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 . 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