FUZZY P-LIMIT SPACES

Transcription

1 Bull. Korean Math. Soc. 28 (1991), No. 2, pp. 191 FUZZY P-LIMIT SPACES KYUNG CHAN MIN 1. Introduction The notion of convergence in a fuzzy topological space was introduced in various aspects as a local theory of fuzzy topologies [2, 9, 10, 11, 12, 15, 19, 20]. In [12] a notion of fuzzy limitierung on a set is defined in terms of prefilters convergent to a fuzzy point and allows natural function structure in it. In this paper using fuzzy x-filters we introduce a notion of convergence to a point in a set, called afuzzy p-limitierung, generalizing that of fuzzy neighborhood system introduced by Warren [18], which characterizes fuzzy topology. The notion of fuzzy p-limitierung enables us to form a convenient class containing all fuzzy topological spaces and all limit spaces. In section 4 we show that in fuzzy topology there is no natural function space structure. We show that fuzzy p-limit spaces form a quasitopos and give explicitly natural function space structure in it, providing an exponential law C(X Y, Z) = C(X, C(Y, Z)). Form a categorical point of view the category FpLim of fuzzy p-limit spaces andfuzzy continuous maps is shown to be a quasitopos (=topological universe) containing the category FTop of fuzzy topological spaces as a bireflective subcategory and the category Lim of limit spaces as a bicoreflective subcategory. Throughout this paper, we use Lowen s notion of fuzzy topology [7]. For categorical background we refer to [4, 6] 2. Fuzzy p-limitierung Warren [18] characterized a fuzzy topology in terms of a fuzzy neighborhood system, which defines a concept of convergence to a point in a fuzzy Received July 14, Revised May 9, Thes research was supported by the Korea Science and Engineering Foundation.

2 Kyung Chan Min topological space. A fuzzy set N in X is called a neighborhood of a point x in (X,δ)iff there exists U δ such that U N and 0 <µ U (x)=µ N (x). A fuzzy set U is open in (X,δ) iff for every x X with µ U (x) >0, U is a neighborhood of x. Thus this neighborhood system determines a fuzzy topology. A function f : (X,δ) (Y,δ ) is fuzzy continuous at x iff for every neighborhood N of f (x) there exists a neighborhood M of x such that f (M) N and µ M (x) = µ N ( f (x)). DEFINITION 2.1. Let X be a set and x X. A fuzzy x -filter F on X is a family of fuzzy sets in X subject to the following axioms: (F1) if A F and A x B, then B F,(A x B means A B and µ A (x) = µ B (x)) (F2) for all A, B F, we have A B F, (F3) 0 / F, where 0 is the constant map with value 0, (F4) for every subfamily {A i } I of F, i I A i F. The family of all neighborhoods of x in a fuzzy topological space X is a fuzzy x-filter on X, called the neighborhood fuzzy x-filter. Let B be a family of fuzzy sets in X. If B satisfies the conditions, 0 / B and for any B 1, B 2 B, there exists B B with B 1 B 2 x B, then B generates afuzzyx-filter[b] ={A I X : i I B i x A for some family {B i } I in B}. Denote F x (X) = the collection of all fuzzy x-filter on X. For F, G F x (X), we denote F G in F x (X) iff for each A F there exists B G with B x A. For F, G F(X) x, the family {A B : A F, B G,µ A (x)=µ B (x)} generates a fuzzy x-filter, denoted by F G in F x (X). We note that if F i G i in F x (X) for i = 1, 2,,n, then n i=1 F i n i=1 G iin F x (X). For a function f : X Y and F F x (X), wedefineafuzzy f(x)-filter f (F) = {B I Y : f (A) B with 0 <µ A (x)=µ B (f(x)) for some A F} on Y. In general, for A F, f (A) needs not be a member of f (F). However if µ A (x) = α>0,then f (A α) f (F). Hence we can show that for f : X Y, g : Y Z and F F x (X), g f (F) = g( f (F)). DEFINITION2.2. A fuzzy p-limit space (fps for short) is a set X structured with a function, called a fuzzy p-limitierung, which assigns to every x X aset (x) of fuzzy x-filters on X subject to the following axioms: (L0) F (x) α F for all α>0, (L1) [x] ={A I X :µ A (x)>0} (x), 192

3 Fuzzy p-limit spaces (L2) F (x) and F G in F x (X) G (x), (L3) F, G (x) F G (x). If F (x), we say that F converges to x and sometimes write F x instead of F (x). REMARK. A subset A of a set X can be identified with the characteristic function χ A. Thus if we substitute I by {0, 1}, the two point chain, then it is easy to see that the notion of fuzzy p-limitierung is equivalent to that limitierung. We introduce the notion of initial and final fuzzy p-limitierung. If the identity map 1 X : (X, 1 ) (X, 2 is fuzzy continuous, then we say that 1 is finer than 2 and that 2 is coarser than 1. THEOREM 2.3. (Existence of initial structure) Let X be a set, let {(Y i, i )} I be a family of fps s, and for each i I let f i : X Y i be a map. Define a function on X as follows; for each x X a fuzzp x-filter F (x) iff f i F i ( f i (x)) for each i I. Then is the initial fuzzy p-limitierung on X w.r.t. the family { f i } I, which is the coarsest fuzzy p-limitierung on X making each map f i fuzzy continuous. Proof. It follows from the fact that f ([x]) = [ f (x)] and f(f G) = f(f) f(g) fuzzy x-filters F, G. The existence of initial structures guarantees that of final structures, However, we present here an explicit form of final fuzzy p-limitierung: Let Y be a set, let {(X i, i )} I be a family of fps s and for each i I let f i : X i Y be a map. Define a function on Y as follows, for each y Y a fuzzy y-filter F (y) iff F [y]orf n k=1 f ik(f k ), where F k i k (x k ) 1 for some x k f i k (y), i k I and k = 1,,n.Then is the final fuzzy p- limitierung on Y w.r.t. the family { f i } I, which is the finest fuzzy p-limitierung on Y making each f i fuzzy continuous. In a usual way, the notions of subspace, product, quotient space and coproduct can be defined. 3. Relationship with fuzzy topology and limitierung Let (X,δ) be a fuzzy topological space (fts for short). For each x X, let δ (x) ={F F x (X): N x F},where N x is the neighborhood 193

4 Kyung Chan Min fuzzy x-filter. Then δ is a fuzzy p-limitierung on X. We call a fuzzy p- limitierung fuzzy topological if the convergent fuzzy x-filters are precisely those of a fuzzy topology. We show that a fuzzy p-limitierung is not fuzzy topological in general. Let (X, )be a fps. A fuzzy set U in X is said to be open iff for every x X with µ U (x) >0,F ximplies U F. Then the collection δ of all open fuzzy sets in (X, )forms a fuzzy topology on X. Moreover, by routine work, we have the following. PROPOSITION 3.1. (1) For any fuzzy topology δ on X,δ =δ δ. (2) A fuzzy p-limitierung on x is fuzzy topological iff = δ. (3) For every x X, (x) δ (x). Let X and Y be fps s. A function f : X Y is said to be fuzzy continuous at x X iff f (F) f (x) in Y, whenever F x in X. A function f : X Y is fuzzy continuous iff it is fuzzy continuous at every x X. Note that the identity map 1 X : X X is fuzzy continuous since 1 X (F) = F [x] for a fuzzy x-filter F. Recall that for fts s X, Y, amap f :X Y is fuzzy continuous at x X iff N f (x) f (N x ) in F f (x) (Y ). Thus by definitions we have PROPOSITION 3.2. (1) A function f : (X,δ) (Y,δ )is fuzzy continuous iff f : (X, δ ) (Y, δ )is fuzzy continuous. (2) If f : (X, ) (Y, ) is fuzzy continuous, then f : (X,δ ) (Y,δ )is fuzzy continuous. Let (X, ) be a limit space and F x in X. Then the set F F of all fuzzy sets A in X such that µ A (F) µ A (x) in I r (=I with the right topology) and µ A (x) >0 form a fuzzy x-filter on X. Let (x) ={F F x (X): F F F for some F (x)}then is a fuzzy p-limitierung on X. Note that F F G = F F F G for F, G (x). Let (X, )be a fps. For each F (x) let F be the set of all filters F on X such that µ A (F) µ A (x) in I r for each A F with µ A (x) >0.Then (x) = F (x) F defines a limitierung on X, since F G in F G for F F, G G. PROPOSITION 3.3. (1) For any limitierung on X, =. (2) For any fuzzy p-limitierung on X, (x) (x). Proof. (1) By defintions, (x) (x). Let F (X). Then F F for some F (x) and F F G for some G (x). We show that 194

5 Fuzzy p-limit spaces F G <x> :LetV G <x >.Then the characteristic function χ V F G and hence χ V (F) χ V (x) in I r. Thus V F for some F F. (2) It is immediate from definitions. PROPRSITION 3.4. (1)Ifamap f :(X, ) (Y, )is fuzzy continuous, then f : (X, ) (Y, De ) is continuous. (2) A map f : (X, ) (Y, ) is continuous iff f : (X, ) (Y, )is fuzzy continuous. Proof. It is easy to show by definitions and Proposition 5.1. Categorical comments The category FTop is known to be a topological category [8]. A category FpLim is formed by all fuzzy p-limit spaces and fuzzy continuous maps. By Propositions 3.1 and 3.2, FTop is shown to be a bireflective subcategry of FpLim in a natural way. By Theorem 2.3 and the definition of fuzzy p- limitierung, FpLim is a topological category. By Propositions 3.3 and 3.4 the category Lim is shown to be a bicoreflectivesubcategory offplim. We recall that the category Top of topological spaces is a bicoreflective subcategory of FTop[7] and Top is a bireflective subcategory of Lim[3]. Let L( ˆL) be the embedding functor from Top(FTop) into Lim(FpLim) with reflectors R( ˆR), respectively. Let F( ˆF) be the embedding functor form Top(Lim) into FTop(pLim) with coreflector T ( ˆT ), respectively. We note that F(X) = (X, C(X, I r )) and T (X,δ) = (X,T δ ), where T δ is initial topology w.r.t. the family {µ : X I r : µ δ}. Then we have the following diagram: THEOREM 3.5. (1) ˆT ˆL = L T, (2) ˆR ˆF = F R, (3) ˆL F = ˆF L. 195

6 Kyung Chan Min Proof. (1) and (2) are equivalent by the uniqueness of adjoint functor. We show (1). F x in ˆT ˆL(X,δ)iff F Nx, where N x is the neighborhood fuzzy x-filter in (X,δ), iff F N x in T (X,δ), where N x is the neighborhood filter of x in T (X,δ)iff F x in L T (X,δ). (3) F x in ˆL F(X, T ) iff F N x, where N x is the neighborhood fuzzy x-filter in F(X, T ) = (X,δ T ). Note that if µ A : X I r is continuous at x, then there exists a continuous map µ B : X I such that B A and µ B (x) = µ A (x). Hence N x = F Nx, where N x is the neighborhood filter of x in (X, T ). Therefore F x in ˆL F(X, T ) iff F x in ˆF L(X, T ). REMARK. This Theorem shows that ˆT and ˆF are extensions of T and F by (1) and (3), repectively, and ˆL and ˆR are extensionsof L and R by (3) and (2), respectively. COROLLARY. If a fps (X, ) is fuzzy topological, then ˆT (X, ) is topological. Proof. By Proposition 3.1 and Theorem 3.5, ˆT (X, ) = = LT ˆR(X, ). ˆT ˆL ˆR(X, ) REMARK. It is well known [16] that there exists a limit space (X, ), which is not equal to L(X,T ) for any topology T on X. Hence ˆF(X, ) = ˆL(X, δ) for any fuzzy topology δ on X, i.e., ˆF(X, ) is not fuzzy topological, by Proposition 3.3(1) and Theorem 3.5(1). Moreover there exists a fts (X,δ) whichisnotequalto F(X, T ) foranytopologyt on X [7]. Hence ˆL(X,δ)= ˆF(X, )for any limitierung on X by Proposition 3.1(1) and Theorem 3.5(2). 4. Function spaces Since the category Top is a bicoreflective subcategory of FTop and the full embedding functor from Top into FTop preserves initial sources, by Proposition 2.4 in [13] FTop is not cartesian closed. This means that there is no natural function space structure in FTop. However we can show that FpLim has a natural function space structure. In fact, FpLim is shown to be a quasitopos(= topological universe [14]) in the sense of Herrlich [5], equivalently cartesian colsed and hereditary. 196

7 Fuzzy p-limit spaces THEOREM 4.1. Final epi-sinks in FpLim are preserved by pullbacks. Proof. Let { f i : X i Y } I be a final epi-sink in FpLim, g : Z Y be a fuzzy continuous map and for each i I the diagram k i W i X i h i f i Z g Y a pullback. We will show that {h i : W i Z} I is a final epi-sink in FpLim. Note that for each i I, W i ={(z,x i ): f i (x i )=g(z)}is a subspace of the product Z X and h and k are projections. Clearly {h i } I is epi-sink. Let H b in Z. Then g(h [b]) g(b) in Y and hence there exists F i a i in X i (i = 1,,n)such that g(h [b]) n i=1 f i(f i [a i ]inf g(b) (Y)and f i (a i ) = g(b). Let G i be the fuzzy (b, a i )-filter on W i generated by {[H, F i ]: H H [b],f i F i [a i ]},where µ [H,Fi ](z, x i ) = µ H (z) µ Fi (x i ). Then h i (G i ) b in Z, since h i (G i ) H [b] (Note that h i ([H, F i ] t) H, where t = µ [H,Fi ](a i )), k i (G i ) a i in X and hence G i (b, a i ) in W i. We claim that H [b] n i=1 h i(g i ). Take A n i=1 h i(g i ). Then there exist H i H [b] andf i F i [a i ] such that n i=1 h i([h i, F i ]) A with µ Hi (b) µ Fi (a i ) = µ A (b) >0.for each i = 1,,n. Denote α = µ A (b). Note that µ fi (F i α )( f i(a i )) = α for each i = 1,,n and hence n i=1 f i(f i α ) n i=1 f i(f i [a i ]). Thus there exists H H [b] such that g(h) U n i=1 f i (F i α ) with 0 <µ H (b)=α.now we show that K = ( n i=1 H i) H b A. Take any z Z. Then µ H (z) µ g(h) (g(z)) n i=1 sup x i f 1 i (g(z)) µ F i (x i ) α and hence µ K (z) = ( n i=1 µ H i (z)) µ H (z) ( n i=1 µ H i (z)) ( n i=1 sup (z,x i ) W i n i=1 (µ H i (z) sup (z,x i ) W i µ Fi (x i )) = n i=1 sup (z,x i ) W i µ [Hi,F i ](z, x i ) = n i=1 µ h i ([H i,f i ])(z) µ A (z). (µ Fi (x i ) α )) 197

8 Kyung Chan Min Obviously µ K (b) = µ A (b). Hence the result follows. This Theorem means that FpLim is a quasitopos and hence cartesian closed and hereditary. (So, for any fps X the comma category FpLim/ X is cartesian closed. [5]) However there is no subobject classifier in FpLim, sincea bimorphism is not an isomorphism in general. Cartesian closedness implies the existence of a natural function space structure, which is now introduced explicitly. For fps s X, Y, let C(X, Y ) be the set of all fuzzy continuous maps from X intoy. ForafuzzysetLin C(X, Y ) and a fuzzy set A in X, we define a fuzzy set L(A) in Y by µ L(A) (y) = sup (x,g) cυ 1 (y) µ L(g) µ A (x), if eυ 1 (y) =, and 0, otherwise, where eυ is the evaluation map. Let H be a fuzzy f- filter on C(X, Y ) and A be a fuzzy x-filter on X. The family {B I Y : L(A) B, 0 <µ L (f)=µ A (x)=µ B (f(x)) for some L H, A A} generates a fuzzy f (x)-filter H(A) on Y. Define a function on C(X, Y ) as follows; for each f C(X, Y ) a fuzzy f-filter H ( f ) iff for each x X, H(A) f (x) in Y, whenever A x in X. Then is the natural fuzzy p-limitierung on C(X, Y ), i.e., with respect to this function space structure, eυ : X C(X, Y ) Y is fuzzy continuous and for a fuzzy continuous map h : X Z Y there exists a unique fuzzy continuous map h : Z C(X, Y ) such that eυ (1 X h ). (Cf. [12]) Thus we have the following convenient properties [4]: (1) C(X Y, Z) = C(X, C(Y, Z)) (first exponential law) (2) C(X, Y i ) = C(X, Y i ) (second exponential law) (3) C( X i, Y ) = C(X i, Y ) (third exponential law) (4) X X i = (X X i ) (distributive law) (5) Finite products of quotient maps are quotient maps. 198

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