FRANCISCO GALLEGO LUPI ´A ˜NEZ
Received 22 September 2004 and in revised form 29 March 2005
The introduction of intuitionistic fuzzy sets is due to K. T. Atanassov, who also proposed some problems about this subject. D. C¸oker defined the intuitionistic fuzzy topological spaces and, with some coworkers, studied these spaces. In this paper, we define and study the notion of quasicoincidence for intuitionistic fuzzy points and obtain a characteriza- tion of continuity for maps between intuitionistic fuzzy topological spaces
The introduction of “intuitionistic fuzzy sets” is due to Atanassov [1], and this theory has been developed in many papers [2,3,4]. This author proposed as an open prob- lem “to investigate the topological and geometric properties of the IFSs” remarking that
“some first steps in this direction are made” [4].
In this paper, we define for intuitionistic fuzzy sets the notion of quasicoincidence and the corresponding neighborhood structure (see [9]). These concepts allow us to obtain a characterization of continuity for maps between two intuitionistic fuzzy topological spaces.
(For notions on ordinary fuzzy topology used in this paper, see [7,8].) First, we list some previous definitions.
Definition 1[1]. LetXbe a nonempty set. An intuitionistic fuzzy set (IFS)AofXis an object having the form
A=
x,µA,γA
| x∈X, (1)
where the functions µA:X→I andγA:X→I denote the degree of membership and the degree of nonmembership of each elementx∈X to an ordinary subset ofX, and 0≤µA(x) +γA(x)≤1 for eachx∈X.
Notation 2. 0∼= x, 0, 1and 1∼= x, 1, 0.
Definition 3[2]. LetXbe a nonempty set, and letAandBbe two IFSs ofX. Then, (a)A⊆BifµA(x)≤µB(x) andγA(x)≥γB(x);
(b)A=BifA⊆BandB⊆A;
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:10 (2005) 1539–1542 DOI:10.1155/IJMMS.2005.1539
1540 Quasicoincidence for intuitionistic fuzzy points (c)A∪B= {x,µA∨µB,γA∧γB |x∈X};
(d)A∩B= {x,µA∧µB,γA∨γB |x∈X}.
Definition 4[5]. Let{Aj|j∈J}be an arbitrary family of IFSs ofX. Then, (a)Aj= {x,∧µAj,∨γAj |x∈X};
(b)Aj= {x,∨µAj,∧γAj |x∈X}.
(For other definitions concerning IFSs used in this paper, see [5,6].)
Definition 5. LetA= {x,µA,γA |x∈X}andB= {x,µB,γB |x∈X}be two IFSs. Say thatAquasicoincides withB, denoted byAqBifµAquasicoincides withµBandγAquasi- coincides withγB.
Remark 6. If AqB, we have that A∩B=0∼. (µAqµB implies that µA∧µB=0, then A∩B=0∼).
Remark 7. IfAandBverify thatγA=µAandγB=µB, thenAqBif, and only if,µAqµB. Proposition 8. Let A and B be two IFSs of X, let f :X→Y be a map between two nonempty setsXandY, then ifAqB, f(A)q f(B).
Proof. AqB if and only if µAqµB and γAqγB. Then, we have that f(µA)q f(µB) and f(γA)q f(γB), that is, (1−f(γA))q(1−f(γB)). Thus f(A)q f(B).
Proposition9. LetXandYbe two nonempty sets, let f :X→Ybe a map, letAbe an IFS ofX, and letCbe an IFS ofY. If f(A)qC,Aq f−1(C).
Proof. f(A)qC if and only if f(µA)qµC and f(γA)qγC. Then, µAq f−1(µC) and γAq f−1(γC), that is,γAq(f−1(γC))(because f−1(γC)=f−1(γC)).
Remark 10. IfA,B, andCare IFSs ofX, such thatAqB, andB⊆C, thenAqC.
Definition 11. Let (X,τ) be an IFTS, and letpbe an IFP ofX. Say that an IFSNofXis a Q-neighborhood ofpif there exists an IFOSAof (X,τ) such thatpqAandA⊆N.
Theorem12. Let(X,τ)be anIFTS, letpbe anIFPofX, and letᐁQ(p)be the family of all theQ-neighborhoods ofpin(X,τ), then,
(1)N∈ᐁQ(p)implies thatpqN,
(2)N1,N2∈ᐁQ(p)imply thatN1∩N2∈ᐁQ(p), (3)ifN∈ᐁQ(p)andN⊆M, thenM∈ᐁQ(p),
(4)ifN∈ᐁQ(p), there exists M∈ᐁQ(p),M⊆N, such that, for everyIFPe which quasicoincides withM,M∈ᐁQ(e).
Proof. (1)N∈ᐁQ(p) if and only if there exists an IFOSAsuch that pqAandA⊆N, thenpqN (byRemark 10).
(2)N1,N2areQ-neighborhoods ofpif and only if there exist two IFOSsAisuch that pqAi,Ai⊆Ni(i=1, 2), then, ifp=c(α,β), we have thatcαqµAi,µAi≤µNi,c1−βqγAi,γAi≥ γNi(i=1, 2), thencαqµA1∩A2,µA1∩A2≤µN1∩N2,c1−βqγA1∩A2,γA1∩A2≥γN1∩N2, andpq(A1∩ A2),A1∩A2⊆N1∩N2, withA1∩A2an IFOS.
(3) It is obvious.
(4)N∈ᐁQ(p) if and only if there exists an IFOSAsuch thatpqAandA⊆N, thenA is also aQ-neighborhood ofp, and for each IFPesuch thateqA,Ais aQ-neighborhood
ofe.
Proposition13. LetXbe a nonempty set, for eachIFPpofX, letᐁQ(p)be a family of IFSsverifying (1), (2), and (3) of the theorem, thenτ= {UIFS|U∈ᐁQ(p)ifpqU}is an IFTinX. If also the family verifies (4), thenᐁQ(p)is the system ofQ-neighborhoods ofpin (X,τ).
Proof. 1∼∈τby (3).
Ui∈τ(i=1, 2), andpq(U1∩U2), thenUi∈ᐁQ(p) andU1∩U2∈ᐁQ(p) by (2).
{Uj}j∈J ∈τ,pq∪Uj with p=c(α,β) if and only ifcαqµ∪Uj andc1−βqγ∪Uj and it is equivalent tocαqµ∪Uj0 (for somej0ofJ) andc1−βqγ∪Uj (for all jofJ). Then pqUj0 for some j0∈J,Uj0∈ᐁQ(p) for some j0∈J, and∪Uj∈ᐁQ(p) by (3).
Finally, ifN∈ᐁQ(p), there existsM∈ᐁQ(p),M⊆N, such that for every IFPewhich quasicoincides withM, we have thatM∈ᐁQ(e), thenM∈τ,pqM,M⊆N, andN is a Q-neighborhood of pin (X,τ). Conversely, for everyQ-neighborhoodNofpin (X,τ), there is anA∈τsuch thatpqA,A⊆N, then for every IFPewhich quasicoincides with A, we have thatA∈ᐁQ(e), thusN∈ᐁQ(p).
Proposition14. LetX,Y be two nonempty sets, let f :X→Y be a map, letτbe anIFT inX, and letsbe anIFTinY. Then, f : (X,τ)→(Y,s)is continuous if, and only if, for each IFPpofX, and for eachQ-neighborhoodVof f(p), there exists aQ-neighborhoodUofp such that f(U)⊆V.
Proof. IfV is aQ-neighborhood of f(p), there exists an IFOSGsuch that f(p)qGand G⊆V, then pq f−1(G) (byProposition 9), and f−1(G) is an IFOS such that f−1(G)⊆
f−1(V). Thus,f−1(V) is aQ-neighborhood ofpand f(f−1(V))⊆V.
Conversely, for each G∈s, we have that, for every IFP p such that pq f−1(G) is f(p)q f(f−1(G)) (byProposition 8), then f(p)qG, andGis aQ-neighborhood of f(p).
By the hypothesis, there exists aQ-neighborhoodUofpsuch that f(U)⊆G, thenU⊆ f−1(G) and f−1(G)∈ᐁQ(p). FromProposition 13, it follows that f−1(G)∈τ.
References
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[4] ,Intuitionistic Fuzzy Sets. Theory and Applications, Studies in Fuzziness and Soft Com- puting, vol. 35, Physica-Verlag, Heidelberg, 1999.
[5] D. C¸oker,An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems88 (1997), no. 1, 81–89.
[6] D. C¸oker and M. Demirci,On intuitionistic fuzzy points, Notes IFS1(1995), no. 2, 79–84.
[7] Y.-M. Liu and M.-K. Luo,Fuzzy Topology, Advances in Fuzzy Systems—Applications and The- ory, vol. 9, World Scientific, New Jersey, 1997.
[8] N. Palaniappan,Fuzzy Topology, CRC Press, Florida, 2002.
1542 Quasicoincidence for intuitionistic fuzzy points
[9] P. M. Pu and Y.-M. Liu,Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore- Smith convergence, J. Math. Anal. Appl.76(1980), no. 2, 571–599.
Francisco Gallego Lupi´a˜nez: Departamento de Geometr´ıa y Topolog´ıa, Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
E-mail address:fg [email protected]
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