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Fuzzy nets and their application
Fuzzy Sets and Systems 51 (1992) 41-51 North-Holland
41
Fuzzy nets and their application Ratna Dev Sarma Department of Mathematics, University of Delhi, India
Naseem Ajmal Department of Mathematics, Zakir Hussain College (University of Delhi), Delhi, India Received April 1991 Revised October 19!)1
Abstract: Fuzzy net-theory is enriched by the introduction of the fuzzy net of fuzzy sets. The limsup, liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed. Alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets, fuzzy continuity, maps with closed fuzzy graph, open fuzzy mapping, etc. Thus the net theoretic approach is shown to be a promising tool in fuzzy topology.
Keywords: Fuzzy point; quasi-coincidence; neighbourhood; Q-neighbourhood; fuzzy net.
1. Introduction
The net-theoretic approach in fuzzy topology was first introduced by Pu and Liu [10]. Lowen [7] further extended its scope by establishing the equivalence of a fuzzy net and a pre-filter. In [1,2,3], the notion of fuzzy net is used in support of the theory of localization of fuzzy continuity and its weaker forms. Recently, Liu [11] has again thrown light upon this topic by discussing convergence classes of fuzzy nets. In this paper, we further enrich this theory by (i) introducing the concept of a fuzzy net of fuzzy sets, and (ii) showing the applications of fuzzy nets in fuzzy topology, particularly in relation to several fuzzy mappings. In [5], Chang has defined sequences of fuzzy sets. But our approach in defining a fuzzy net of fuzzy sets is different from that of Chang. Io fact, ours is a generalization of the fuzzy net of fuzzy points defined by Pu and Liu [10] and an Correspondence to: Dr. N. Ajmal, B85, Pandara Road, New Delhi 110 003, India.
extension of the notion of net of sets defined by Mrowka [12] in general topology. In the first section, we define the limsup, liminf and limit of a fuzzy net of fuzzy sets and discuss their various properties. With suitable examples, we demonstrate the effect of fuzzy topology on the convergence of a fuzzy net. The notion of open fuzzy mapping, initially introduced by Wong [14], has been studied by different authors. In [8] and [9], its properties and behaviour with respect to various fuzzy topological concepts are studied. Several weaker forms of open fuzzy mapping are discussed in [4] and [15]. While in [2], their relationships with other fuzzy topological mappings are studied, in [3] and [13], functions with closed fuzzy graph have been studied and used to show the interplay among various fuzzy topological concepts. Here, in this paper, we provide characterizations of an open fuzzy mapping and of a mapping with closed fuzzy graph with the help of fuzzy nets of fuzzy points as well as that of fuzzy sets. We characterize fuzzy continuity in two different ways: using fuzzy nets of fuzzy sets and that of fuzzy points respectively. In establishing these equivalences, the study remains consistent and analogous to its counterpart in general topology, wherever the later exists. Thus this paper establishes the nettheoretic treatment as a powerful tool for analysis in fuzzy topology, perhaps as successful as in general topology.
2. Preliminaries
Throughout the paper, X, Y, Z, etc. denote ordinary sets while/t, r/, a etc. denote fuzzy sets defined on an ordinary set. The fuzzy sets in this paper are defined with respect to the closed unit interval I = [0, 1]. The union and intersection of a family {/~i}i of fuzzy sets are denoted by ~/i/~i and / ~ ~i respectively. The constant fuzzy sets which take each member of X to zero and one respectively are denoted by 0x and lx
0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
42
respectively. A fuzzy point with support x and value a~, where 0 < tr ~< 1, is denoted by x~. The support of a fuzzy set ~ is denoted by supp ~. While x~ ~
Definition 2.1 [10]. Two fuzzy sets /, and r/ defined on a set X are said to be overlapping or quasi-coincident if there exists an x in X such that / z ( x ) + r / ( x ) > 1. For such an x, we say, /z and 7/ overlap at x. Usually we denote it by /zqr/. Otherwise /z and ~/ are said to be non-overlapping and denoted by/z ~ ~/. For the definition of a fuzzy topological space (or briefly fts), we follow Chang [5]. The notation for a f t s is (X, T(x)) (or, simply X), where T(x) is the fuzzy topology on X. The closure of a fuzzy set/~ is denoted by ft.
Definition 2.2 [10]. A fuzzy set /~ in a fts X is said to be a neighbourhood of a fuzzy point x~, if there exists an open fuzzy set T/in X, such that
overlap frequently with ~u if for every m in D, there exists an nm in D such that nm/>m and x~7~ q ~-
Definition 2.7 [10]. A fuzzy net {X%n}neD in a fts X is said to converge to a fuzzy point x~ iff {X~n}~D eventually overlaps with every Q-nbd of x~. We denote it by x~--->x~ and say that x~ is a limit of {x~n}~o. On the other hand, x,~ is called a clusterpoint of {x~,,}n~o iff {x~,}~o frequently overlaps with every Q-nbd of x~. Theorem 2.1 [10]. A fuzzy point x~ <<-fz iff # overlaps with every Q-nbd of x~. Theorem 2.2 [10]. A fuzzy point x , <~ft iff there is a fuzzy net {xnn}neD in l~ such that x n*o--->x~. Theorem 2.3 [10]. A fuzzy set Iz in a fts X is closed iff no fuzzy net in ~ can converge to a fuzzy point not contained in I~. Definition 2.8 [5]. A mapping f:X---~ Y from a fts X to fts Y is called fuzzy continuous i f f f - l ( r / ) is open in X for every open fuzzy set ~/in I". Definition 2.9 [14]. A mapping f:X--->Y is called an open fuzzy mapping iff f(/u) is open in Y for every open fuzzy set/z in X.
Definition 2.3 [10]. A fuzzy set ~ is said to be a quasi-neighbourhood (Q-nbd, in short) of a fuzzy point x~, if there exists an open fuzzy set r/ in X, such that x~ q r/~
Definition 2.10 [4]. Let/~ and r / b e fuzzy sets in X and Y respectively. T h e n / z × ~/ is a fuzzy set in X x Y, defined by
Definition 2.4 [10]. A fts X is called a T2-fts or a Hausdorff space if for any two fuzzy points xo~
for every (x, y) in X x Y.
and Yt~, where x =/=y, there exist Q-nbds/~ and r/ of x~ and ya respectively, such t h a t / z ^ ~/= 0x.
Definition 2.5 [10]. Let (D, >i) be a directed set and ~ be the collection of all fuzzy points in a set X. Then the function S:D---> ~ is called a fuzzy net in X. For n ~ D, its image under S is denoted by x n and hence the fuzzy net S is usually denoted by {X~}n~O. Ot n
Definition 2.6 [10]. A fuzzy net {x~},eo in a fts X is said to overlap eventually with a fuzzy set/~ if there exists an m in D such that x ~,, q/~ for all n ~>m. On the other hand, {X~o}n~D is said to
(/z × ~/)((x, y)) = min{/~(x), r/(y)}
3. Fuzzy net of fuzzy sets In this section, we introduce the notion of fuzzy net of fuzzy sets which extends the notion of net of sets by Mrowka [12] in general topology. We also discuss various aspects of the convergence of fuzzy nets of fuzzy sets. Definition 3.1. Let (D, >1) be a directed set and X be any non-empty set. Let ,~ be the collection of all fuzzy sets in X. The function S : D---> ,~ is called a fuzzy net of fuzzy sets in X. For n c D, we denote its image under S by /~n and hence a
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
fuzzy net of fuzzy sets is usually denoted by
{fn}n~O"
(XT,°}noO, limsupx],
and
liminfx]°
nED
In [10], a fuzzy net in X is defined to be a function from a directed set to the collection of all fuzzy points in X. Since a fuzzy point is a particular type of fuzzy set, the above definition generalizes the definition of a fuzzy net given in [10]. For clarity, we denote a fuzzy net of fuzzy sets by {f~}nEo, while that of fuzzy points by {x]o},Eo respectively, where (D, />) is the directed set. D e f i n i t i o n 3.2. Let S = {f,}n~o be a fuzzy net of fuzzy sets in a fts X. For any fuzzy point x~ in X, we say (i) x~ ~< lim supnED fn iff S frequently overlaps with every Q-nbd of x , ; that is, given a Q-nbd r/ of x , and any m c D, there exists an n/> m in D such that f , q T/. (ii) x~ <~ lim infnEo fn iff S eventually overlaps with every Q-nbd of x~; that is, given a Q-nbd r/ of x~, there exists an m • D such that fn q 7/ for every n/> m. In case such an x , does not exist, in the first and second case respectively, we define
limsupfn=0x nED
and
liminffn=0x.
43
nED
are simply the union of the cluster points and that of the limits of {X no~.},Eo respectively. D e f i n i t i o n 3.3. Let S = {fn}nED be a fuzzy net of fuzzy sets in a f t s X such that
lim sup f~ = lim inf fn nED
(--- f , say).
nED
Then we say that the fuzzy net S converges to f and denote it by limned f . = f . We also denote it by fn--> f . In the following examples, we demonstrate how the fuzzy topology can effect the convergence of a fuzzy net of fuzzy sets. Example 3.1. Let X = {x,y}. Consider the fuzzy net {fn}n~D, where D is the set of natural numbers under usual ordering, defined by 1 /~n(x) = - , n
1 fn(Y) = 1 - - , n
for every n • D. Case (i): Tl(X)= {0x, xo~, Ix}, where a~ is a real number, 0 < cr <~ 1. Then
nED
lim inf fn = xl_o~ v Yl Thus for a fuzzy net of fuzzy sets S = {f,}nED, lim SUpnED fin is the supremum of the collection of all the fuzzy points x~ such that S frequently overlaps with each Q-nbd of all these fuzzy points. On the other hand, liminfn~ofn is the supremum of the collection of all those fuzzy points x , such that S eventually overlaps with each Q-nbd of every fuzzy point of the collection. The justification of the above definitions lies in the facts that if x~ is any fuzzy point such that x ~ <~ lim sup f .
nED
=
lim sup fn. nED
Thus lim.Eo #. = x l _ . v y l in the fts (X, Tl(x)). Case (ii): Tz(x)={Ox, lx}U{x~:OXl_o~ v y l . Case (v): Ts(x)={Ox, l x } U { x , vYt~:0
n~D
and x~_E is any fuzzy point contained in x~, then x , _ , ~< lim sup fn. n~D
Similar is the case with lim infn~D fn- It can also be noticed that for a fuzzy net of fuzzy points
(-1) n
U.(x)=½+
2n
1
'
U.(Y)=I
n
for every n e D. Case (i): Tl(x) = {0x, Xl/2, lx}. Then lim inf/~. = xl/2 v Yt, nED
44
R.D. Sarma, N. Ajmal
/ F u z z y nets a n d their application
while lim supneo #~ = l x in the fts (X, Tl(X)). Case (ii): Tz(x) = {0x, x~, l x } , ½< a: ~< 1. H e r e fn --~ lx. Case (iii): T3(x) = {0x, x,~, l x } , 0 < o( < ½. Then lim f~ = Xl_or V Yl.
Again, if y# ~ #, then there exists a Q-nbd r/o of y~ and an mo in D such that #n ~ r/o for all n/> mo in D. Since E is confinal, there exists m~ in E such that #n~r/o for all n / > m ~ in E. Consequently, y# ~ lim SUpneE fin" This implies that
ned
Case (iv): T4(x) = {0x, x , v YI~, l x } , 0 < a', fl ~< 1. Then limn~n #n = lx. Case (v): Ts(x) = {0x, lx} U {x~:/3 < t r ~< 1}, where /3 • [1, 1]. Then lim, eD #, = I x in the fts
(x, Case (vi): T 6 ( x ) = {Ox, u 0< where fl ~ (0, ½). Then limn~o #n =Yl in the fts
lim sup #n ~< #. neE
Hence by Theorem 3.1, limnee#n = #. Lemma 3.1. For any fuzzy set I~o and a collection {#i}i, if ~'~0 ~ f i for each i, then
Uo Vi fi.
(x, Case (vii): T7(x) = {0x, lx} U {y/~: fll ~< fl ~< f12}, where ill, ~2 • (0, 1]. Here limneo #n = lX. Case (viii): T s ( x ) = { 0 x , l x } U {x,~ v yt~:O< a:, fl ~< 1}. Then limned #n = lx.
The proof is omitted.
Theorem 3.3. For any fuzzy net {fn}neO in a fts X, limsup#n-- A
Here we notice that in the first example, although limneo #, exists in all the cases, it is different for different fuzzy topologies, while in the second example, Case (i), lim~eo [~n does not even exist. Any fuzzy net defined on an indiscrete fts is always convergent. In the following theorems, we discuss some properties of limsup and liminf of a fuzzy net of fuzzy sets.
Theorem 3.1. For any f u z z y net {#n}neO in a fts X, lim inf #n <~ lim sup ~'/n" neD
neD
neD
V #~.
m ~ D n>~m
ProoL If x~ ~m such that #n, q r/, whence Vn~>m #n q q. Thus x~ ~< Vn~m #n, for each m • D. Therefore, it follows that
V
meD n~m
Again, if x~ ~ lim supneo #n, then there Q-nbd r/o of xo~ and mo • D such that fn each n ~>m0, whence Vn~m,, #n ~ r/o in the above lemma. Therefore, x~ ~ V From this,
Proof. The proof follows from the definitions directly.
exists a ~ r/o for view of . . . . . #,-
rneD n ~ m
This completes the proof.
Theorem 3.2. If a fuzzy net {#n}neO converges to # in a f t s X and E is a confinal subset of D, then the fuzzy net {#n}nee /S also convergent and limned kin = f .
Theorem 3.4. For any fuzzy net {~'~n}neD in a fts X, V
Proof. Let x~ ~<# and r / b e a Q-nbd of x~. Then { # , } , c o and hence { # , } , e e eventually overlaps with r/, whence #~
A #n ~< lim inf •n.
r e e D n>-rn
neD
Proof. Let xo, ~< V
A btn
meO n~m
45
R.D. Sarma, N. Ajmal / Fuzzy nets and their application and r/be a Q-nbd of x~. Then there exists some y in X, such that
rl(y) + (,,VD,An # n ) ( y ) > 1. In view of Lemma 3.1, this implies that there exists some m0 • D such that
m0 • D and a Q-nbd r/0 of x~ such that #, ~1r/0 for every n/> mo. If r/ is any Q-nbd of x~, then n ^ r / 0 is also a Q-nbd of x~, so that {#n v r / . } , , o frequently overlaps with r / ^ r/o. But #n~r/0 for every n>~mo implies that {r/,}~eo frequently overlaps with ~/. Thus x~ ~ lim sup r/,. neD
r/(y) + (hAm # n ) ( y ) > 1. Therefore, Again /~,~m #, q r/ implies that g, q r/ for every n ~>m0. Thus {#,}neo eventually overlaps with every Q-nbd of xo~. Therefore,
lim sup(#, v r/n) ~ lim sup #n v lim sup r/n.
x~ ~
The reverse inequality can easily be verified. This completes the proof.
neD
Consequently, V
neD
neD
nED
Theorem 3.7. For two f u z z y nets {#n},~o and {r/n}nEo in a fts X,
A #n ~
mED n~m
neD
lim inf(#n v r/n) = lim inf #. v lim inf r/n. neD
for n, n ' E D with n ~ n ' , then the f u z z y net {#n},Eo is convergent and
neD
neD
Theorem 3.5. If gn ~ # " ' ,
lim #n = A /~n. ned
The proof is on the same lines of the above proof. The following result is a direct consequence of the above two theorems.
neD
Theorem 3.8. If the f u z z y nets {#,),eD and {r/n}.Eo converge in a f t s X, then the f u z z y net
Proof. Clearly
{#, v tln},eO also converges and /~ /in ~< lim inf #n. neD
nED
lim(#n v ~/,) = lim #, v lim r/,. Also by Theorem 3.3,
ned
lim sup #, = A /~,. neD
ned
4. Application of fuzzy nets
neD
Theorem 3.6. For two f u z z y nets {#,},co and {T/,},EO in arts X,
lim sup(#n v r/,) = lim sup #, v lim sup r/n. neD
ned
neD
This section manifests some applications of the notion of a fuzzy net and in the process, provides alternative characterizations of several concepts in fuzzy topology. First, we start with the following result about a closed fuzzy set.
ned
Theorem 4.1. Let # be any f u z z y set in fts X. Proof. Let
x~, ~ lim sup(#, v r/,) ned
and x~ ~ l i m s u p , Eo#,.
Then there exists an
Then the following are equivalent: (i) # is a closed f u z z y set. (ii) For every f u z z y net {#, },eo in # (that is, #, <<,# for each n • D ), lim sup, co #n ~< #. (iii) For every f u z z y net {#n},eO in #, lim inf, eo #, ~< #.
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
46
Proof. Let /~ be a closed fuzzy set in X and {/~}nEO be a fuzzy net in p. Consider any fuzzy point x, ~
(ii) For each fuzzy net {x~.}nEo in X, x ] --~ x~ then f (x n~.) f (x~) in Y.
if
Here, following a technique provided in [1], we offer another net-theoretic characterization of fuzzy continuity in terms of fuzzy net of fuzzy sets. Lemma 4.1. For any mapping f :X---> Y from a fts X to a f t s Y, Iz q r! in X implies f(lz) qf(r/) in Y. Theorem 4.4. Let f :X--> Y be a mapping from a fts X to a fls Y. Then the following are equivalent: (i) f is fuzzy continuous. (ii) For any fuzzy net {/~n}~EO in X,
x~ ~
whence by condition (iii), x~ ~<~. Therefore, by Theorem 2.3, # is closed. Thus, condition (iii) implies condition (i). This completes the proof. As a consequence of this theorem, we obtain the following result. Theorem 4.2. Let g be any f u z z y set in a fts X. Then the following are equivalent: (i) ju is an open f u z z y set. (ii) For every f u z z y net S = {/~n}nEO in X, if limsupnEo/z, overlaps with It, then S overlaps frequently with It. (iii) For every fuzzy net S = {#n}nEO in X, if liminfnEo/~n overlaps with Iz, then S overlaps eventually with tt. Corollary 4.1. A fuzzy set Iz in a fts X is open iff whenever a fuzzy net of fuzzy points converges to some f u z z y point overlapping with Iz, the f u z z y net overlaps eventually with It.
f ( l i m inf #,,) ~
neD
/
n~D
(iii) For any net {gn},,cO in X, f(lim sup #n) ~
nED
nED
Proof. Let condition (i) hold. Consider any fuzzy net {~n}nED in X. Let x~ ~
and r/ be a Q-nbd of f ( x ~ ) in Y. As f is fuzzy continuous, f-~(r/) is a Q-nbd of x~, in view of Theorem 4.2. As a result, {/~n}nEO eventually overlaps withf-l(r/). Consequently, by Lemma 4.1, {f(#n)}nEo eventually overlaps with f ( f - l ( q ) ) and hence with r/, as f(f-l(r/))~< 7/. Now, 77 being an arbitrary Q-nbd of f ( x , ) , this implies that f(x~) <- lim inff(/~n). nED
In [10], Pu and Liu have characterized fuzzy continuity in terms of fuzzy net and Q-nbds of fuzzy points respectively as follows.
Thus, f ( l i m inf ~t,) ~ lim inf f(/~.). \
Theorem 4.3. A mapping f :X ~ Y from a fts X to a fts Y is f u z z y continuous iff any one of the following holds: (i) For each fuzzy point x~ in X and each Q-nbd 71 of f(x~) in Y, there is a Q-nbd i~ of x~ such that f ( # ) <~~1.
nED
/
nED
In this discussion, if we replace 'eventually' by 'frequently' then it becomes clear that the condition (i) implies condition (iii) also. Finally, we show that if condition (i) does not hold, neither (ii) nor (iii) can hold. So, let f be
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
not fuzzy continuous. T h e n there exists an x~ in X and a Q-nbd T/ of f ( x ~ ) in Y, such that f(#~) $ r/, for every Q-nbd/z~ of x~. Let A be the collection of all Q-nbds of x~ in X. Then (A, >/) is a directed set, where '~>' is the inverse relation of 'fuzzy set inclusion'. This is clear from the fact that reflexivity and transitivity are obvious in A. Also for any l-g1 and ~2 in A, #x ^ ~2 is also in A with #1 A #2 ~ #1 and ~-~1A ~'~2~ ~'~2, where '~<' is the 'fuzzy set inclusion relation'. Now as f(#~) $ r/, there exists some yg in Y such that (f(g~))(y~) > r/(y~). Then there exists a real positive number try, such that
(f(i, ti))(y i) > oc; "> r/(yi), where ol; = 1 - oil. Now
( f(~ / ))(y' ) =
sup
(~i(x')} >
.;.
x~f-~(y 9
Therefore, f-~(y~) is non-empty and there exists x i in f - ~ ( y i ) such that #i(x i) > 1 - vii. Consider the fuzzy net S = {S~,}t,:a, where s~,, -- xZ~,. It is clear that x~ ~< lim inf s~, ~.ieA
such that (i) T = S o N , yN(i)
that is, for each l e E ,
47
x~,=
etN(i)•
(ii) For each n ~ D, there exists some m ~ E such that for every p ~ E with p >i m, N ( p ) >! n. Theorem 4.5. A f u z z y net (x].},~o converges to
a f u z z y point x~ iff each of its f u z z y subnet converges to x~. Proof. Let the fuzzy net { x ~ } , ~ o converge to x~ and let {y#'nJmeE be a fuzzy subnet of {x~.),~o. Consider any Q-nbd 1/ of x~. Then there exists m e D, such that x~,~q r/ for every n t> m in D. For this m, there exists t in E such that for every p>~t in E, N ( p ) > ~ m , where N:E--->D is the function stated in Definition 4.1. Now, y ~ = xNt in E, N ( p ) > ~ m , implies that y~, q 7/for every p >i t. Thus {Y~'m},,~E eventually overlaps with r/. Consequently, {y#rn}m~ e converges to x~. The other part of the proof is obvious as every fuzzy net is a fuzzy subnet of itself. This completes the proof. From now onward, a fuzzy subnet of a fuzzy net {x~.},~o will be denoted by ~..N(m)l the l."" OCN(m) J m c E ~ justification for which is evident from the definition.
and hence x~ ~< lim sup~,:A S,,, by T h e o r e m 3.1. But due to the existence of r/ as a Q-nbd of f ( x , ) , it is clear that
Theorem 4.6 [10]. A f t s X is a T2-fts iff no f u z z y
f(x~) ~ lim i n f f ( s . )
net can converge to two f u z z y points with distinct support.
gtifiA
as well as
f(x~) ~ lim sup f(s~,,). Thus if the condition (i) is violated, both condition (ii) and condition (iii) are violated. This completes the proof. In fact, if the range space is a Te-fts, the condition for fuzzy continuity of f given in Theorem 4.2(ii) can be relaxed to a good extent. We shall discuss this in a theorem which is in the spirit of Theorem 5.1 [6] of general topology. Definition 4.1 [10]. A fuzzy net T = {X mflm}meE in X is called a f u z z y subnet of a fuzzy net S={y%o}n~o iff there is a function N:E--->D
Lemma 4.2. Let (A, >a) and (B, >b) be two disjoint directed sets which are isomorphic (that is, there is a one-to-one function h f r o m A to B such that o6 >~ re2 iff hO¢l >bhOl2). Then there is a directed set (C,
{ymr.},,~a be f u z z y nets in X with disjoint directed sets A and B which are isomorphic. Then there is a f u z z y net {z~p}p~c in X such that {X~}n~a and {Y'~,}m~B are f u z z y subnets of {z~p}p~C and z~['-~ to, iff x~ --->too and Yom m to#" We omit the proofs of the lemmas as they are either identical or similar to their counterparts in general topology.
48
R . D . Sarma, N. A j m a l [ F u z z y nets a n d their application
Now we come to our proposed theorem. Theorem 4.7. Let f :X--+ Y be a mapping from a fts X to a T2-fts Y. Then the following are equivalent: (i) f is continuous. (ii) If any fuzzy net {X~.},~D converges to x~ in X, there exists y~ in Y such that the fuzzy net {f( X n~.)},~o converges to y~. (iii) If {XT,.},~D converges to x~ in X, there exists a fuzzy subnet {Xo,~,..,}.,~E N(m) n of {X,~.}..O in X such that {f(x:~<:>)}m+E N(m) converges to f(x~) in Y. (iv) For each x: in X, there exists y: in Y such that whenever {x~:},+D converges to x:, there exists a fuzzy subnet f"t.4, ~ NOtlV,mlJm~E (m)~l n o f {Xotn}n~ D such that { f (x~N~m~) N(m) }m,e converges to f (x~). Proof. Since each of the conditions (ii), (iii) and (iv) is clearly implied by fuzzy continuity in view of Theorem 4.3, we only need to show that each of these conditions implies fuzzy continuity. Assume that condition (ii) holds. Suppose, if possible, f is not fuzzy continuous. Then by Theorem 4.2, there exists a fuzzy n e t { X n n } n e A in X such that x ,~, --+ xo, in X, but f(x$~) - ~ f ( x , ) in Y. Now, by condition (ii), as x$---~x~, it follows that f(x$~)--+y~ for some Yo, in Y. Let {Yo~}m~B be the fuzzy net in X such that yom = x~ for all m in B, where B is a directed set disjoint but isomorphic to A. Thus by Lemma 4.3, there is a fuzzy net {z~}p~c in X such that {X~.}.~A and {Y'~.,},,,,n are fuzzy subnets o f (zPflp}pEC and z~'-*xo,. Then {f(x~,.)},~a and {f(yom)},~n are fuzzy subnets of {f(z~,)}p~c. However f ( X n~.)-~ y~ and f(y'ffm)-~ f(xo, ). Since Y is T2, {f(z~,)}p,c cannot converge unless f(x,~)=y~, in view of Theorem 4.5 and Theorem 4.6, which contradicts condition (ii). Thus condition (ii) implies condition (i). Assume that condition (iv) does hold. We show that condition (iii) also holds. Let x~.-~ x~ in X and y~ be the fuzzy point in Y corresponding to x~ in X. Then, if any fuzzy net converges to x~, the image fuzzy net of some fuzzy subnet of this will converge to y,~. Let {Y,~.}meBm be a fuzzy net in X where Yo~mm_- Xc, for m ~ trl all m in B. Then y ~ x~ and f(y~,)--->f(x~). Therefore, the image fuzzy net of any fuzzy subnet of {Y'~.}m~8 converges to f(x~), by Theorem 4.4. Since Y is T2, therefore by Theorem 4.6, f(x~) = y~.
Finally we show that condition (iii) implies fuzzy continuity. Let condition (iii) hold and if possible, let f be not fuzzy continuous. Then there exists an x~ in X and a Q-nbd 0 of f(x~) such that f ( # ~ ) ~ r/, for every Q-nbd #i of xo, Then proceeding as in the last part of Theorem 4.4, we can construct a fuzzy net S = {Su,},:A in X. This fuzzy net converges to x~ in X. But no image fuzzy net of any fuzzy subnet of S converges to f(x~), contradicting condition (iii). Consequently f is fuzzy continuous. This completes the proof. Lemma 4.4. For any mapping f :X--+ Y from a fts X to afts Y, if {Y~.}neD is a fuzzy net of fuzzy points in Y, {f -1 (Y~.)},~o is a fuzzy net of fuzzy sets in Y. The proof is trivial. Lemma 4.5. Let f :X--+ Y be any mapping and {Y].}~D be a fuzzy net in Y. If za ~ lim sup f - l ( y ] , ) , nED
then there exists a fuzzy net in X which converges to z, and image fuzzy net of which is a fuzzy subnet of {Y].},,o. Proof. Let {Y~.}.~o be a fuzzy net in Y and zt3 ~
-I
n (Yo,,).
Then for each n • D and for each Q-nbd r/of za, there exists an m(n, rl)>-n in D such that ,Y~m<,.,> I q r/. Let
D O= {m(n, r/): n • D, r/ is a Q-nbd of zt~}. Define '~>' in D O by re(n, rl)>~m(no, r/0) iff n i> no and r / ~ r/0. It easily follows that (D °,/>) is a directed set. Now for every re(n, 77)• D °, f-l(ym(~,~n)) q r/. Let f-~(" ,,(n,o)x and r/ overlap j ,y~(..,~) at X m(n'n). Clearly,
f (~.x m(n'n)'l Ogm(n,vl)fl =vm(n'n) .70~rn(n,~) for every m(n, q) in D °. It can be easily verified that I,,m(n.n)~ [ Y oLm(n.n) J m(n,rl)c:D 0 is a fuzzy subnet of {Y~.)~¢D. Then {.¢m(n,r/)] "*~O6n(n,,O f m ( n , ~ ) ~ D 0 is a fuzzy net in X such that its image fuzzy net is a fuzzy subnet of the given fuzzy net.
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
Next we show that t~mtn'~)/ ~..~ot,,(,.,D I m(n.n )ED, converges to z#. Let r/o be a Q-nbd of z#. Due to our choice of X m(n'~l) ~,,~,.,~, X m(n'"°) ~,,,.,~ q r/o, where n is arbitrary but fixed in D. Let m ( n ' , rl')>~ m(n, r/o). Then r/'<~r/o and as before, xm(n',rl') ,'1 ~ t . hono~ "m(n"rl') " " Thus Ogm(n" rl') %1 #~ "~ IJt~ltl~¢~ ~'~ O(m(n' O, } ~'~ r i O " {xm(n'r/.)'~ ..... ~ D o eventually overlaps with every Q-nbd of z~. Therefore this fuzzy net converges to za. This completes the proof. °
49
{• Y nJ ~ o converge to z, in Y and w be an element in
supp{ limEsupf - 1(y~,,) } • That is, there exists some positive real number o, where 0 < o ~< 1, such that wo <~lim supf-l(y~.). NED
Theorem 4.8. Let {x~.},Eo and {Y],},Eo be f u z z y nets in fts X and Y respectively. Then n ~ n x o,. x~ in X and y o~.---~Yt3 in Y implies that Ix",
[x,
in the product fis X × Y. Conversely, [x", y"]~o---> [x, y], in X × Y implies that xo~ ~ ---~x , in g and n ----9, Yo~. y, in Y re,wectively.
For the proof, we refer the reader to [13]. In what follows we provide net-theoretic characterizations of mappings with closed fuzzy graph, and open fuzzy mapping respectively. For detailed investigations of these two types of mappings, see [3, 13] and [2, 4, 8, 9, 15] respectively. The following theorem and the last two in fact, extend Theorem 1.2, Theorem 6.2 and Theorem 6.3 of Fuller [6]. At first, we recall the following definition from [13]. Definition 4.2. Let f : X - - - , Y be a function from
a fts X to a f t s Y. f is said to have closed f u z z y graph if the characteristic function Zc,, where G = {(x, f ( x ) ) : x • X},
is a closed fuzzy set in the product fts X x Y. Theorem 4.9. Let f : X ~ Y be a function f r o m a fts X to a fts }I. Then the following are equivalent: (i) f has closed f u z z y graph. (ii) I f {Y~.},ED converges to zo, in Y, then supp{ limsup f - ' ( y ~ . ) } m_supp{f-'(z~)}. (iii) If {Y~°}.Eo converges to z~ in #, then
Then by Lemma 4.5, there exists a fuzzy net {x~'.},,,,e in X such that x'~---~wo and n {f( X m#,.)}m~e is a fuzzy subnet of {Y~.},Eo. Thus, x'~,---*wo and f(x~.~)---~z~, by Theorem 4.5. Therefore, by Theorem 4.8, [X m, f (xm)]flm -'-~ [W,
Z]min{o,.o}.
As %c is closed by condition (i), it follows that [w, Z]mi,~o.~ <~Xc, in view of Theorem 2.3. Or, in other words, f ( w ~ ) = z ~ . Thus w, <~f-t(z~). Consequently, w ~ supp{f-~(z~)}. Hence supp{limsup f-l(y~,.) } c supp {f-l(z~)}. Therefore, condition (i) implies condition (ii). That condition (ii) implies condition (iii) is obvious in view of Theorem 3.1. Let condition (iii) hold. We show that f has closed fuzzy graph. Let {[x~, f(x")]~,}~,o be any fuzzy net in X~ which converges to [w, z]~ in X x Y. Then, by Theorem 4.8, x~°--->wa and n n _._> f(x~.)--~z#. Write f ( X n~°)=y~. Then y~. ya. n _~ n Again, in view of the fact that x~. ~ f - - 1 (y..) and n x.~ w#, it follows that for every Q-nbd ~ of w~, there exists too• D, such that f - t ~y~°) . . . . q It for all n I> too; hence w# ~
Therefore, by condition (iii), w e supp{f-~(zt0}. That is, f ( w ) = z. Thus [w, z]a <~Xc.. Consequently Xc is a closed fuzzy set in X × Y, by Theorem 2.3. This completes the proof. Theorem 4.10. A one-to-one surjection f : X --* Y
supp{ lira ionff - ' ( y : . ) } c supp{f-'(z.)}.
is an open f u z z y mapping iff for every f u z z y net {x~°},Eo in X with f(x~.)--*f(x~) in Y, x ~°--+ x~, in X.
Proof. Suppose that condition (i) holds. Let
Proof. Let f be an open fuzzy mapping and
50
R . D . S a r m a , N. A j m a l
/ F u z z y nets a n d their application
{X~,}neO be a fuzzy net in X such that f(x~)---~f(x~), for some x~ in X. Let /z be an
(f(/~0))(y ~°) > 1, SO that
open Q-nbd of x~ in X. Then f(/~) is an open fuzzy set in Y and in view of Lemma 4.1, f(x~) qf(/,). That is, f(t~) is a Q-nbd of f(xo~) in Y. As f ( x ~n ) ~ f(x~), the fuzzy net {f( X nJ } n e o eventually overlaps with f(p). Now f being one-to-one, this implies that x no~n eventually overlaps with/*. Consequently, x~,--* x~ in X. Conversely, let t~ be an open fuzzy set in X. Let {Y~}n~o be a fuzzy net in Y converging to some fuzzy point y~, where Yo, qf(/*). Now f being onto, there exist x n and x~ in X, such that f ( x n J - Y_ _o , ,n for every n in D and f(x~)=y~. Thus f(x~n)---~f(x~) and f(x~) qf(/z). Since f is one-to-one, this gives that x~ q/~. Thus /~ is a Q-nbd of x~. Now, following the given condition, x ~ ___>x~. Therefore {x~}n~D n eventually overlaps with/~. That is, there exists m in D, such that x ~,~q/~ for all n / > m ; hence n f ( ~X )n q f ( / ~ ) for all n~>m. That is, {Y~}neo eventually overlaps with f(/,). Hence by Corollary 4.1, f(/z) is an open fuzzy set in Y. Consequently, f is an open fuzzy mapping. This completes the proof.
/3no+ sup {#o(Xn)}>l. x~e/-J(y,.O)
Theorem 4.11. Let f :X--> Y be a function from
a f t s X to a f t s Y. The the following are equivalent: (i) f is an open fuzzy mapping. (ii) If {Y~,)n~o converges to YIJ in Y, then
But then, flno+PO(xn°)> l for some x n° f-l(yno), whence x~°0q ~o and x~° <~f-l(v~O). v-no
~
x~
~-n0~
no That is, f - - 1 (Ya~,) q/~o, which is a contradiction. Now, f being open, f(/*o) is an open fuzzy set in Y. Also, as x,, q/~0 in X, by Lemma 4.1 we have f(x~,)qf(ito) in Y. Thus f(/~o) is a Q-nbd of f(x~). Again, x~ ~
x~ ~
Consequently,
f-l(ya) <~lim inff-l(y~,). neD
Clearly, if condition (ii) holds, then condition (iii) holds, in view of Theorem 3.1. Finally, we assume that condition (iii) holds. Suppose f is not open. Then there exists an open fuzzy set /~ in X such that l x - f ( t z ) is not a closed fuzzy set in Y. Thus there exists ya ~< l r - f ( ~ ) such that ya ~ lv - f ( # ) . Then by Theorem 2.2, there exists a fuzzy net {Y~n}neO in ly --f(/*) converging to ya. Now, ya ~ l r - f ( / ~ ) implies that
f-l(y#) ~ lim inff-~(y~). ned
(iii) If { Y ~ } ~ o converges to Yt~ in Y, then
f-'(y#) <-lim supf-l(y~,). ned
Proof. Let condition (i) hold. Let {Y~n}~eo converge to Yt~ and x~ <~f-~(YtJ). Suppose, if possible, x~ ~ lim inff-l(y~,). ned
Then there exists an open Q-nbd /Zo of x~ such that {f-~(Y~,)}n~o is not eventually overlapping with it0. From this it follows that {Y~)neo is not eventually overlapping with f(/Zo). For if n0 Ya~0qf(#o) for some no in D, then fin0+
/3+
sup {/~(x)} > 1. xe/-l(y)
Therefore, f - l ( y ) is non-empty. Tiros there exists x ef-~(y) such that x a q/z, or, in other words, xa ~
so that {f-X(y~n)}ne D frequently overlaps with /~, and hence {Y~)n~o frequently overlaps with f(/~). Thus for a given n e D, there exists m in D such that
tim + (f(lt))(Y m) > 1. That is, y m ~ l r - f ( b * ) , which contradicts the choice of {Y~n}n~O in l r - f ( / * ) . Hence our
R.D. Sarma, N. A]mal / Fuzzy nets and their application
assumption is wrong and consequently f is an open fuzzy mapping. Theorem 4.12. If the mapping f : X--> Y is open, having closed fuzzy graph, then Y~.-->Yt3 in Y implies that
supp{fim f-l(y~.)} : supp{f-l(yt~)}. Conversely, if y~ --->yt~ in Y implies that
lim f-~(y~.) = f-l(yt3 ),
neD
then f is an open fuzzy mapping and has closed fuzzy graph. In both the cases it is assumed that lim,~o f-l(y~.) exists. Proof. By Theorem 4,9 and Theorem 4.11, if f is open and has closed fuzzy graph, then y~---~yo in
Y implies that supp{f-'(Yt3)} ~_supp{lim ionff-~(Y~.)}
~_supp{ limsup f-'(y;.)} ~_supp{f-l(y,)}, in view of Theorem 3.1. By the same theorems, the converse part follows immediately. Acknowledgement
The first author is highly indebted to one of his supervisors, Professor K.C. Gupta for his encouragement during the preparation of the paper.
51
References [1] N. Ajmal, Fuzzy continuity and its pointwise characterization by dual points and fuzzy nets, communicated. [2] N. Ajmal and S.K. Azad, Fuzzy almost continuity and its pointwise characterization by dual points and fuzzy nets, Fuzzy Sets and Systems 34 (1990) 81-101. [3] N. Ajmal and R.D. Sarma, Various fuzzy topological notions and fuzzy almost continuity, Fuzzy Sets and Systems 50 (1992) 209-223. [4] K.K. Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weak continuity, J. Math. Anal. Appl. 82 (1981) 14-31. [5] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [6] R.V. Fuller, Relations among various continuous and non-continuous functions, Pacific J. Math. 253 (1968) 495-509. [7] R. Lowen, Relation between filter and net convergence in fuzzy topological spaces, Fuzzy Math, 4 (1983) 41-52. [8] S.R. Malghan and S.S. Benchalli, On fuzzy topological spaces, Glas. Mat. 16 (1981) 313-225. [9] S.R. Malghan and S.S. Benchalli, Open maps, closed maps, and local compactness in fuzzy topological spaces, J. Math. Anal. Appl. 99 (1984) 338-342. [10] Pu Pao Ming and Liu Ying Ming, Fuzzy topology I: Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl 76 (1980) 571-599. [11] Liu Ying Ming, On fuzzy convergence classes, Fuzzy sets and Systems 30 (1989) 47-51. [12] S. Mrowka, On convergence of nets of sets, Fund. Math. 45 (1958) 237-246. [13] R.D. Sarma and N. Ajmal, Functions with closed fuzzy graph in fuzzy topological spaces, Communicated. [14] C.K. Wong, Fuzzy topology: product and quotient theorems, J. Math. Anal. Appl. 45 (1973) 697-704. [15] T.H. Yalvac, Fuzzy sets and function on fuzzy spaces, J. Math. Anal. Appl. 126 (1987) 409-423.
41
Fuzzy nets and their application Ratna Dev Sarma Department of Mathematics, University of Delhi, India
Naseem Ajmal Department of Mathematics, Zakir Hussain College (University of Delhi), Delhi, India Received April 1991 Revised October 19!)1
Abstract: Fuzzy net-theory is enriched by the introduction of the fuzzy net of fuzzy sets. The limsup, liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed. Alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets, fuzzy continuity, maps with closed fuzzy graph, open fuzzy mapping, etc. Thus the net theoretic approach is shown to be a promising tool in fuzzy topology.
Keywords: Fuzzy point; quasi-coincidence; neighbourhood; Q-neighbourhood; fuzzy net.
1. Introduction
The net-theoretic approach in fuzzy topology was first introduced by Pu and Liu [10]. Lowen [7] further extended its scope by establishing the equivalence of a fuzzy net and a pre-filter. In [1,2,3], the notion of fuzzy net is used in support of the theory of localization of fuzzy continuity and its weaker forms. Recently, Liu [11] has again thrown light upon this topic by discussing convergence classes of fuzzy nets. In this paper, we further enrich this theory by (i) introducing the concept of a fuzzy net of fuzzy sets, and (ii) showing the applications of fuzzy nets in fuzzy topology, particularly in relation to several fuzzy mappings. In [5], Chang has defined sequences of fuzzy sets. But our approach in defining a fuzzy net of fuzzy sets is different from that of Chang. Io fact, ours is a generalization of the fuzzy net of fuzzy points defined by Pu and Liu [10] and an Correspondence to: Dr. N. Ajmal, B85, Pandara Road, New Delhi 110 003, India.
extension of the notion of net of sets defined by Mrowka [12] in general topology. In the first section, we define the limsup, liminf and limit of a fuzzy net of fuzzy sets and discuss their various properties. With suitable examples, we demonstrate the effect of fuzzy topology on the convergence of a fuzzy net. The notion of open fuzzy mapping, initially introduced by Wong [14], has been studied by different authors. In [8] and [9], its properties and behaviour with respect to various fuzzy topological concepts are studied. Several weaker forms of open fuzzy mapping are discussed in [4] and [15]. While in [2], their relationships with other fuzzy topological mappings are studied, in [3] and [13], functions with closed fuzzy graph have been studied and used to show the interplay among various fuzzy topological concepts. Here, in this paper, we provide characterizations of an open fuzzy mapping and of a mapping with closed fuzzy graph with the help of fuzzy nets of fuzzy points as well as that of fuzzy sets. We characterize fuzzy continuity in two different ways: using fuzzy nets of fuzzy sets and that of fuzzy points respectively. In establishing these equivalences, the study remains consistent and analogous to its counterpart in general topology, wherever the later exists. Thus this paper establishes the nettheoretic treatment as a powerful tool for analysis in fuzzy topology, perhaps as successful as in general topology.
2. Preliminaries
Throughout the paper, X, Y, Z, etc. denote ordinary sets while/t, r/, a etc. denote fuzzy sets defined on an ordinary set. The fuzzy sets in this paper are defined with respect to the closed unit interval I = [0, 1]. The union and intersection of a family {/~i}i of fuzzy sets are denoted by ~/i/~i and / ~ ~i respectively. The constant fuzzy sets which take each member of X to zero and one respectively are denoted by 0x and lx
0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
42
respectively. A fuzzy point with support x and value a~, where 0 < tr ~< 1, is denoted by x~. The support of a fuzzy set ~ is denoted by supp ~. While x~ ~
Definition 2.1 [10]. Two fuzzy sets /, and r/ defined on a set X are said to be overlapping or quasi-coincident if there exists an x in X such that / z ( x ) + r / ( x ) > 1. For such an x, we say, /z and 7/ overlap at x. Usually we denote it by /zqr/. Otherwise /z and ~/ are said to be non-overlapping and denoted by/z ~ ~/. For the definition of a fuzzy topological space (or briefly fts), we follow Chang [5]. The notation for a f t s is (X, T(x)) (or, simply X), where T(x) is the fuzzy topology on X. The closure of a fuzzy set/~ is denoted by ft.
Definition 2.2 [10]. A fuzzy set /~ in a fts X is said to be a neighbourhood of a fuzzy point x~, if there exists an open fuzzy set T/in X, such that
overlap frequently with ~u if for every m in D, there exists an nm in D such that nm/>m and x~7~ q ~-
Definition 2.7 [10]. A fuzzy net {X%n}neD in a fts X is said to converge to a fuzzy point x~ iff {X~n}~D eventually overlaps with every Q-nbd of x~. We denote it by x~--->x~ and say that x~ is a limit of {x~n}~o. On the other hand, x,~ is called a clusterpoint of {x~,,}n~o iff {x~,}~o frequently overlaps with every Q-nbd of x~. Theorem 2.1 [10]. A fuzzy point x~ <<-fz iff # overlaps with every Q-nbd of x~. Theorem 2.2 [10]. A fuzzy point x , <~ft iff there is a fuzzy net {xnn}neD in l~ such that x n*o--->x~. Theorem 2.3 [10]. A fuzzy set Iz in a fts X is closed iff no fuzzy net in ~ can converge to a fuzzy point not contained in I~. Definition 2.8 [5]. A mapping f:X---~ Y from a fts X to fts Y is called fuzzy continuous i f f f - l ( r / ) is open in X for every open fuzzy set ~/in I". Definition 2.9 [14]. A mapping f:X--->Y is called an open fuzzy mapping iff f(/u) is open in Y for every open fuzzy set/z in X.
Definition 2.3 [10]. A fuzzy set ~ is said to be a quasi-neighbourhood (Q-nbd, in short) of a fuzzy point x~, if there exists an open fuzzy set r/ in X, such that x~ q r/~
Definition 2.10 [4]. Let/~ and r / b e fuzzy sets in X and Y respectively. T h e n / z × ~/ is a fuzzy set in X x Y, defined by
Definition 2.4 [10]. A fts X is called a T2-fts or a Hausdorff space if for any two fuzzy points xo~
for every (x, y) in X x Y.
and Yt~, where x =/=y, there exist Q-nbds/~ and r/ of x~ and ya respectively, such t h a t / z ^ ~/= 0x.
Definition 2.5 [10]. Let (D, >i) be a directed set and ~ be the collection of all fuzzy points in a set X. Then the function S:D---> ~ is called a fuzzy net in X. For n ~ D, its image under S is denoted by x n and hence the fuzzy net S is usually denoted by {X~}n~O. Ot n
Definition 2.6 [10]. A fuzzy net {x~},eo in a fts X is said to overlap eventually with a fuzzy set/~ if there exists an m in D such that x ~,, q/~ for all n ~>m. On the other hand, {X~o}n~D is said to
(/z × ~/)((x, y)) = min{/~(x), r/(y)}
3. Fuzzy net of fuzzy sets In this section, we introduce the notion of fuzzy net of fuzzy sets which extends the notion of net of sets by Mrowka [12] in general topology. We also discuss various aspects of the convergence of fuzzy nets of fuzzy sets. Definition 3.1. Let (D, >1) be a directed set and X be any non-empty set. Let ,~ be the collection of all fuzzy sets in X. The function S : D---> ,~ is called a fuzzy net of fuzzy sets in X. For n c D, we denote its image under S by /~n and hence a
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
fuzzy net of fuzzy sets is usually denoted by
{fn}n~O"
(XT,°}noO, limsupx],
and
liminfx]°
nED
In [10], a fuzzy net in X is defined to be a function from a directed set to the collection of all fuzzy points in X. Since a fuzzy point is a particular type of fuzzy set, the above definition generalizes the definition of a fuzzy net given in [10]. For clarity, we denote a fuzzy net of fuzzy sets by {f~}nEo, while that of fuzzy points by {x]o},Eo respectively, where (D, />) is the directed set. D e f i n i t i o n 3.2. Let S = {f,}n~o be a fuzzy net of fuzzy sets in a fts X. For any fuzzy point x~ in X, we say (i) x~ ~< lim supnED fn iff S frequently overlaps with every Q-nbd of x , ; that is, given a Q-nbd r/ of x , and any m c D, there exists an n/> m in D such that f , q T/. (ii) x~ <~ lim infnEo fn iff S eventually overlaps with every Q-nbd of x~; that is, given a Q-nbd r/ of x~, there exists an m • D such that fn q 7/ for every n/> m. In case such an x , does not exist, in the first and second case respectively, we define
limsupfn=0x nED
and
liminffn=0x.
43
nED
are simply the union of the cluster points and that of the limits of {X no~.},Eo respectively. D e f i n i t i o n 3.3. Let S = {fn}nED be a fuzzy net of fuzzy sets in a f t s X such that
lim sup f~ = lim inf fn nED
(--- f , say).
nED
Then we say that the fuzzy net S converges to f and denote it by limned f . = f . We also denote it by fn--> f . In the following examples, we demonstrate how the fuzzy topology can effect the convergence of a fuzzy net of fuzzy sets. Example 3.1. Let X = {x,y}. Consider the fuzzy net {fn}n~D, where D is the set of natural numbers under usual ordering, defined by 1 /~n(x) = - , n
1 fn(Y) = 1 - - , n
for every n • D. Case (i): Tl(X)= {0x, xo~, Ix}, where a~ is a real number, 0 < cr <~ 1. Then
nED
lim inf fn = xl_o~ v Yl Thus for a fuzzy net of fuzzy sets S = {f,}nED, lim SUpnED fin is the supremum of the collection of all the fuzzy points x~ such that S frequently overlaps with each Q-nbd of all these fuzzy points. On the other hand, liminfn~ofn is the supremum of the collection of all those fuzzy points x , such that S eventually overlaps with each Q-nbd of every fuzzy point of the collection. The justification of the above definitions lies in the facts that if x~ is any fuzzy point such that x ~ <~ lim sup f .
nED
=
lim sup fn. nED
Thus lim.Eo #. = x l _ . v y l in the fts (X, Tl(x)). Case (ii): Tz(x)={Ox, lx}U{x~:O
n~D
and x~_E is any fuzzy point contained in x~, then x , _ , ~< lim sup fn. n~D
Similar is the case with lim infn~D fn- It can also be noticed that for a fuzzy net of fuzzy points
(-1) n
U.(x)=½+
2n
1
'
U.(Y)=I
n
for every n e D. Case (i): Tl(x) = {0x, Xl/2, lx}. Then lim inf/~. = xl/2 v Yt, nED
44
R.D. Sarma, N. Ajmal
/ F u z z y nets a n d their application
while lim supneo #~ = l x in the fts (X, Tl(X)). Case (ii): Tz(x) = {0x, x~, l x } , ½< a: ~< 1. H e r e fn --~ lx. Case (iii): T3(x) = {0x, x,~, l x } , 0 < o( < ½. Then lim f~ = Xl_or V Yl.
Again, if y# ~ #, then there exists a Q-nbd r/o of y~ and an mo in D such that #n ~ r/o for all n/> mo in D. Since E is confinal, there exists m~ in E such that #n~r/o for all n / > m ~ in E. Consequently, y# ~ lim SUpneE fin" This implies that
ned
Case (iv): T4(x) = {0x, x , v YI~, l x } , 0 < a', fl ~< 1. Then limn~n #n = lx. Case (v): Ts(x) = {0x, lx} U {x~:/3 < t r ~< 1}, where /3 • [1, 1]. Then lim, eD #, = I x in the fts
(x, Case (vi): T 6 ( x ) = {Ox, u 0< where fl ~ (0, ½). Then limn~o #n =Yl in the fts
lim sup #n ~< #. neE
Hence by Theorem 3.1, limnee#n = #. Lemma 3.1. For any fuzzy set I~o and a collection {#i}i, if ~'~0 ~ f i for each i, then
Uo Vi fi.
(x, Case (vii): T7(x) = {0x, lx} U {y/~: fll ~< fl ~< f12}, where ill, ~2 • (0, 1]. Here limneo #n = lX. Case (viii): T s ( x ) = { 0 x , l x } U {x,~ v yt~:O< a:, fl ~< 1}. Then limned #n = lx.
The proof is omitted.
Theorem 3.3. For any fuzzy net {fn}neO in a fts X, limsup#n-- A
Here we notice that in the first example, although limneo #, exists in all the cases, it is different for different fuzzy topologies, while in the second example, Case (i), lim~eo [~n does not even exist. Any fuzzy net defined on an indiscrete fts is always convergent. In the following theorems, we discuss some properties of limsup and liminf of a fuzzy net of fuzzy sets.
Theorem 3.1. For any f u z z y net {#n}neO in a fts X, lim inf #n <~ lim sup ~'/n" neD
neD
neD
V #~.
m ~ D n>~m
ProoL If x~ ~
V
meD n~m
Again, if x~ ~ lim supneo #n, then there Q-nbd r/o of xo~ and mo • D such that fn each n ~>m0, whence Vn~m,, #n ~ r/o in the above lemma. Therefore, x~ ~ V From this,
Proof. The proof follows from the definitions directly.
exists a ~ r/o for view of . . . . . #,-
rneD n ~ m
This completes the proof.
Theorem 3.2. If a fuzzy net {#n}neO converges to # in a f t s X and E is a confinal subset of D, then the fuzzy net {#n}nee /S also convergent and limned kin = f .
Theorem 3.4. For any fuzzy net {~'~n}neD in a fts X, V
Proof. Let x~ ~<# and r / b e a Q-nbd of x~. Then { # , } , c o and hence { # , } , e e eventually overlaps with r/, whence #~
A #n ~< lim inf •n.
r e e D n>-rn
neD
Proof. Let xo, ~< V
A btn
meO n~m
45
R.D. Sarma, N. Ajmal / Fuzzy nets and their application and r/be a Q-nbd of x~. Then there exists some y in X, such that
rl(y) + (,,VD,An # n ) ( y ) > 1. In view of Lemma 3.1, this implies that there exists some m0 • D such that
m0 • D and a Q-nbd r/0 of x~ such that #, ~1r/0 for every n/> mo. If r/ is any Q-nbd of x~, then n ^ r / 0 is also a Q-nbd of x~, so that {#n v r / . } , , o frequently overlaps with r / ^ r/o. But #n~r/0 for every n>~mo implies that {r/,}~eo frequently overlaps with ~/. Thus x~ ~ lim sup r/,. neD
r/(y) + (hAm # n ) ( y ) > 1. Therefore, Again /~,~m #, q r/ implies that g, q r/ for every n ~>m0. Thus {#,}neo eventually overlaps with every Q-nbd of xo~. Therefore,
lim sup(#, v r/n) ~ lim sup #n v lim sup r/n.
x~ ~
The reverse inequality can easily be verified. This completes the proof.
neD
Consequently, V
neD
neD
nED
Theorem 3.7. For two f u z z y nets {#n},~o and {r/n}nEo in a fts X,
A #n ~
mED n~m
neD
lim inf(#n v r/n) = lim inf #. v lim inf r/n. neD
for n, n ' E D with n ~ n ' , then the f u z z y net {#n},Eo is convergent and
neD
neD
Theorem 3.5. If gn ~ # " ' ,
lim #n = A /~n. ned
The proof is on the same lines of the above proof. The following result is a direct consequence of the above two theorems.
neD
Theorem 3.8. If the f u z z y nets {#,),eD and {r/n}.Eo converge in a f t s X, then the f u z z y net
Proof. Clearly
{#, v tln},eO also converges and /~ /in ~< lim inf #n. neD
nED
lim(#n v ~/,) = lim #, v lim r/,. Also by Theorem 3.3,
ned
lim sup #, = A /~,. neD
ned
4. Application of fuzzy nets
neD
Theorem 3.6. For two f u z z y nets {#,},co and {T/,},EO in arts X,
lim sup(#n v r/,) = lim sup #, v lim sup r/n. neD
ned
neD
This section manifests some applications of the notion of a fuzzy net and in the process, provides alternative characterizations of several concepts in fuzzy topology. First, we start with the following result about a closed fuzzy set.
ned
Theorem 4.1. Let # be any f u z z y set in fts X. Proof. Let
x~, ~ lim sup(#, v r/,) ned
and x~ ~ l i m s u p , Eo#,.
Then there exists an
Then the following are equivalent: (i) # is a closed f u z z y set. (ii) For every f u z z y net {#, },eo in # (that is, #, <<,# for each n • D ), lim sup, co #n ~< #. (iii) For every f u z z y net {#n},eO in #, lim inf, eo #, ~< #.
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
46
Proof. Let /~ be a closed fuzzy set in X and {/~}nEO be a fuzzy net in p. Consider any fuzzy point x, ~
(ii) For each fuzzy net {x~.}nEo in X, x ] --~ x~ then f (x n~.) f (x~) in Y.
if
Here, following a technique provided in [1], we offer another net-theoretic characterization of fuzzy continuity in terms of fuzzy net of fuzzy sets. Lemma 4.1. For any mapping f :X---> Y from a fts X to a f t s Y, Iz q r! in X implies f(lz) qf(r/) in Y. Theorem 4.4. Let f :X--> Y be a mapping from a fts X to a fls Y. Then the following are equivalent: (i) f is fuzzy continuous. (ii) For any fuzzy net {/~n}~EO in X,
x~ ~
whence by condition (iii), x~ ~<~. Therefore, by Theorem 2.3, # is closed. Thus, condition (iii) implies condition (i). This completes the proof. As a consequence of this theorem, we obtain the following result. Theorem 4.2. Let g be any f u z z y set in a fts X. Then the following are equivalent: (i) ju is an open f u z z y set. (ii) For every f u z z y net S = {/~n}nEO in X, if limsupnEo/z, overlaps with It, then S overlaps frequently with It. (iii) For every fuzzy net S = {#n}nEO in X, if liminfnEo/~n overlaps with Iz, then S overlaps eventually with tt. Corollary 4.1. A fuzzy set Iz in a fts X is open iff whenever a fuzzy net of fuzzy points converges to some f u z z y point overlapping with Iz, the f u z z y net overlaps eventually with It.
f ( l i m inf #,,) ~
neD
/
n~D
(iii) For any net {gn},,cO in X, f(lim sup #n) ~
nED
nED
Proof. Let condition (i) hold. Consider any fuzzy net {~n}nED in X. Let x~ ~
and r/ be a Q-nbd of f ( x ~ ) in Y. As f is fuzzy continuous, f-~(r/) is a Q-nbd of x~, in view of Theorem 4.2. As a result, {/~n}nEO eventually overlaps withf-l(r/). Consequently, by Lemma 4.1, {f(#n)}nEo eventually overlaps with f ( f - l ( q ) ) and hence with r/, as f(f-l(r/))~< 7/. Now, 77 being an arbitrary Q-nbd of f ( x , ) , this implies that f(x~) <- lim inff(/~n). nED
In [10], Pu and Liu have characterized fuzzy continuity in terms of fuzzy net and Q-nbds of fuzzy points respectively as follows.
Thus, f ( l i m inf ~t,) ~ lim inf f(/~.). \
Theorem 4.3. A mapping f :X ~ Y from a fts X to a fts Y is f u z z y continuous iff any one of the following holds: (i) For each fuzzy point x~ in X and each Q-nbd 71 of f(x~) in Y, there is a Q-nbd i~ of x~ such that f ( # ) <~~1.
nED
/
nED
In this discussion, if we replace 'eventually' by 'frequently' then it becomes clear that the condition (i) implies condition (iii) also. Finally, we show that if condition (i) does not hold, neither (ii) nor (iii) can hold. So, let f be
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
not fuzzy continuous. T h e n there exists an x~ in X and a Q-nbd T/ of f ( x ~ ) in Y, such that f(#~) $ r/, for every Q-nbd/z~ of x~. Let A be the collection of all Q-nbds of x~ in X. Then (A, >/) is a directed set, where '~>' is the inverse relation of 'fuzzy set inclusion'. This is clear from the fact that reflexivity and transitivity are obvious in A. Also for any l-g1 and ~2 in A, #x ^ ~2 is also in A with #1 A #2 ~ #1 and ~-~1A ~'~2~ ~'~2, where '~<' is the 'fuzzy set inclusion relation'. Now as f(#~) $ r/, there exists some yg in Y such that (f(g~))(y~) > r/(y~). Then there exists a real positive number try, such that
(f(i, ti))(y i) > oc; "> r/(yi), where ol; = 1 - oil. Now
( f(~ / ))(y' ) =
sup
(~i(x')} >
.;.
x~f-~(y 9
Therefore, f-~(y~) is non-empty and there exists x i in f - ~ ( y i ) such that #i(x i) > 1 - vii. Consider the fuzzy net S = {S~,}t,:a, where s~,, -- xZ~,. It is clear that x~ ~< lim inf s~, ~.ieA
such that (i) T = S o N , yN(i)
that is, for each l e E ,
47
x~,=
etN(i)•
(ii) For each n ~ D, there exists some m ~ E such that for every p ~ E with p >i m, N ( p ) >! n. Theorem 4.5. A f u z z y net (x].},~o converges to
a f u z z y point x~ iff each of its f u z z y subnet converges to x~. Proof. Let the fuzzy net { x ~ } , ~ o converge to x~ and let {y#'nJmeE be a fuzzy subnet of {x~.),~o. Consider any Q-nbd 1/ of x~. Then there exists m e D, such that x~,~q r/ for every n t> m in D. For this m, there exists t in E such that for every p>~t in E, N ( p ) > ~ m , where N:E--->D is the function stated in Definition 4.1. Now, y ~ = xN
and hence x~ ~< lim sup~,:A S,,, by T h e o r e m 3.1. But due to the existence of r/ as a Q-nbd of f ( x , ) , it is clear that
Theorem 4.6 [10]. A f t s X is a T2-fts iff no f u z z y
f(x~) ~ lim i n f f ( s . )
net can converge to two f u z z y points with distinct support.
gtifiA
as well as
f(x~) ~ lim sup f(s~,,). Thus if the condition (i) is violated, both condition (ii) and condition (iii) are violated. This completes the proof. In fact, if the range space is a Te-fts, the condition for fuzzy continuity of f given in Theorem 4.2(ii) can be relaxed to a good extent. We shall discuss this in a theorem which is in the spirit of Theorem 5.1 [6] of general topology. Definition 4.1 [10]. A fuzzy net T = {X mflm}meE in X is called a f u z z y subnet of a fuzzy net S={y%o}n~o iff there is a function N:E--->D
Lemma 4.2. Let (A, >a) and (B, >b) be two disjoint directed sets which are isomorphic (that is, there is a one-to-one function h f r o m A to B such that o6 >~ re2 iff hO¢l >bhOl2). Then there is a directed set (C,
{ymr.},,~a be f u z z y nets in X with disjoint directed sets A and B which are isomorphic. Then there is a f u z z y net {z~p}p~c in X such that {X~}n~a and {Y'~,}m~B are f u z z y subnets of {z~p}p~C and z~['-~ to, iff x~ --->too and Yom m to#" We omit the proofs of the lemmas as they are either identical or similar to their counterparts in general topology.
48
R . D . Sarma, N. A j m a l [ F u z z y nets a n d their application
Now we come to our proposed theorem. Theorem 4.7. Let f :X--+ Y be a mapping from a fts X to a T2-fts Y. Then the following are equivalent: (i) f is continuous. (ii) If any fuzzy net {X~.},~D converges to x~ in X, there exists y~ in Y such that the fuzzy net {f( X n~.)},~o converges to y~. (iii) If {XT,.},~D converges to x~ in X, there exists a fuzzy subnet {Xo,~,..,}.,~E N(m) n of {X,~.}..O in X such that {f(x:~<:>)}m+E N(m) converges to f(x~) in Y. (iv) For each x: in X, there exists y: in Y such that whenever {x~:},+D converges to x:, there exists a fuzzy subnet f"t.4, ~ NOtlV,mlJm~E (m)~l n o f {Xotn}n~ D such that { f (x~N~m~) N(m) }m,e converges to f (x~). Proof. Since each of the conditions (ii), (iii) and (iv) is clearly implied by fuzzy continuity in view of Theorem 4.3, we only need to show that each of these conditions implies fuzzy continuity. Assume that condition (ii) holds. Suppose, if possible, f is not fuzzy continuous. Then by Theorem 4.2, there exists a fuzzy n e t { X n n } n e A in X such that x ,~, --+ xo, in X, but f(x$~) - ~ f ( x , ) in Y. Now, by condition (ii), as x$---~x~, it follows that f(x$~)--+y~ for some Yo, in Y. Let {Yo~}m~B be the fuzzy net in X such that yom = x~ for all m in B, where B is a directed set disjoint but isomorphic to A. Thus by Lemma 4.3, there is a fuzzy net {z~}p~c in X such that {X~.}.~A and {Y'~.,},,,,n are fuzzy subnets o f (zPflp}pEC and z~'-*xo,. Then {f(x~,.)},~a and {f(yom)},~n are fuzzy subnets of {f(z~,)}p~c. However f ( X n~.)-~ y~ and f(y'ffm)-~ f(xo, ). Since Y is T2, {f(z~,)}p,c cannot converge unless f(x,~)=y~, in view of Theorem 4.5 and Theorem 4.6, which contradicts condition (ii). Thus condition (ii) implies condition (i). Assume that condition (iv) does hold. We show that condition (iii) also holds. Let x~.-~ x~ in X and y~ be the fuzzy point in Y corresponding to x~ in X. Then, if any fuzzy net converges to x~, the image fuzzy net of some fuzzy subnet of this will converge to y,~. Let {Y,~.}meBm be a fuzzy net in X where Yo~mm_- Xc, for m ~ trl all m in B. Then y ~ x~ and f(y~,)--->f(x~). Therefore, the image fuzzy net of any fuzzy subnet of {Y'~.}m~8 converges to f(x~), by Theorem 4.4. Since Y is T2, therefore by Theorem 4.6, f(x~) = y~.
Finally we show that condition (iii) implies fuzzy continuity. Let condition (iii) hold and if possible, let f be not fuzzy continuous. Then there exists an x~ in X and a Q-nbd 0 of f(x~) such that f ( # ~ ) ~ r/, for every Q-nbd #i of xo, Then proceeding as in the last part of Theorem 4.4, we can construct a fuzzy net S = {Su,},:A in X. This fuzzy net converges to x~ in X. But no image fuzzy net of any fuzzy subnet of S converges to f(x~), contradicting condition (iii). Consequently f is fuzzy continuous. This completes the proof. Lemma 4.4. For any mapping f :X--+ Y from a fts X to afts Y, if {Y~.}neD is a fuzzy net of fuzzy points in Y, {f -1 (Y~.)},~o is a fuzzy net of fuzzy sets in Y. The proof is trivial. Lemma 4.5. Let f :X--+ Y be any mapping and {Y].}~D be a fuzzy net in Y. If za ~ lim sup f - l ( y ] , ) , nED
then there exists a fuzzy net in X which converges to z, and image fuzzy net of which is a fuzzy subnet of {Y].},,o. Proof. Let {Y~.}.~o be a fuzzy net in Y and zt3 ~
-I
n (Yo,,).
Then for each n • D and for each Q-nbd r/of za, there exists an m(n, rl)>-n in D such that ,Y~m<,.,> I q r/. Let
D O= {m(n, r/): n • D, r/ is a Q-nbd of zt~}. Define '~>' in D O by re(n, rl)>~m(no, r/0) iff n i> no and r / ~ r/0. It easily follows that (D °,/>) is a directed set. Now for every re(n, 77)• D °, f-l(ym(~,~n)) q r/. Let f-~(" ,,(n,o)x and r/ overlap j ,y~(..,~) at X m(n'n). Clearly,
f (~.x m(n'n)'l Ogm(n,vl)fl =vm(n'n) .70~rn(n,~) for every m(n, q) in D °. It can be easily verified that I,,m(n.n)~ [ Y oLm(n.n) J m(n,rl)c:D 0 is a fuzzy subnet of {Y~.)~¢D. Then {.¢m(n,r/)] "*~O6n(n,,O f m ( n , ~ ) ~ D 0 is a fuzzy net in X such that its image fuzzy net is a fuzzy subnet of the given fuzzy net.
R.D. Sarma, N. Ajmal / Fuzzy nets and their application
Next we show that t~mtn'~)/ ~..~ot,,(,.,D I m(n.n )ED, converges to z#. Let r/o be a Q-nbd of z#. Due to our choice of X m(n'~l) ~,,~,.,~, X m(n'"°) ~,,,.,~ q r/o, where n is arbitrary but fixed in D. Let m ( n ' , rl')>~ m(n, r/o). Then r/'<~r/o and as before, xm(n',rl') ,'1 ~ t . hono~ "m(n"rl') " " Thus Ogm(n" rl') %1 #~ "~ IJt~ltl~¢~ ~'~ O(m(n' O, } ~'~ r i O " {xm(n'r/.)'~ ..... ~ D o eventually overlaps with every Q-nbd of z~. Therefore this fuzzy net converges to za. This completes the proof. °
49
{• Y nJ ~ o converge to z, in Y and w be an element in
supp{ limEsupf - 1(y~,,) } • That is, there exists some positive real number o, where 0 < o ~< 1, such that wo <~lim supf-l(y~.). NED
Theorem 4.8. Let {x~.},Eo and {Y],},Eo be f u z z y nets in fts X and Y respectively. Then n ~ n x o,. x~ in X and y o~.---~Yt3 in Y implies that Ix",
[x,
in the product fis X × Y. Conversely, [x", y"]~o---> [x, y], in X × Y implies that xo~ ~ ---~x , in g and n ----9, Yo~. y, in Y re,wectively.
For the proof, we refer the reader to [13]. In what follows we provide net-theoretic characterizations of mappings with closed fuzzy graph, and open fuzzy mapping respectively. For detailed investigations of these two types of mappings, see [3, 13] and [2, 4, 8, 9, 15] respectively. The following theorem and the last two in fact, extend Theorem 1.2, Theorem 6.2 and Theorem 6.3 of Fuller [6]. At first, we recall the following definition from [13]. Definition 4.2. Let f : X - - - , Y be a function from
a fts X to a f t s Y. f is said to have closed f u z z y graph if the characteristic function Zc,, where G = {(x, f ( x ) ) : x • X},
is a closed fuzzy set in the product fts X x Y. Theorem 4.9. Let f : X ~ Y be a function f r o m a fts X to a fts }I. Then the following are equivalent: (i) f has closed f u z z y graph. (ii) I f {Y~.},ED converges to zo, in Y, then supp{ limsup f - ' ( y ~ . ) } m_supp{f-'(z~)}. (iii) If {Y~°}.Eo converges to z~ in #, then
Then by Lemma 4.5, there exists a fuzzy net {x~'.},,,,e in X such that x'~---~wo and n {f( X m#,.)}m~e is a fuzzy subnet of {Y~.},Eo. Thus, x'~,---*wo and f(x~.~)---~z~, by Theorem 4.5. Therefore, by Theorem 4.8, [X m, f (xm)]flm -'-~ [W,
Z]min{o,.o}.
As %c is closed by condition (i), it follows that [w, Z]mi,~o.~ <~Xc, in view of Theorem 2.3. Or, in other words, f ( w ~ ) = z ~ . Thus w, <~f-t(z~). Consequently, w ~ supp{f-~(z~)}. Hence supp{limsup f-l(y~,.) } c supp {f-l(z~)}. Therefore, condition (i) implies condition (ii). That condition (ii) implies condition (iii) is obvious in view of Theorem 3.1. Let condition (iii) hold. We show that f has closed fuzzy graph. Let {[x~, f(x")]~,}~,o be any fuzzy net in X~ which converges to [w, z]~ in X x Y. Then, by Theorem 4.8, x~°--->wa and n n _._> f(x~.)--~z#. Write f ( X n~°)=y~. Then y~. ya. n _~ n Again, in view of the fact that x~. ~ f - - 1 (y..) and n x.~ w#, it follows that for every Q-nbd ~ of w~, there exists too• D, such that f - t ~y~°) . . . . q It for all n I> too; hence w# ~
Therefore, by condition (iii), w e supp{f-~(zt0}. That is, f ( w ) = z. Thus [w, z]a <~Xc.. Consequently Xc is a closed fuzzy set in X × Y, by Theorem 2.3. This completes the proof. Theorem 4.10. A one-to-one surjection f : X --* Y
supp{ lira ionff - ' ( y : . ) } c supp{f-'(z.)}.
is an open f u z z y mapping iff for every f u z z y net {x~°},Eo in X with f(x~.)--*f(x~) in Y, x ~°--+ x~, in X.
Proof. Suppose that condition (i) holds. Let
Proof. Let f be an open fuzzy mapping and
50
R . D . S a r m a , N. A j m a l
/ F u z z y nets a n d their application
{X~,}neO be a fuzzy net in X such that f(x~)---~f(x~), for some x~ in X. Let /z be an
(f(/~0))(y ~°) > 1, SO that
open Q-nbd of x~ in X. Then f(/~) is an open fuzzy set in Y and in view of Lemma 4.1, f(x~) qf(/,). That is, f(t~) is a Q-nbd of f(xo~) in Y. As f ( x ~n ) ~ f(x~), the fuzzy net {f( X nJ } n e o eventually overlaps with f(p). Now f being one-to-one, this implies that x no~n eventually overlaps with/*. Consequently, x~,--* x~ in X. Conversely, let t~ be an open fuzzy set in X. Let {Y~}n~o be a fuzzy net in Y converging to some fuzzy point y~, where Yo, qf(/*). Now f being onto, there exist x n and x~ in X, such that f ( x n J - Y_ _o , ,n for every n in D and f(x~)=y~. Thus f(x~n)---~f(x~) and f(x~) qf(/z). Since f is one-to-one, this gives that x~ q/~. Thus /~ is a Q-nbd of x~. Now, following the given condition, x ~ ___>x~. Therefore {x~}n~D n eventually overlaps with/~. That is, there exists m in D, such that x ~,~q/~ for all n / > m ; hence n f ( ~X )n q f ( / ~ ) for all n~>m. That is, {Y~}neo eventually overlaps with f(/,). Hence by Corollary 4.1, f(/z) is an open fuzzy set in Y. Consequently, f is an open fuzzy mapping. This completes the proof.
/3no+ sup {#o(Xn)}>l. x~e/-J(y,.O)
Theorem 4.11. Let f :X--> Y be a function from
a f t s X to a f t s Y. The the following are equivalent: (i) f is an open fuzzy mapping. (ii) If {Y~,)n~o converges to YIJ in Y, then
But then, flno+PO(xn°)> l for some x n° f-l(yno), whence x~°0q ~o and x~° <~f-l(v~O). v-no
~
x~
~-n0~
no That is, f - - 1 (Ya~,) q/~o, which is a contradiction. Now, f being open, f(/*o) is an open fuzzy set in Y. Also, as x,, q/~0 in X, by Lemma 4.1 we have f(x~,)qf(ito) in Y. Thus f(/~o) is a Q-nbd of f(x~). Again, x~ ~
x~ ~
Consequently,
f-l(ya) <~lim inff-l(y~,). neD
Clearly, if condition (ii) holds, then condition (iii) holds, in view of Theorem 3.1. Finally, we assume that condition (iii) holds. Suppose f is not open. Then there exists an open fuzzy set /~ in X such that l x - f ( t z ) is not a closed fuzzy set in Y. Thus there exists ya ~< l r - f ( ~ ) such that ya ~ lv - f ( # ) . Then by Theorem 2.2, there exists a fuzzy net {Y~n}neO in ly --f(/*) converging to ya. Now, ya ~ l r - f ( / ~ ) implies that
f-l(y#) ~ lim inff-~(y~). ned
(iii) If { Y ~ } ~ o converges to Yt~ in Y, then
f-'(y#) <-lim supf-l(y~,). ned
Proof. Let condition (i) hold. Let {Y~n}~eo converge to Yt~ and x~ <~f-~(YtJ). Suppose, if possible, x~ ~ lim inff-l(y~,). ned
Then there exists an open Q-nbd /Zo of x~ such that {f-~(Y~,)}n~o is not eventually overlapping with it0. From this it follows that {Y~)neo is not eventually overlapping with f(/Zo). For if n0 Ya~0qf(#o) for some no in D, then fin0+
/3+
sup {/~(x)} > 1. xe/-l(y)
Therefore, f - l ( y ) is non-empty. Tiros there exists x ef-~(y) such that x a q/z, or, in other words, xa ~
so that {f-X(y~n)}ne D frequently overlaps with /~, and hence {Y~)n~o frequently overlaps with f(/~). Thus for a given n e D, there exists m in D such that
tim + (f(lt))(Y m) > 1. That is, y m ~ l r - f ( b * ) , which contradicts the choice of {Y~n}n~O in l r - f ( / * ) . Hence our
R.D. Sarma, N. A]mal / Fuzzy nets and their application
assumption is wrong and consequently f is an open fuzzy mapping. Theorem 4.12. If the mapping f : X--> Y is open, having closed fuzzy graph, then Y~.-->Yt3 in Y implies that
supp{fim f-l(y~.)} : supp{f-l(yt~)}. Conversely, if y~ --->yt~ in Y implies that
lim f-~(y~.) = f-l(yt3 ),
neD
then f is an open fuzzy mapping and has closed fuzzy graph. In both the cases it is assumed that lim,~o f-l(y~.) exists. Proof. By Theorem 4,9 and Theorem 4.11, if f is open and has closed fuzzy graph, then y~---~yo in
Y implies that supp{f-'(Yt3)} ~_supp{lim ionff-~(Y~.)}
~_supp{ limsup f-'(y;.)} ~_supp{f-l(y,)}, in view of Theorem 3.1. By the same theorems, the converse part follows immediately. Acknowledgement
The first author is highly indebted to one of his supervisors, Professor K.C. Gupta for his encouragement during the preparation of the paper.
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