Vo lu m e 6 2 , N u m b e r 2 , P a g e s 6 7 –7 4 (2 0 1 3 ) IS S N 1 3 0 3 –5 9 9 1
GENERALIZED NEIGHBOURHOOD SYSTEMS OF FUZZY POINTS
SEVDA SA ¼GIRO ¼GLU, ERDAL GÜNER AND EDA KOÇYI ¼GIT
Abstract. We de…ne the generalized fuzzy neighbourhood systems on the set of fuzzy points in a nonempty setX and investigate their prop-erties by using a new interior operator. With the help of these concepts we introduce generalized fuzzy continuity, which include many of the variations of fuzzy continuity already in the literature, as special cases.
1. Introduction
A neighbourhood system assigns each object a (possibly empty, …nite or in…nite) family of nonempty subsets. Such subsets, called neighbourhoods, represent the semantics of near. Formally, neighbourhoods play the most fundemantel role in mathematical analysis. Informally, it is a common and intuitive notion. It is in databases [10,20], in rough sets [27], in logic [5], in texts of genetic algorithms [14], and many others. This paper introduces generalized neighbourhood systems on the set of fuzzy points of a nonempty set.
The fundemantal idea of fuzzy sets was …rst introduced by Zadeh [35]. Chang [9] is known as the initiator of the notion of fuzzy topology. In 1976, the fuzzy topology was redi…ned in somewhat di¤erent way by Lowen [15]. Then many attempts have been made to extend various branches of mathematics to the fuzzy settings. We focus our work to extend the notions of the generalized neighbourd system to the fuzzy settings. To generalize the notions of topolgy, the initial attempts can be seen in [18] and [16], respectively, i.e., supratopologies and minimal structures. Recently, Császár [11] introduced the notions of generalized topologies (brie‡y GT) and generalized neighbourhood systems (brie‡y GNS). In [1], fuzzy supratopology and, recently, in [26], generalized fuzzy topology were de…ned as generalizations of fuzzy topology introduced by Chang. In addition, as a generalization of fuzzy topology introduced by Lowen, fuzzy minimal structure was de…ned in [3]. The neighbourhood and q-neighbourhood of a fuzzy point in a fuzzy topological space
Received by the editors Octob. 21, 2012; Accepted: Dec. 08, 2013. 2000 Mathematics Subject Classi…cation. 54A05, 54A40, 54C05.
Key words and phrases. Generalized fuzzy topology, generalized fuzzy neighbourhood system, fuzzy( ; 0)-continuity.
c 2 0 1 3 A n ka ra U n ive rsity
in the sense of Chang was introduced by Pu and Liu [19]. An earlier study on neighbourhood of fuzzy points can be found in [13].
In this paper, we de…ne the generalized fuzzy neighbourhood systems on the set of fuzzy points in a nonempty set X and investigate their properties by using a new interior operator which corresponds to the notion of the interior operator in general form and gives us the way to show that every generalized fuzzy topology can be generated by a generalized fuzzy neighbourhood system. In addition, we introduce generalized fuzzy continuity with the help of generalized fuzzy neighbourhood sys-tems. These notions lead us to give a general form to various concepts discussed in the literature.
2. Preliminaries
Let X be an arbitrary nonempty set. A fuzzy set A in X is a function on X into the interval I = [0; 1] of the real line. The class of all fuzzy sets in X will be denoted by IX and symbols A; B; ::: is used for fuzzy sets in X: The complement of a fuzzy
set A in X is 1X A: The fuzzy sets in X taking on respectively the constant
values 0 and 1 are denoted by 0X and 1X, respectively. A fuzzy set A is nonempty
if A 6= 0X: For two fuzzy sets A; B 2 IX; we write A B if A (x) B (x) for
each x 2 X: For a family fAjgj2J IX; the union C = [jAj and the intersection
D = \jAj; are de…ned by C (x) = sup
J fA
j(x)g and D (x) = inf
J fAj(x)g for each
x 2 X: For a fuzzy set A in X; the set fx 2 X : A (x) > 0g is called the support of A: A fuzzy singleton or a fuzzy point with support x and value (0 < 1) is denoted by x : The fuzzy point x is said to be contained in a fuzzy set A; denoted by x 2 A; i¤ A (x) ; whereas the notion x qA means that x is quasi-concident with A; i.e., x qA implies + A (x) > 1.
Let f be a function from X to Y; A 2 IX and B 2 IY: Then f 1(B) and f (A)
are de…ned as; f 1(B) (x) = B (f (x)) for x 2 X and
f (A) (y) = ( sup x2f 1(y) A (x) ; f 1(y) 6= ; 0 ; otherwise for y 2 Y; respectively.
Throughout this paper, by a fuzzy topological space (shortly fts) we mean a fts (X; o) ; as initiated by Chang [9], i.e., o IX satisfy (a) 0
X, 1X 2 o; (b) If Aj2 o
for each j 2 J 6= ;; then [j2JAj 2 o and (c) If A; B 2 o; then A \ B 2 o: The
elements of o are called fuzzy open sets and their complements are called fuzzy closed sets. We shall denote the fuzzy interior and fuzzy closure of a fuzzy set A 2 IX with i
oA and coA; respectively, i.e. ioA = [ fU : U A; A 2 og and
coA = \ fF : F A; 1X A 2 og : A fuzzy set V is called a neighbourhood of
fuzzy point x i¤ there exists U 2 o such that x 2 U V and V is called a q-neighbourhood of x i¤ there exists U 2 o such that x qU V: The fuzzy set
theoretical and fuzzy topological concepts used in this paper are standard and can be found in Zadeh [35], Chang [9], Pu and Liu [19].
The family of all fuzzy semiopen [4] (resp. fuzzy preopen [32], fuzzy - [32], fuzzy -open [7], fuzzy semi-preopen [34], fuzzy regular open [4]) sets of (X; o) shall be denoted by F So (resp. F P o; F o; F o; F SP o; F Ro).
Fuzzy minimal structures are de…ned and investigated in [3]. A subfamily m IX is said to be a fuzzy minimal structure on X i¤ 1
X 2 m for each 2 I and
the elements of m are called fuzzy m-open sets. A fuzzy supratopology [1] is a subfamily g of IX; satisfying 0
X, 1X 2 g and arbitrary union of members of g
belongs to g. In addition, if g satis…es these conditions except 1X 2 g; then g is
said to be a generalized fuzzy topology (brie‡y GFT) in [26]. In the sequal, the set of all fuzzy points in X is denoted by P:
3. Generalized Fuzzy Neighbourhood Systems Let us de…ne
: P ! 2IX satisfy V (x) for V 2 (x )
Then we shall say that V 2 (x ) is a generalized fuzzy neighbourhood (brie‡y GFN) of the fuzzy point x and is a generalized fuzzy neighbourhood system (brie‡y GFNS) on the set of fuzzy points in X. We denote by (P) the collection of all GFNS’s on P:
For an arbitrary fuzzy set A 2 IX, write x 2 P
;A i¤ there exists V 2 (x )
satisfying V A:
De…nition 3.1. Let 2 (P) and A 2 IX: Then de…ne the fuzzy set { A as:
({ A) (x) = ( sup x 2P ;A ; 9 2 I satisfying x 2 P ;A 0 ; otherwise for all x in X: { A is called the interior of A on : Lemma 3.2. Let 2 (P) : Then
(a) { 0X = 0X;
(b) { A A; for A 2 IX;
(c) A B implies { A { B; for all A; B 2 IX:
Proof. (a) Since P ;0X = ;; we have ({ 0X) (x) = 0 for all x in X: Thus { 0X= 0X:
(b) Clearly { 0X 0X. Let A 6= 0X and an arbitrary x in X: If ({ A) (x) = 0;
then { A A: If ({ A) (x) = sup
x 2P ;A
:= t > 0; then there exists 2 I satisfying x 2 P ;A. In this case, let xtj 2 P ;Afor j 2 J 6= ;; then there exists Vj2 xtj
satisfying tj Vj(x) A (x) for each j 2 J: Thus t = supj2Jtj A (x) : Therefore
{ A A:
Proposition 1. Let an arbitrary (Aj)j2J IX. Then S j2J { Aj { S j2J Aj ! :
Proof. Clearly Aj [j2JAj for each j 2 J: Then { Aj { S j2J Aj ! for each j 2 J by Lemma 3.2(c) : Hence S j2J { Aj { S j2J Aj ! :
Lemma 3.3. Let 2 (P) and g = G 2 IX: G = { G : Then g is a GFT on X:
Proof. Clearly, 0X 2 g by Lemma 3.2(a) : Let G = [j2JGj and Gj 2 g for j 2
J 6= ;. Then { G G is clear by Lemma 3.2(b) : On the other hand we have
G S
j2J
{ Gj since Gj 2 g for each j 2 J: In addition, S j2J
{ Gj { G is clear by
Proposition1 Therefore G { G: Hence G 2 g:
So it is clear that every GFNS generates a GFT. In this case we shall write g for this g.
Lemma 3.4. If g is a GFT on X; then there is a 2 (P) satisfying g = g : Proof. Let us de…ne V 2 (x ) i¤ x 2 V 2 g. Then clearly 2 (P). Now we have to prove that g = g :
(a) Let G 2 g and G (x) = t > 0 for an arbitrary x in X: Thus we have xt2 G 2 g
and so xt 2 P ;G for t 2 I: Therefore ({ G) (x) t = G (x) : Hence G 2 g by
Lemma 3.2(b) :
(b) Let G 2 g : If G = 0X; then G 2 g: If G 6= 0X; then there exists x in X such
that G (x) = t > 0: Then ({ G) (x) = sup
x 2P ;G
:= t > 0: In this case, let xtj 2 P ;G
for j 2 J 6= ;; then there exists Vj2 xtj satisfying Vj G for each j 2 J: Thus
Vx= [j2JVj G for (Vj)j2J g and Vx(x) = t: Now if we write V =
S
G(x)>0
Vx;
then V 2 g and V G: Now let an arbitrary z in X: If G (z) = 0; then clearly G V: If G (z) := l > 0; then there exists Vz 2 g satisfying Vz(z) = l V (z) :
Thus G V: Hence G 2 V:
Note that we shall write = g for the GFNS de…ned as: (x ) = fV : x 2 V 2 gg
for each x 2 P:
The following result is clear by Lemma 3.4.
Then we shall say that each fuzzy generalized topology on X can be generated by some generalized fuzzy neighbourhood system on X:
If g is a GFT on X and A 2 IX; then in the sense of [1] and [26] the interior
of A (we shall write igA) on g is de…ned as the union of all G A; G 2 g: Then
similarly we shall de…ne i for 2 (P) as i := ig :
Proposition 2. Let 2 (P) and A 2 IX: Then i A { A:
Proof. Let J 6= ; and (Gj)j2J IX denotes the elements of g satisfying Gj A:
Then i A = [j2JGj { S j2J Gj ! = { (i A) { A by Proposition 1 and Lemma 3.2(c) :
The following example shows that i A 6= { A; in general.
Example 3.5. Let X = fa; bg ; A and B be fuzzy subsets of X de…ned as follows: A (a) = 0:1; A (b) = 0:5
B (a) = 0:3; B (b) = 0:3 Now de…ne the GFNS as:
(a ) = fAg ; for 0 0:1 and (b ) = fBg ; for 0 0:3 (a ) = f1Xg ; for 0:1 1 (b ) = f1Xg ; for 0:3 < 1:
Then clearly ({ A) (a) = 0:1; ({ A) (b) = 0; while g = f0X; 1Xg and so i A = 0X:
Note that, for a GFT g IX; we shall write 2
g(P) i¤ V 2 g for V 2 (x ) ;
x 2 P:
Proposition 3. If 2 g(P) for the GFT g = g ; then { = i :
Proof. Let ({ A) (x) := t > 0 for an arbitrary x in X: Thus there exists 2 I such that x 2 P ;A: If we write xtj 2 P ;Afor j 2 J 6= ;; then there exists Vj2 xtj
satisfying Vj A for each j 2 J: Thus Vx = [j2JVj A for (Vj)j2J g : So
Vx= { Vx { A and this implies that Vx(x) = t: Hence { A i A:
4. Generalized fuzzy continuity
In this section we de…ne generalized fuzzy continuity with the help of the concepts introduced above.
De…nition 4.1. Let X and X0 be two sets, 2 (P), 02 (P0) and a function
f : X ! X0: Then f is said to be fuzzy ; 0 -continuous i¤, for each x 2 X and
V 2 0(f (x )) ; there is U 2 (x ) satisfying f (U) V:
Now let X be a set, g be GFT on X and suppose that : IX ! IX satis…es
Then de…ne
( ; g) (x ) = fV : V = G for some G 2 g such that x 2 Gg for each x 2 X. Clearly ( ; g) 2 (P) :
In the literature, various examples of fuzzy ; 0 -continuity can be found. Let o and o0 be fuzzy topologies on X and X0; respectively. The case = o; 0 = (co0; o0) is called fuzzy weak continuity in [22], while = o; 0 =
(cF So0; o0) gives fuzzy weakly semi-continuous maps in the sense of [12], =
(cF So; F So) ; 0= F So0gives fuzzy strongly irresolute maps of [17], = o; 0=
(cF So0; F So0) gives fuzzy semi-irresolute maps of [17], = o; 0 = (io0co0; o0)
gives fuzzy almost continuous maps of [2], = F o; 0= (io0co0; o0) gives fuzzy
almost -continuous maps of [25], = (ioco; o) ; 0 = (io0co0; o0) gives fuzzy
-continuous maps of [33], = F So; 0 = F Ro0 gives fuzzy almost semi-continuous
maps of [32].
Let X and X0 be two sets, g and g0 be two generalized fuzzy topologies on X
and X0; respectively. In the sense of fuzzy supra-continuity [1], we shall say that
f : X ! X0 is fuzzy (g; g0)-continuous i¤ G02 g0 implies f 1(G0) 2 g.
Proposition 4. A fuzzy ; 0 -continuous map is fuzzy g ; g 0 -continuous.
Proof. Let G0 2 g 0 and f 1(G0) (x) := t > 0 for an arbitrary x in X: Then
there exists 2 I such that f (x) 2 P00;G0 since f 1(G0) (x) = G0(f (x)) =
{ 0G0 (f (x)) : Therefore there is a GFN V 2 0(f (x) ) such that V G0: Since
f is fuzzy ; 0 -continuous, we have a GFN U 2 (x ) satisfying f (U) V and so U f 1(G0) : Thus x 2 P
;f 1(G0): Therefore t { f 1(G0) (x) : Hence
f 1(G0) = { f 1(G0) :
Proposition 5. If f is fuzzy g ; g 0 -continuous, = g and 0= g0 for some
GFT’s g IX and g0 IX0; respectively, then f is fuzzy ; 0 -continuous.
Proof. Let an arbitrary x 2 P and V0 2 0(f (x )) : Then V0(f (x)) and V0 2 g0 = g 0 by Corollary 1 Therefore f 1(V0) (x) and f 1(V0) 2 g : Thus
0 < { f 1(V0) (x) : Then there exists t such that x
t 2 P ;f 1(V0),
and so there is a GFN U 2 (xt) satisfying U f 1(V0) : Also U 2 (x ) since
= g: Therefore we have a GFN U 2 (x ) satisfying f (U) V0: Hence f is
fuzzy ; 0 -continuous.
Now let o and o0 be fuzzy topologies on X and X0; respectively. Then clearly
the case = o; 0 = o0 is fuzzy continuity [9] in the classical sense, = F o;
0 =
o0 is called fuzzy -continuity in [32], = F So; 0 = o0 is called fuzzy
semi-contunity in [4], = F P o; 0 = o0 is called fuzzy pre-continuity in [24],
= F o; 0 = o0 is called fuzzy -contunity in [7], = F o; 0 = F So0
semi-precontinuity in [34], = F o; 0= F o0 is called M - -fuzzy continuity in
[7], = F P o; 0 = F P o0 is called M -fuzzy precontinuity in [8], while = F So;
0 =
F So0 gives fuzzy irresolute maps of [21], = F o; 0 = F o0 gives fuzzy
-irresolute maps of [28], = F P o; 0= F P o0 gives fuzzy pre-irresolute maps of
[6], = F So; 0 = F o0 gives fuzzy semi- -irresolute maps of [31], = F o;
0 =
F P o0 gives fuzzy -pre-irresolute maps of [30].
5. Conclusions
This paper presents an extended study of fuzzy topology and fuzzy continuity with respect to generalized fuzzy neighbourhood systems. Similar results can be found by considering generalized q-neighbourhood system of fuzzy points in X, i.e.
q : P ! 2IX satisfy x qV for V 2 q (x ) :
Then a member of q (x ) shall be called as a generalized fuzzy q-neighbourhood (brie‡y GFQN) of the fuzzy point x .
References
[1] M.E Abd El-Monsef, A.E. Ramadan, On fuzzy supra topological spaces, Indian J. Pure and Appl. Math. 18(4)(1987), 322–329.
[2] N. Ajmal, S. K. Azad, Fuzzy almost continuity and its pointwise characterization by dual points and fuzzy sets, Fuzzy Sets and Systems, 34(1990), 81-101.
[3] M. Alimohammady, M. Roohi, Fuzzy minimal structure and fuzzy minimal vector spaces, Chaos, Solutions and Fractals, 27(2006), 599-605.
[4] K.K. Azad, On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82(1981) 14-32.
[5] T. Back, Evolutionary Algorithm in Theory and Practice, Oxford University Press. 1996. [6] S.Z. Bai, W.W. Liang, Fuzzy non-continuous mapping and fuzzy pre-semi-seperation axioms,
Fuzzy Sets and Systems, 94(1998), 261-268.
[7] G. Balasubramanian, On fuzzy -compact spaces and fuzzy -extremally disconnected spaces, Kybernetika, 33(3)(1997), 271-277.
[8] G. Balasubramanian, On fuzzy preseperation axioms, Bull. Cal. Math. Soc., 90 (1998), 427-434.
[9] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24(1968) 182-190.
[10] W.W. Chu, Neighbourhood and associative query answering, Journal of Intelligent Informa-tion Systems, 1(1992), 355-382.
[11] Á. Császár, Generalized topology, generalized continuity, Acta Math. Hungar., 96 (4)(2002), 351-357.
[12] S. Dang, A. Behera, S. Nanda, On fuzzy weakly semi-continuous functions, Fuzzy Sets and Systems, 67(1994), 239-245.
[13] C. De Mitri, E. Pascali, Characterization of fuzzy topologies from neighbourhoods of fuzzy points, J. Math. Anal. Appl. 93(1983), 1-14.
[14] K. Essenger K. Some connections between topological and Modal Logic. Mathematical Logic Quarterly, 41(1995), 49-64.
[15] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56(1976), 621-633.
[16] H. Maki, On generalizing semi-open and preopen sets, Report for Meeting on Topological Spaces Theory and its Applications, Yatsushiro College of Technology, (1996), 13-18. [17] S. Malakar, On fuzzy semi irresolute and strong irresolute functions, Fuzzy Sets and Systems,
45(1992), 239-244.
[18] A.S. Mashhour, A.A. Allam, F.S. Mahmoud, F.H. Khedr, On supratopological spaces, Indian J. Pure and Appl. Math. 14(4)(1983), 502–510.
[19] P.P. Ming, L.Y. Ming, Fuzzy topology. I. Neighbourhood structure of a fuzzy point and Moore -Smith convergence, J. Math. Anal. Appl. 76(1980) 571-579.
[20] A. Motro, Supporting goal queries in relational databases, Expert Database Systems, Pro-ceedings of the …rst International Conference, L. Kerchberg, Institute of Information Man-agement, Technology and Policy, University of S. Carolina., 1986.
[21] M.N. Mukherjee, S.P. Sinha, Irresolute and almost open functions between fuzzy topological spaces, Fuzzy Sets and Systems, 29(1989), 381-388.
[22] M.N. Mukherje, S.P. Sinha, On some weaker forms of fuzzy continuous and fuzzy open map-pings on fuzzy topological spaces, Fuzzy Sets ans Systems, 32 (1989), 103-114.
[23] M.N. Mukherjee, S.P. Sinha, On some near-fuzzy continuous functions between fuzzy topo-logical spaces, Fuzzy Sets and Systems, 34(1990), 245-254.
[24] A. Mukherjee, Fuzzy almost semi-continuous and almost semi-generalized continuous map-pings, Acta Cienc. Indica Math. 32(4)(2006), 1363–1368.
[25] H.A. Othman, S. Latha, Some weaker forms of fuzzy almost continuous mappings, Bull. of Kerela Math. Assoc. 5(2)(2009), 109-113.
[26] G. Palani Chetty, Generalized fuzzy topology, Ital. J. Pure and Appl. Mat. 24 (2008), 91-96. [27] Z. Pawlak, Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic
Pub-lishers, 1991.
[28] R. Prasol, S.S. Thakur, R.K. Saraf, Fuzzy -irresolute mappings, J. Fuzzy Math. 2(1994), 335-339.
[29] R.K. Saraf, S. Mishra, G. Navalagi, On fuzzy strongly -continuous functions, Bull. Greek Math. Soc. 47(2003), 153–159.
[30] K.K. Saraf, M. Caldas, N. Navalagi, On strongly fuzzy -preirresolute functions, Advances in Fuzzy Mathematics, 3(1)(2008), 19-25.
[31] V. Seenivasan, G. Neyveli, G. Balasubramanian, Fuzzy semi -irresolute functions, Mat. Bohemica, 132(2)(2007), 113-123.
[32] A.B. Shahna, On fuzzy strong semi continuity and fuzzy precontinuity, Fuzzy Sets and Sys-tems, 44(2)(1991), 303-308.
[33] S. Supriti, Fuzzy -continuous mappings, J. Math. Anal. Appl. 126(1987), 130-142. [34] S.S. Thakur, S. Singh, On fuzzy semi-preopen sets and fuzzy semi-precontinuity, Fuzzy Sets
and Systems, 98(1998) 383-391.
[35] L.A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353.
Current address : Ankara University, Faculty of Sciences, Dept. of Mathematics, Tando¼ gan-Ankara, TURKEY
E-mail address : ssagir@science.ankara.edu.tr, guner@science.ankara.edu.tr
, eyazar@science.ankara.edu.tr
