={a,b,c,d},wherez={

,

X,A,A2,A3,A4},

A

=<x,{a},{b,c}>,

A:=<x,{b,c},{a}>, Aa

=<x,,{a,b,c}>,

A

=<x,{a,b,c},> The IFSS N=<x,{a},{b}> in X is C4-connectcd, but not

C-connccted

^[Namely,

A

^and

A2

do satisfy the propertiesin(C).]

COUNTEREXAMFLE 3.6. Consider the IFST z on

X={ a,b,c},

where

={

,X,AI,A2,A},A=<x,,{a}>,A==<x,{a},{b,c}>,A3=<x,{a},>.

The IFSS

N=<x,{a},3>

in X isC4-connected, butnot

C2-connected.

[Namely,

A

^and

A=

^{do satisfythepropertiesin(C2).}

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