THE STUDY OF NESTED TOPOLOGIES
THROUGH FUZZY TOPOLOGY
by
~6lix ~chulte'
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
in the
Mathematics
Program
-
Adv
(Signature) .b,
G j B J
Dean of the Graduate Skhool
Date-
YOUNGSTOWN SfATE UNIVERSITY
March, 1984
'TABLE OF CON'tENffS
PAGE
I11 . L-F.i'P FAMILIES OF 'TOPOLOGIES ....... 3
V . COMPAC. INESS AND THE DIFFERENT NOTIONS OF
.......... FUZZY coriiPAc ~~ESS 11
VI . COIiNEC.i'IVIII'Y AND 4 -CONNEC1fIVITY . 13
VII . CON'I'INUI1l'Y AND L-FUZZY CONTINUITY . . . . 15
............ VIII . SU141'ABILI'iY 20
IX . 16. AS A SEMI-CLOSURE OPERAPR OR A
....... CLOSURE OPERAafOR FORd&L-L 25
............... EIBLIOGRAPHY 33
-
CHAPTER I
Preliminaries
In this paper L will denote a complete, completely
distributive lattice with an order reversing involution.
We will distinguish the two following subsets of L:
LC= )eLlaa=>*44 or 4~43 , LP=)~LC:~<~ and*< r=>d~h.*~].
O~L~ will be assumed throughout this paper. Let X be
a set. An L-fuzzy topology 'I' is a collection of L-fuzzy
sets (mappings from X into L) which is closed under arbitrary
suprema and finite infima. Let Y(X) be the set of crisp
subsets of X , c : P(X)-7Y(X) is a semi-closure operator
provided that coif)=@, c(X)-X, Acc(~) for A€P(X), and
C(AU B)=C(A)~C(B) for A,B in P(X). For A€F(X), Y(A) denotes -
the characteristic function of A. By an L-fuzzy topology in
the Lowen sense, we mean a fuzzy topology including the -
constant maps as open sets. For further references on basic
definitions and properties of L-fuzzy topologies see[9].
CHAPTER I1
Introduction
Using Rodabaugh's definition of 4 -closure [6], Klein
has defined L-fuzzy topology producing collection of operators
(L-Fti'P). Using this concept and the related results, we will
study how a finite family of nested topologies indexed by a
lattice generates a fuzzy topology, Moreover, we will examine
how topological properties are transmitted from the nested
topologies to the induced fuzzy topologies and vice-versa.
In particular, we will refine Kleinjresult inb]about the equi-
valence of fuzzy continuity and level continuity and prove it
to hold for the Lowen topology. We will show that plenty of
suitable closed (open) sets are at our disposition, a result-
that may be significant with regard to the ~ietze'extension
problem. Finally, we will categorically embed topologies -
generated by a finite lattice in topologies generated by the
unit interval, generalising the results obtained so far.
As a general remark, it will appear that properties involving
- -
only open sets (closed) behave relatively well (e.g. compactness,
hausdorff) but that properties requiring in their definition
both open and closed sets are somewhat more elusive to track
down.
CHAP'I'ER I11
L-FTE families of topologies
We now give a summary of the results obtained by
Klein in [3,4]
Definition 3.1. ~etd'~-&\and let A be a crisp subset of X.
The 4-closure of A, denoted by cH(A), is given by:
if GLT and G(x)>4 , then GA~(A)#o .
3
It was shown in [3] that *is a semi-closure operator
Definition 3.2. Let 4CLa- [l\and let GCT. 4(~)=[x:~(x))d 1.
By lemma 2.2 in[3], [&(G)I GCti'] is a topology for X which
we denote by '14.
Definition 3.3. Let X be a set and let C= Q :deLa-[I\\ be
-
a collection of operators on Y(X). C is L-fuzzy topology
producing (L-FY'P) provided :
(a) for every dc La- ill , k4 is a semi-closure operator,
d, =hlh :4e~] , and ACP(X) , then --
(c) if A,BEP(x) and Acko(B), then k4(A)cQ(B) for every
A6 ~~-p'j.
Definition 3.4. Let C be an L-FfP collection.
-
(a) For A€P(X), GAis the L-fuzzy set defined by
G,+ (x) =A~A:X k4(A)]. (By convention Ap(=I).
(b) ll(C) is the L-fuzzy topology with basis [G~: Atp(X)].
Theq -closure operators generated by T(C) are the operators
The class of d-closure operators induced by an L-fuzzy
topological space was shown to be an L-FTP collection in[4].
We are ready now to prove our first lemma.
Lemma 3.5. Let C= (4 : t L" -[l)j with la finite be a
family of closure operators such that if o( h 4 then 'P4GT4
(where Td , Ta are the topologies with k4 ,k4 as closure
operators respectively.) We have :
(a) if 4$4 arheL and AeP(X) then kd(A)c&(A), -
(b) if fl # 7cLa- I11 ,d =h[4 :qe~\and AeP(X), then
~*(A)=O~~(A) he^)
(c) if A,B and Acko (B) then b(A)C&(B).
Proof. (a) Suppose x( kh(a). Then xcX-ly(A)=Ue , an open set
in q. Since 'qC_Td , Ur) is open in Q and X-Qj is a closed
set in #f* . Thus we have x# X-U43A and so xk k (A).
(b) sinceT is finite.de 4 :&el and the conclusion -
I 'J
follows from part (a).
(c) Let A,B in P(X) and ACk,(B). We have kd(A~kA(~(B>).
As an immediate consequence of lemma 3.5 we have
the following
theorem.
meorem 3.6. ~f &=lk la- [I\\ is a finite family of
closure operators generating topologies T8 such that if 4(L
-
then TaY3 , we have thatdis an L-F'PP family.
Definition 3.7. Let L be a lattice with ~~-[i]=~~,d~,.-*~dn~
and O=*tc*r <.no<*n
. Bfamily of topologies h is L-FTP iff :
(l)&= [%;)with a;e~~-{l\
(2) W&,DW&~~*-O~ wah
As a notational convention, C= ka; will denote the associated
family of closure operators.
I I
Remark. Given an L-FTP family of topologies, the fuzzy
topology T(C) will be used on X unless another topology is
explicitly mentioned.
We are now going to prove a computational lemma, which will
introduce a construction used throughout this paper.
Lemma 3.8. Let d be an L-F'I'P family of topologies. Then for any
A in P(X) we have :
GA(x)=l iff XCX-~,(A)
ti,,(x)=&h iff XE&JA)-%,
xbk,(~) (=> x6X-kh(A)
B=E~](=)G~(~)=A~(=> XGIQ~(A)-~~,-$A)
B= dn-K ,ah.,+, , . . . <=> Gh(x)=dh-~<=) x6 hh-y(~)-& h.l, )(A)
C
B=[A,, . . . ,L\ (=). G&(x)=~I =O (=> xek*, (A)
-
Theorem 3.9 Let dbe an L-FTP family of topologies. Then
vdi e La-t4 WA. c = ,IA; (where WA; , Q; are the topologies of
Def.3.7 and Def.3.2 respectively).
Proof. By Theo . 2.4 in [3] we have for A; €L~- ill : W4 C .
So it is sufficient to show ,&CW4; . Let AeP(X). By lemma 3.8
A~(G)=[x : CA(x)>44 = *-%;(A) , therefore '&;c W+ and
ty~~ =W,; for each A; e L~- b\ .
Using the definition of the A -property in [6] :
Definition 3.10. Let A~L- [l\ . (X) has the &-property provided,
for ArX, ca(A)=A if and only if there is U 'I' with
Applying Theorem 2.4 in [3] we have the following corollary:
Corollary 3.11. Let &be an L-F1iP family of topologies. Then
(C) has the &-property for all &la-[l\.
We will need several notions first introduced in L4j.
n
Definition 3.12. Let c=[~ La- [I\\ be an L-FTP collection -
on P(X). SC) denotes the set of fuzzy topologies for X
which induce Q as d -closure operator when &ela- M*
$c) need not be a singleton or closed under finite
intersections but is closed under suprema. For further
details refer to [43.
Definition 3.13. (a) denotes the collection of fuzzy sub-
sets of X which are constant maps from X into L.
(b) p denotes the collection of fuzzy sub- -
sets of X which are either y($) or a map from X into L-
Lo\
a
GJ
We have for TCJ(C) I supp,'Q] is inqc) (corollary 2.8 in [4])
and supYc)= SU~[T'(C),~) (theorem 2.9 in [4]) .
Proposition 3.14. Let d. =b; : &aa- [Ill be an L-FTP family of
topologies. Then Wdi= h;(G) : GCT(C)\I Tc] .
I
Proof. Any basis element of T(C)VTc can be written as Gp,)\b,
for G,,€T(C) and b€L (b also represents the constant map with
value b) . Let F=~L c such that *i>/b] and let I\F= or^ .
Applying lemma 3.8, we have :
~~Ab(x)= b iff x- X-k.S-,(A)
-c;AA~(x)=~,-, iff x~k,,, (A)-~~-~(A)
GAAb.(x)= 0 iff xcko(A)
Thus, for lci4; =j?f since diab.
Remark. For L linearly ordered, sup ~T(c) ,TC 1 is the smallest.
L I
topology in the Lowen sense in %c) . In this paper, this
-
topology will be called the Lowen minimum.
Corollary 3.15. Let &.be an L-FTP collection of topologies.
Then the Lowen minimum has the d -property for all d CL~- 111 .
- -
f'/
In general, sup)(c) does no-c have the d-property for din L~ -[I\.
It may in some cases. For example, if &; is discrete for all i
sup$ ) does have the A-property for all &La - ll\ .
Example. Let X = 8, 'I1= - usual topology, To = discrete topology
2
and La=[O,&,l] . We have :
-
G = & (=) xt [O , 1)
Gco,~) = 0 C=> xc(0,l)
Take Ht Tp such that H(x)=* on X- [2,3] and H(x)=l on [2,3J .
t
: G,o,,)nH (x))b j =(2,3]fi~-~,l] = 3 which is not
open in 'I?&.
2
Remark, GA=GB does not imply A=b .
Example. X=R, TI - =indiscrete topology, (I', =usual topology then
2
- -
i;(,g=kof,~ . This fact, of course, stems from : A=B
/) A=B.
We will need the following fact in a later section:
Lemma 3.16. iX : G(x), IX : G(x)~*. .
I
Lemma 3.17. Let be an L-FIF collection of topologies.
Then for 'P in$C) we have T(C)C'I' if L is linearly ordered.
Proof. This follows directly from Theorem 2.3 in [4'j .
Hausdorff and+ -Hausdorff properties
We will need the following notion first defined in [6].
Definition 4.1. (X,li') is o( -Hausdorff (q*-~ausdorff) for EL
if for each x,y~.X such that x#~, there are u,veY such that
u(x>)< ux (4 (v(y)b4) and uAv=O.
Proposition 4.2. Let &be an L-FTP family of topologies.
If Q, is Hausdorff then for any d; in La-{I), T(C) is
*
d; -Hausdorff (e(.t-~ausdorff and also 1-Hausdorff) .
Proof. Let xfy. Since 'Id, is Hausdorff, there are U(x),U(y)
in I' such that xeU(x) ,y€U(y) and U(x) fl u(~) =$. Since X-~(x)
and X-U(y) are closed in each I&; ,
We have GX-U(x) (XI '1 9 GX-U(y) (y)=l and G~-~(~)/\G~-~(~~=o,
theref ore f (C ) is 4-Hausdorff (d: -~ausdorff) for any &;La-
\.
Proposition 4.3. If T(C) is 4;-Hausdorff for a;*in L~-[I\ then
q; is Hausdorff. If T(C) is 4:-~ausdorff for A; in La- [0,1\
then :Gi_, is Hausdorff.
Proof. Let %# yobe in X. Since T(C) is d;-Hausdorff (df-~ausdorff)
there are G,H in T(C) such that G(x~)>~(; , H(X~)>~; (G(%))/~;,
H(x, )+hi) and GAH=O. Consider
1. Suppose 26
Then G(z))hi
have a contradiction with the fact,
that GI\H=O. (~(z ))I& , H(z)>td;, and therefore we have G(z)AH(z)Za; ,
a contradiction with GAH=O). Since, clearly x,E x:G(x)>di
C
and both of these sets are open in I&; , 8ii(C)
having the 4-property for all 4 , we have '&; is Hausdorff.
y,ekiH(x)h4) and these sets are open in GL-,
by Lemma 3.16, hence a;-, is Hausdorff).
Remark. This result depends in an essential way on the fact
that li(C) has the 4 -property for all 4 .
One can weaken the hypothesis of proposition 4.2 and prove
a slightly more general result.
Proposition 4.4. Let &be an L-FTY family of topologies.
Let T4j be Hausdorff.Then for d;c~~ 1 with ct;Ab for some G5 in T'(C).
'1
Obviously, the G4 constituteand;-shading of X, which is
reducible to a finite &-subshading since 'i'(C) is d; -compact,
and therefore 1 qj,, is reducible to a finite subcovering.
-
For necessity, let be an dlshading of X. Consider
I[ x : "(x)>)*L\I, . Clearly, it is an open covering of X.
Since is compact, it is reducible to a finite subcovering
and therefore (G,~]~ is reducible to a finite A' -subshading.
Corollary 5.3 T(C) is dl -compact for all *; in L~-{I\ iff
, is compact.
Remark 1. This proposition and its ensuing corollary depend
on T(C) having the X -property.
Remark 2. A closely related functorial proof of Froposition
5.2 can be found in Theorem 3.1 of[6].
From now on we suppose 8 , an L-FTP family of topologies,
given.
rF
Proposition 5.4. (X,'I'(C)) is&% -compact iff (X,T(C)) is
Ap.1 -compact for 4 in L~- [o\ .
Proof. To prove sufficiency, let G be an at.,-shading of T(C) . -
Then we have V Ul'n a& and since P(C) is df -compact, there exists
'3
a finite subfamily d of such that VG~ >id& that is VG,~ >d kt. -
7 d
L
Hence, 3 is reducible to a finite 44 -subshading.
I 3,
*
For necessity, let K5 be an dl-shading of 'il(C). hen VH~)/~&
3
implies V H37*k.,,hence [H+\ is an c$*rshading of T(C). Since - - T(C)
'3
i~d~-~compact, there exists a finite subfamily L such that
*
YH4 71% . Therefore, [H~\~ is a finite &&-subshading of T(C).
Remark 3. By Corollary 3.15 , Corollary 5.3 and proposition 5.4
are still true fQr the Lowen minimum.
CHAPTER VI
Connectivity and4 -connectivity
In this chapter I use Rodabaugh's definition of
&-connectivity from L7].
Definition 6.1. Let (X,'i') be a fuzzy topological space.
(X,!) is &-connected if there do not exist U,V in T -@,1{
such that uvv >I' and UI\V = 0 . (X,I') is d-disconnected
if there are U,V in T -p,J such that U VV)X and UAV=O.
Proposition 6.2. Let &be an L-FTP family of topologies.
Then for &in L~ -tl\ , if P(C) is J;-disconnected, then
Td; is disconnected.
Proof. Suppose T(C) is d;-disconnected. Therefore there
exist G,H in '1(C) such that 2\r H)h; and G A H=O. By theorem
3.9, the sets U = x : G , V = [x : H(x)X(' are open.
'1
Obviously, U U V=X. Suppose z is in Un V. G(z)W; and H(z))d~
and since A;E L~ -{l( , H(z)AG(z)% which contradicts the
fact that GAH=o. therefore U nV=d , which proves that-
(X, T&) is disconnected.
Proposition 6.3. ~etkbe an L-FTP family of topologies.
Then I'd, disconnected implies l(C) not 1-connected.
Proof. Suppose 'Id, is disconnected. 'There exist U,V in 'I'
such that UVY=X, UnV=0. By theorem 3.9, U=[x : G(x)>o~
-
and V = )xi H(x)@\ for some G,H in P(C). Clearly ~V~>gand
GhH=O. Thus T(C) is not 1-connected.
Remark 1. fhis proposition depends in an essential way
upon 'i(C) having the 4 -property.
Proposition 6.4. Let =\Ii),, be an L-PYP family of topologies
-
with IfA, connected. If G~J(C) and G=G, -then G=O or G=i.
Proof. Let G=% and G#O, 1. xr~ (x))0] is open in 4, , which
-
I
is closed in Td, . Since G=G, G is open
is open in 7JhG Td, . Hence,
in !I&, . Therefore, TA, is disconnected.
Remark 2. These propositions are still true for any topologies
0'
having the -property in JC ) , including Lowen's minimum.
CHAPTER VII
Continuity and L-fuzzy continuity
Definition 7.1. Let (x,?),(Y,T) be two topological spaces.
A function F:(x,~)+(Y,~) is said to be L-fuzzy continuous
P -'
-1
if for any H in1 F(H) is inr. (F(H)=HoF).
Theorem 7.2. Let d= I(X,qi )\ (Y be two L-F1iP families
1
of topological spaces. Let cA,k, be theird-closure operators
in X,Y, respectively, and T(C),T(D) the generated fuzzy
topologies.
Let fs (x,%;)->(Y,%;) . We have:
(1) If f:(x,~(C))->(Y,T(D)) is L-fuzzy continuous then
r
f r (x, %)-> (Y, b;) is continuous for all oc; in L~
- 113.
(2) The converse is true if f: (X,')->(Y,T~) is a homeo-
morphism for all r; in L~- tl).
-
Proof. (1) Since f is L-fuzzy continuous, we can use
Lemma 2.11 in [k]rBt X and (Y,T) be L-fuzzy topologies
and let f:X+Y be L-fuzzy continuous. For o(; in L ,
let ca; and k4;- be the 4 -closure operators in X,Y resp-ectively.
Then for every A in P(X), f(cAi(A))C k4;(f(A)). Hence, f is
continuous at each level.
(2) It is sufficient to show that for HA, a basis element
-1
of T(D), f(H,) is an open set in .i'(C). For y in Y and A
-
in P(Y) the general form of a basis element in T(D) is :
HA(y)=l iff y6Y-kbh(A)
HA(yIzd;iff ytkdi(A)-kdi-\(A)
HA(y)=O iff yeko(A).
Now let xu. We have HA(f(x))=l iff x&~-f-l(k*,,(A))
H,(~(x) )=o iff x~-'(q, (A)).
Since f is a homeomorphism for each d; in L~- [I\, we have
,I
f (kAl (A) )=c4; (f4(A) ) , therefore HA(f (x) )=Hf-'(A) (x) , which
shows that f is L-fuzzy continuous.
Corollary 7.3. Let =[X,% i, %=[Y,c; 4 be two L-FP families
of topological spaces, we have :
f is an L-fuzzy homeomorphism iff for each 4; in L~-{~J
(L n
f: (X, (A; ) ->(Y, I A; ) is a
homeomorphism.
In Pheorem 2.12 in[b], it was shown that level continuity
was equivalent to fuzzy continuity if instead of T(C) and T(D),
r' P
we take Sup(Q(c) ) and SU~(~D) ) . We shall see in the following
example that it is possible to find a smaller topology in 5~)
such that this conclusion still holds.
Example. Let X=Y=I and let L=(o,~, 11 .
$-level (X;iil - ) =usual topology on P (Y,T,)=indiscrete
2 2
0-level ( X, Po ) =usual topology on L
(Y,To)=usual topology
on 1
For A€P(x) and BeP(Y) and B#@, we have;
GA(x)=l iff x&-k, - (A) GB(y) =1 never
2
-
GA(x) -3 never
GB(r)=f iff ~EY-P~(B)
G,(x)=O iff xdco(A)=kl(A)
GB(y)=O iff ywo(B)
2
Let f be any function from X into Y. i'hen f-'(~~) of
will take only two values: 0,3. Hence, the inverse image
of GB cannot be written as a supremum of characteristic
functions. In other words, no map is fuzzy continuous from
T(C) into T(D). Suppose now, that f is continuous at each
+P
level. We claim that for any Bhin P(Y) , GBof =G
~-'(c,(B)
43.
GBof (x) -1 never
G~O~(X) -3
iff xex-f"( co(B) )
G~O~(X)=O iff xei1(c0(~)
Therefore, it is easy to see that GB0f.G A&. Hence,
f"( CO(A) ) "
f is fuzzy continuous from (X,'Y(C)VT~) into (Y, P(D) ) .
s/
To conclude, let us show that TCVT(C) # SU~~C). Let A=[o,~).
Define G(x) =1 iff xeX-A and ~(x) = 6 iff XEA. 'Then G€Tp .
Suppose G is in i'cT(C), then [xiG(x)=l\ is open in To= 1, 2
because T, \l T(c) has thed -property. To summarize, we have
b
exhibited a fuzzy topology different from Sup(qC)) for
which level continuity is equivalent to fuzzy continuity.
-
Our next theorem will generalize this example. From now on,
we will denote Tc\lr(C) by T(K).
Theorem 7.4. Let d,a be two L-P1P families of topological - -
spaces as given in Theorem 7.2. Then continuity at each
level is equivalent to fuzzy continuity from (X,T(K)) into
(Y,T(D)
Proof. We only need to show sufficiency. Let f be continuous
at each level di, and let GA be in T(D) . We claim
where H = Gf-l(4JA) )\/[G f -'(kn.,(~)
Note that df' (kAh(A)) is the characteristic function of
-1 -1
X-f (kd,(A)) because f (kA,(A)) is closed in each T4; .
Moreover, for any r such that l$r(n-1, an easy computation
shows that G -'
f (kah(A))
Ad,+, takes only two values db+,and 0.
More precisely,
~d+~l(x) =dp+,iff xtx-f-'(hc (A)
Gf? (kp (A))
Gf*'(k@+( A) )Abp+l (x) =O iff xef-'(kb,(~) 1
on ~-"(4,, (A)), G~o~(x)=H(x) =l. On f-'(k*,,(~) )-f-'(kL, (A) )=
f-\(k,++lA)-~,(~)), ~~of(x)=zr+t and for any jlr,
f-'(k~~ (A) )~f-'(k,,, (A)), that is:
On f-'(ko(A) ) , everything is 0. In conclusion, H=tiAof.
Level continuity is equivalent to fuzzy continuity using the
Lowen minimum for both domain and range,
Remark. We can slightly generalize this result by using a
countable chain for L~ (rather than a finite chain) in the
-
definition of an L-F'YP family of topological spaces, It is
easy to see that Lemma 3.8, 'Theorem 3.9 are still true
and that Theorem 7.2, 7.4 still hold. - -
Corollary 7.5. Let d.5 be two L-FPP families of topological
spaces as given in rheorem 7.2. Let f t X-'>Ye Let TI ,%be
L-fuzzy topologies such that T(C )s~',c~'(K) and .I'(D)I'I; .
If there is an A in ~a&uch that f: (x,%) -? (Y ,%) is discontinuous,
-
then f t (X,'X)->(Y ,'fd is fuzzy discontinuous.
Proof. Suppose there is an A in L~- )l\ such that f is dis-
continuous. By 'Theorem 7.4, f:(X,Y'(K))-7(Y,i'(D)) is fuzzy
discontinuous. Therefore, f:(X,'I:)-a(Y,~(D)) is fuzzy dis-
continuous and so f t (X, 3)-+ (Y ,I%) is fuzzy discontinuous.
CHAPTER VIII
Suitability
First, recall from [71 the definitions of a
suitable space and of a fuzzy retract. For both (X,T) is
an L-fuzzy topological space.
Definition 8.1. If ACX, then A is non-trivial iff P($A$x.
A is a suitable open set in (X,T) iff A is non-trivial and
q(A) is an L-fuzzy open set in (X,). (X,T) is suitable iff
(X,T) has a suitable open set.
Definition 8.2. Let ACX. A is an L-fuzzy retract of X in
(1 if there is a function r: (X,T)-~(A,T~) such that
r(x)=x for each x in A and r is fuzzy continuous.
Theorem 8.3. Let &be an L-FTP family of topological spaces.
Then we have3 A is suitable open iff for each d; in ~~-[l\,
-
A is open in '1~;.
Proof. Suppose A is suitable open in I'(C). Let Y(A) =VG
11 3
5 A;'
We have A=U x:GA.(x)=l =u X-c&,(A;) , which is open --
'3 I.
in 4, , therefore open in t; , for each A; in L~- [I\.
Now, for sufficiency, let A in Gi , for each 4; in
Ill.
Denote C=X-A, we have cdj (C)=cd;(C)=C, for each in ~~-[l\.
'Theref ore, we have :
-
GC(x)= 1 iff XEX-cch(C) iff xcX-C.
GC(x) = d; iff xec A;(C)-c4;-,(C) =c-c=P(
GC(x)= 0 iff xfc d,(C)=C
Therefore, GC=y(X-C)=4(A) and A is a suitable open set.
Corollary 8.4. Given A, an L-PPP family of topological spaces,
we have that the set of suitable open sets of T(C) is equal
to tr -\$,x\*
Corollary 8.5. Let (X,T) be a topological space. We can
associate to (X,'?) a fuzzy topological space (x,?") in a
natural way: f is inriff f=y(A) for A in 1'. Let (x,Ch)
be the fuzzy topological space associated with ,&+, , the
w
coarsest topology in &. Then T(C )3 6,.
Remark 8.6. Corollary 8.5 gives us another proof of Proposition
4.2.
F'
Corollary 8.7. Let 1 LI>~. Then sup()(c) )=YCv9?(C)=~(~) iff
'1l4, is discrete.
Proof. For necessity, let O,i = x:G(x)=*;\ , and Q,=[x:G(x)=i~-
P'
C
for some chosenG in T The 04; are pairwise disjoint.
-
Denote H=V( (04; )A*;). We have H(x)=& iff xcG4; (y(O$;) is
4
in l(C) by Theorem 8.3), that is H=G and G is in T(K).
,To prove sufficiency, let A be in P(X). Define G by G(x)=l
iff x€A, dif0 otherwise. G is in T and by Lemma 3.8 --
3
P'
I
x:G(x)=l is open in !I&,, and hence A is open in &,, .
Remark 8.8. 'The condition on the cardinality of L is indis-
-
pensable in the above corollary. If I L I =2, litp= /iX)
and
L- 13
P(C) =sup(J(c) ) for any L-F1P family of topological spaces.
-
CHAPTER VIII
Suitability
First, recall from [71 the definitions of a
suitable space and of a fuzzy retract. For both (x,T) is
an L-fuzzy topological space.
Definition 8.1. If ACX, then A is non-trivial iff P($A$x.
A is a suitable open set in (X,T) iff A is non-trivial and
q(A) is an L-fuzzy open set in (Xi (X,T) is suitable iff
(X,T) has a suitable open set.
Definition 8.2. Let ACX. A is an L-fuzzy retract of X in
(X.'P) if there is a function r:(X,!P)->(A,!PA) such that
r(x)=x for each x in A and r is fuzzy continuous.
Pheorem 8.3. Let Lk be an L-FTP family of topological spaces.
Then we have: A is suitable open iff for each d; in ~~-[l\,
-
A is open in '1~;.
Proof. Suppose A is suitable open in f(C). Let Y(A)=~G
11 3
5 A;'
We have A=U x:G , (x)=l =u X-c&,(A;) , which is open --
5 I A'
in Th, , therefore open in 'Q; , for each A; in L
a
- [l\.
Now, for sufficiency, let A in '&; , for each 4; in ~~-[l\.
Denote C=X-A, we have Cdj (C)=cd;(C)=C, for each a; in ~~-[l\.
'Therefore, we have :
GC(x)= 1 iff x€X-cch(C) iff xcX-C.
-
GC(x)= d; iff xtc A;(C)-c4;-,(C)=C-C=$
GC(x)= 0 iff xecd,(C)=C
Therefore, GC=~(X-C)=~(~) and A is a suitable open set.
Corollary 8.4. Given A, an L-FTP family of topological spaces,
we have that the set of suitable open sets of 'i'(C) is equal
Corollary 8.5. Let (X,T) be a topological space. We can
associate to (X,T) a fuzzy topological space (x,?) in a
natural way: f is inriff f=q(A) for A in L'. Let (x,%,,)
be the fuzzy topological space associated with I&,, , the
ry
coarsest topology in &. Then T (C
6,.
Remark 8.6. Corollary 8.5 gives us another proof of Proposition
4.2,
Corollary 8.7. Let \ ~133. Then sup( IF' (C) )='icv~(c)=r(~) iff
'id, is discrete.
Proof. For necessity, let ={X:G(.)=*~ , and 01= ~:G(X)=I~-
for some chosen G in T The 0d; are pairwise disjoint.
P*
Denote H=V( (04; )Ad;). We have H(x)=dl iff xtG4; (y(04,) is
-
Y
in l(C) by Theorem 8.3), that is H=G and G is in T(K).
'To prove sufficiency, let A be in P(X). Define G by G(x)=l
iff xEA,d;#0 otherwise. G is in T and by Lemma 3.8--
1
P'
I
xrG(x)=l is open in &,, and hence A is open in Td, .
Remark 8.8. The condition on the cardinality of L is indis-
pensable in the above corollary. If I L I =2, Tp= 4X) and
r/ 13
P(C)=SU~(~(C)) for any L-F'lP family of topological spaces.
Let BCX and k= [(X,&; )Il a family of topological spaces be
given. %=[(B ,X OB\ is also an L-FPP family of topological
7
spaces, the generated fuzzy topology will be denoted by ll(B,C).
By T,(C), we understand the fuzzy subspace topology induced
Lemma 8.9.. Let BCX be suitable closed and let &be an L-FTP
family of topological spaces. 'i'hen T(B ,C)~T~(C)<'L'(B,K).
Proof. (1) T(B,C)GTB(C) (P(B,K)&~L'~(K))
Denote by cd the closure operator of & and kL the closure
operator of %= kl3.G: 0 8)) . By definition of a subspace
topology, we have: for AEP(B), kAi(A)=cd; (A)~B. Let AEB,
G~~~(B,c), HA&~(C). For sets U,V,S, s~(u-v)=~~u -snv. Y'hen
H* IB(x) =1 iff xeX-cdh(A)I\B=b-kAh(~)
*
.
-0 iff xec,, (A)hB=kA, (A).
Hence H~I B=~A and T(B,C)CYB(C) (T(B,K)sTB(K)).
(2) 1fB(C)51i'(B,K) (lfB(K)cP(B ,K)).
-
Let ASX, GAcT(C) and such that ifla-[l\ , cdi(A)I\b#P( and
cdL-, (A)~B=$. We have G (x) =l iff xfB-k&,(A)
AIB .
=d; iff xek&~(A)
: never - -
=o
Claim: GA\ B=H= [~(GB~C~~ (A)A~.+I )]I 5 vAi
As in the proof of Pheorem 7.4, GB,,cA,(A) Ad,+,
takes only
two values a,,, and 0. More precisely,
G Bhc,,, (A)~~P+\ I B (x)=&,,, iff xe(X-c,. (BflcAp (A) )hB
-
iff x~B-(c~,(B)~~c~~(A) )AB
iff xtB-kAp(A) (since B is
suitable closed)
A similar computation shows that G
Bncd,*(A)
\ (x)=O iff
Ad,*, B
xckAb(a). Hence, GAIB(x)=H(x) for XCB-kd ;(A). Let xtkd;(A),
then for any r>,i , GBl\cA,(A) AA,,, (x)=o ~~us,H(x)=~;.
Conclusion: H(X)=G*]~(X) on B and since Bnc,,(A)SB for any r,
HEIL~(B ,K) .
Remark. If for AQP(X) and c~;(A)~\B=$ for each i, then
is identically one. Let i:(B,fh;I\b)-7(X,.i'd;) be the injection.
Then GA\ B=GA~ i on B-lk (A).
Corollary 8.10. T(B,K)=YB(K) for B suitable closed.
iheorem 8.11. Let BcX be suitable closed and (X, )I
Q
3 '
q={(B,kOB) be two L-FTP families of toplogical hausdorff
A
spaces. Then we have:
AiEL r: (X,PA;)->(B,'&.AB) is a retraction iff
r: (x,T(K) )->(B ,TB(K) ) is a fuzzy retraction.
Proof. This is a simple application of Theorem 7.4 and
Corollary 8.10.
-
Remark 8.12. Since every problem of extension can be reduced
to a problem of retraction (see ~u[1] for -the ordinary case
for example, and Rodabaugh [7] for the fuzzy case) we have,
in fact, an extension property related to the Tietze extension
property.
Theorem 8.13. Let BSX, &= (X T ) , be an L-FTP family of
r nA.3
topological spaces. If there exists a continuous map
-
r: (x,T~,)-~(B,!&~B) such that r(x)=x for x6B then (B,'IlB(K))
is a fuzzy retract of (X,'Z(K)).
Proof. Let r:(X,P4,)->(B,'&;nB) be a continuous map such that
r(x)=x for x6B. Then for each A2 €ILa-[1(, rr (X,Td; )-7(B,Td; nB)
is a retraction, hence by Theorem 8.11, (B,TB(K)) is a fuzzy
retract of (x,T(K)).
Ka as a Semi-closure Operator or a Closure Operator
T(C) generates A -closure operators ford in L-La.
In Proposition 2 -10 [+I, Klein shows that for in L-1 with
0"
4 4 L~ and T in $(c ) with w , the A -closure operators
generated by 'T,'I'(C) respectively, we have for every A in
P(X) k&(A)Ccd(A). In this chapter, we will find conditions where
this inclusion becomes an equality.
Definition 9.1. In a partially ordered set (P, a), an element
y in Y is said to cover an element x of P if xCy and if
*y.
there does not exist any element zhln P such that xdj
(d; is in L~ since La is finite)
Proof. (i) It is sufficient to show:v A,B in P(X) ,
k4(~UB)Sk4(A)ukg(B). Let x #. k4(A)U.k4 (B) . Then there are
G,H in T(C) such that:
G(X))/~ and GAY(A)=O
H(x))4 and H~~(B)=o
We have G(x)=&, H(x)=$ with A; ,d; eLa- )l\. Without loss
of generality, we have G(x)AH(x) =d;)4 and GAHAq(A) =0,
GAHA~B)=O, which implies (GAHA~(A) )v(GAHA~(B) )=o. Hence,
(GAH)A~(A~B) =0, so x# k4(AUB) and k4 is a semi-closure
operator.
(ii) Let A beEP(X). It is sufficient to show that:
k4(k4(A) )$%(A)
Let %EX-k4 (A). Then there exists G in T(C) such that
G(xo)>4 and GAy(A)=O. Let us consider GA$k (A)). For x
5
in X-k5(A), we have ~A?(ly(~))(x)=o. For x in A, G(x)=O,
hence G/l?(kJ5(A))(x)=0. For x in k4(A)-A, we have G(x)44 ,
that is G(x)(j by Lemma 9.2 and ke~-L~. Since hj covers 0,
G(x)=O. Hence,
xcX, GAy(klj(A))(x)=O. So x,Q %(kl)(A))
and k3 is a closure operator.
A
(iii) Lets; =v[A~EL~-@\ . ByLemma9.2,
- -
we have for any-^ in T(C),
CHAPTER X
Normality
All the topological properties we have considered
so far transfer rather nicely to the fuzzy topology T(C)
generated by an L-FTP family of topological spaces. This
was due to the fact that T(C) had thee(-property. For
fuzzy normality, we do not have, so far, such a direct
correspondence.
Definition 10.1, (X,'f) is pseudo-fuzzy normal iff for any
A,B closed in T such that AhB=O, there exist U,V in T
such that ASU, B,(V and URV=O.
Theorem 10.2. Given an L-FTP family of topological spaces,
we have: T(C) is pseudo-normal iff Td, is normal.
Proof. Let A,B be closed in TA, and such that A~;B=$. We -
1
I
have w(A), &(B) in 'i'(C) (cf. suitability) and +(A)Ay(B) =o.
Hence, there exist H,G in 1(C) such that Y(A)&H , ,q(B)I
be a lattice morphism defined by (I(A; )=/i,. For Y(A; )c!? ,a and for each E C (0,dJ
1
there exists a finite subfamilyh, of/? such that Vu >,~-f.
40
Definition A.9. (X,'i,Lr) is fuzzy compact in the Lowen
I
sense iff (X,I,I) is fuzzy compact in the Lowen sense.
Proposition A.10. (X,'i',L, ) is fuzzy compact in the Lowen
*
sense iff b~,;e L, , (X,'?) is -compact.
Proof. '1'0 show sufficiency, let d; &.A /U(
'2
implies V.u>/&; for some finite subfamily hO of 4.
4c
Let 0(f ,r;-~ .
S 0
case 2.d)A; .~u>A implies vu),~~,, by the Remark A.5.
3 30
Let 0(1/&~~,>;x-2 .
30
Y
For necessity, let f =A; - li-, /2 and [u\+ be an di -shading
of X. I'hen Vu>,&;
implies there exists a finite subfamily
-
'3
& of /3 such that VM. >,A; f >d;-t . By the Remark A. 5, )/u >, AL'
.
/
30 - 30
Now let dL,*tq or L'u>,d .
'20 30
BIBLIOGRAPHY
1 S. 1. Hu, Theory of Retracts, Wayne State University
Press, Detroit, 1965.
2 B. Hutton, Normality in fuzzy topological spaces, J. Math.
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3 A. J. Klein,a -closure in fuzzy topology, Rocky Mountain
J. Math. 11 (1981) 553-560.
4 A. J. Klein, Generating fuzzy topologies with semi-closure
operators, Fuzzy sets and Systems 9 (1983) 267-274.
5 R. Lowen, A comparison of different compactness notions
in fuzz to ological spaces, J. Math. Anal. Appl. 64
(1978) $46-$54.
6 S. E. Rodabaugh, The Hausdorff separation axiom for
fuzzy topological spaces, Top. Appl. 11 (1980) 319-334.
7 S. E. Rodabaugh, Suitability in fuzzy topological spaces,
J. Math. Anal. Appl. 79 (1981) 273-285.
8 S. E. Rodabaugh, Connectivity and the L-fuzzy unit interval-,
Rocky Mountain J. Math. 12 (1982) 113-121.
9 S. E. Rodabaugh, A categorical accomodation of various
notions of fuzzy topology, Fuzzy Sets and Systems 9 (1983)-
241-265.
10 R. H. Warren, Neighborhoods, bases and continuity in
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(1978) 459-4700
11 T. E. ~antner, R. C. Steinlage, andR. H. Warren, compact-
ness in fuzzy topological spaces, J. Math Anal. Appl. 62
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