article

Mathware & Soft Computing 10 (2003) 17-22
Hausdor®ness in Intuitionistic Fuzzy
Topological Spaces
F. Gallego Lupi¶a ~nez
Depto. Geometr¶³a y Topolog¶³a. Fac. Matem¶aticas
Univ. Complutense de Madrid. 28040 Madrid. Spain
fg [email protected]
Abstract
The basic concepts of the theory of intuitionistic fuzzy topological spaces have been de¯ned by D. C
»oker and co-workers.
In this paper, we de¯ne new notions of Hausdor®ness in the
intuitionistic fuzzy sense, and obtain some new properties, in
particular on convergence.
Mathematics Subject Classi¯cation (2000): 54A40, 03E72.
Key words: Intuitionistic topology; separation; Hausdor®ness.
The introduction of "intuitionistic fuzzy sets" is due to K.T. Atanassov [1], and this theory has been developed by many authors [2-4].
In particular D. C
» oker has de¯ned the intuitionistiz fuzzy topological
spaces, and several authors have studied this category [5-14].
Nevertheless, separation in intuitionistic fuzzy topological spaces is
not studied. Only there exists a de¯nition due to D. C
» oker:
De¯nition 1 [5] An IFTS (X; ¿ ) is called Hausdor® i® x1; x2 2 X
and x1 6= x2 imply that there exist G1 =< x; ¹G1 ; °G1 >; G2 =< x; ¹ G2 ;
°G2 >2 ¿ with
¹ G1 (x1) = 1; °G1 (x1 ) = 0
¹ G2 (x2) = 1; °G2 (x2 ) = 0 and G1 \ G 2 = 0».
17
18
F. Gallego Lupi¶an~ez
But this de¯nition is not suitable for to characterize net convergence in intuitionistic fuzzy topological spaces. Then, in this paper,
we will de¯ne two new kinds of intuitionistic fuzzy topological spaces
and obtain various properties.
Firstly, we list some previous de¯nitions:
De¯nition 2 [7] Let X be a nonempty set and c 2 X a ¯xed element
in X. If ® 2 (0; 1] and ¯ 2 [0; 1) are two ¯xed real numbers such that
® + ¯ · 1, then the IFS
c(®; ¯) =< x; c® ; 1 ¡ c1¡¯ >
is called an intuitionistic fuzzy point (IFP for short) in X:
If ¯ 2 [0; 1) is a ¯xed real number, then the IFS
c(¯) =< x; 0; 1 ¡ c1¡¯ >
is called a vanishing intuitionistic fuzzy point (VIFP for short) in
X:
De¯nition 3 [7]
(a) Let c(®; ¯) be an IFP in X such that ®; ¯ 2 (0; 1)
and A =< x; ¹ A; °A > be an IFS in X. c(®; ¯) is said to be properly
contained in A (c(®; ¯) 2 A for short) if ® < ¹A(c) and ¯ > °A(c).
(b) Let c(¯) be a VIFP in X such that ¯ 2 (0; 1) and A =<
x,¹ A; °A > be an IFS in X. c(¯) is said to be properly contained
in A (c(¯) 2 A for short) if ¹A(c) = 0 and ¯ > °A(c):
De¯nition 4 [7]. Let (X; ¿ ) be an IFTS on X and N be an IFS in
X. N is said to be an "¡neighborhood of an IFP c(®; ¯) in X if there
exists an IFOS G in X such that c(®; ¯) 2 G µ N: N is said to be an
"-neighborhood of a VIFP c(¯) in X if ¹N (c) = 0 and there exists an
IFOS G in X such that c(¯) 2 G µ N:
De¯nition 5 [13]. Let X be a non-empty set, P the set of all IFPs
and VIFPs of X and D a directed set. An intuitionistic fuzzy net is a
map s : D ! P. We denote s d = s(d) (for d 2 D), s = (sd )d2D:
De¯nition 6 [13] Let (X; ¿ ) be an IFTS and s be an intuitionistic
fuzzy net in X. s converges to an IFP (or a VIFP) p in (X; ¿) if for
every "¡neighborhood N of p there exists d0 2 D such that sd 2 N for
all d ¸ d 0:
Hausdor®ness in Intuitionistic Fuzzy Topological Spaces
19
De¯nition 7 [13] Let X be a non-empty set and F a non-empty family
of IFSs6= 0». F is an intuitionistic ¯lter in X if
a) for all F 1; F2 2 F is F1 \ F2 2 F.
b) for all F 2 F and each IFS F 0 such that F µ F 0 is F 0 2 F.
De¯nition 8 [13] Let (X; ¿ ) be an IFTS and F be an intuitionistic
¯lter in X. F converges to an IFP p in (X; ¿ ) if every "-neighborhood
of p is member of F:
Remark. If p = c(®; ¯) is an IFP in an IFTS (X; ¿ ) and M is an
IFOS in X such that ¹ M (c) = 1; °M (c) = 0 it does not imply that
M is an "-neighborhood of p (let, for example ® = 1; ¯ = 0). This
fact motives that the C
» oker's de¯nition of Hausdor®ness intuitionistic
fuzzy topological spaces is not suitable for to characterize convergence
in intuitionistic fuzzy topological spaces.
We will introduce some new kinds of Hausdor®ness in intuitionistic
fuzzy topological spaces.
De¯nition 9 An IFTS (X; ¿ ) will be called T2 if for every p; q IFPs
or VIFPs in X such that p 6= q there exists "-neighborhodds M and N
of p and q respectively such that M \ N = 0» :
Clearly, for c(®; ¯) we discard ® = 1; ¯ = 0, and for c(¯) the case
¯ = 0.
Proposition 1. Let (X; ¿ ) be an IFTS. Then (X; ¿ ) is T 2 if and only
if every convergent intuitionistic fuzzy net in (X; ¿) has an only limit.
Proof. If (sd )d2D converges to p and q, two IFPs or VIFPs such that
p 6= q, then for every "-neighborhoods M and N of p and q respectively
there are dM ; dN such that sd 2 M for d ¸ dM and sd 2 N for d ¸ dN .
Thus for some d0 ¸ dM ; dN is s d 2 M \ N for all d ¸ d0 , and
M \ N 6= 0» [13, Proposition 1].
Conversely, if (X; ¿ ) is not T2 there exist two IFPs or VIFPs p and
q such that p 6= q and, for all "-neighborhoods M; N of p and q respectively is M \ N 6= 0».Then, there is either an IFP or VIFP s(M;N) 2
M \ N [13, Proposition 1]. Let D = f(M; N)jM "¡neighborhood of
p and N "-neighborhood of qg directed by µ. Then the intuitionistic
fuzzy net (s(M;N))(M;N)2D converges to p and q:
20
F. Gallego Lupi¶an~ez
De¯nition 10 (15) A fuzzy topological space (X; T ) is said to be fuzzy
Hausdor® if for any two fuzzy points xr ; ys with distinct supports there
exist disjoint ¹; º 2 T with xr 2 ¹ and xs 2 º
(In this de¯nition the value of any fuzzy point xr is such that 0 <
r < 1).
Proposition 2. Let (X; ¿ ) be an IFTS. If (X; ¿) is T2; then (X; ¿1)
is a fuzzy Hausdor® fts (where ¿1 = f¹G jG 2 ¿g).
Proof. For any two fuzzy points xr ; ys with distinct supports and
0 < r; s < 1, we have that p = x(r; 1 ¡ r), q = y(s; 1 ¡ s) are two
distinct IFPs. Then, there exist "-neighborhoods M and N of p and q
respectively such that M \ N = 0». This implies that r < ¹M (x); s <
¹N (y) and xr 2 ¹M ; ys 2 ¹N which are fuzzy neighborhoods with
¹M ^ ¹N = ;:
Proposition 3. Let (X; ¿ ) be an IFTS: If (X; ¿) is T 2 every convergent intuitionistic ¯lter in (X; ¿) has an only limit.
Proof.
If F converges to p and q two distinct IFPs, then the intuitionistic fuzzy net associated to F, sF converges to p and q [13,
Theorem] and (X,¿) is not T2:
(The converse is not true because the convergence of intuitionistic
¯lters is de¯ned only to IFPs).
De¯nition 11 An IFTS (X; ¿ ) will be called q¡T2 if for every distinct
IFPs or VIFPs p and q in X there exist "¡neighborhoods M and N of
0
p and q respectively such that ¹ M · ¹0N and °M ¸ °N
:
Proposition 4. Every T2 IFTS is q ¡ T2:
Proof. Let p; q be two distinct IFPs nor VIFPs then, there exist
"-neighborhoods M; N of p and q respectively such that M \ N = 0» .
Since we have that ¹M ^ ¹N = ;; °M _ °N = 1, this implies that
0
¹M · ¹0N and °M ¸ °N
:
De¯nition 12 A fuzzy topological space (X; T ) will be called q-fuzzy
Haussdor® if for any two fuzzy points xr ; ys with distinct supports there
exist ¹,º 2 T with xr 2 ¹; ys 2 º and ¹ · 1 ¡ º.
Hausdor®ness in Intuitionistic Fuzzy Topological Spaces
21
(In this de¯nition also the value of any fuzzy point xr is such that
0 < r < 1):
Proposition 5. Let (X; ¿ ) be an IFTS. If (X; ¿ ) is q¡T2 then (X; ¿1)
is a q¡fuzzy Hausdor® fts.
Proof. For any two points xr ; ys with distinct supports and 0 < r; s <
1, we have that p = x(r; 1 ¡ r); q = y(s; 1 ¡ s) are two distinct IFPs.
Then, there exist "-neighborhoods M and N of p and q respectively
0 : This implies that x 2 ¹ ; y 2 ¹
such that ¹M · ¹0N and °M ¸ °N
r
M
s
N
which are fuzzy neighborhoods with ¹M · 1 ¡ ¹N :
The author thanks to Professors D. C
» oker and S.J. Lee for making
available their papers to him.
References
[1] K.T. Atanassov: Intuitionistic fuzzy sets, in VII ITKR's Session,
So¯a (June 1983).
[2] K.T. Atanassov: Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20
(1986), 87-96
[3] K.T. Atanassov: Review and new results on intuitionistic fuzzy
sets, preprint IM-MFAIS-1-88, So¯a (1988).
[4] K.T. Atanassov: Intuitionistic fuzzy sets. Theory and applications, Springer-Verlag (Heidelberg, New York, 1999).
[5] D. C
» oker: An introduction to intuitionistic fuzzy topological
spaces, Fuzzy Sets Syst. 88 (1997), 81-89.
[6] D. C
» oker: An introduction to fuzzy subspaces in intuitionistic
fuzzy topological spaces, J. Fuzzy Math. 4 (1996), 749-764.
[7] D. C
» oker and M. Demirci: On intuitionistic fuzzy points, Notes
IFS 1,2 (1995), 79-84.
[8] D. C
» oker and A.H. E»s: On fuzzy compactness in intuitionistic
fuzzy topological spaces, J. Fuzzy Math. 3 (1995), 899-909.
22
F. Gallego Lupi¶an~ez
[9] A.H. E»s and D. C
» oker: More on fuzzy compactness in intuitionistic
fuzzy topological spaces, Notes IFS 2,1 (1996), 4-10.
[10] H. GÄ
ur»cay, D. C
» oker and A.H. E»s: On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math. 5 (1997), 365378.
[11] N. Turanh and D. C
» oker: On some types of fuzzy connectedness
in fuzzy topological spaces, Fuzzy Sets Syst. 60 (1993), 97-102.
[12] Seok Jong Lee and Eun Pyo Lee: The category of intuitionistic
fuzzy topological spaces, Bull. Korean Math. Soc. 37 (2000), 6376.
[13] F.G. Lupi¶an~ez: Nets and ¯lters in intuitionistic fuzzy topological
spaces, preprint.
[14] F.G. Lupi¶a~nez: Covering properties in intuitionistic fuzzy topological spaces, preprint.
[15] R. Srivastava, S.N. Lal and A.K. Srivastava: Fuzzy Hausdor®
topological spaces, J. Math. Anal. Appl 81 (1981), 497-506.