6U, where a E X and 'LA is an ultrafilter. E(X) is atomic as well as dually atomic, In this chapter we investigate the lattice structure

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1 Chapter-3 LATTICE OF FUZZY TOPOLOGIES In 1936 G. Birkhoff [ 4] proved that the family of all topologies on a set forms a complete lattice under the natural order of set inclusion. The least upper bound of a collection of topologies on a set X is the topology which is generated by their union and greatest lower bound is their intersection. The atoms in the lattice E ( X) of all topologies on a set X are of the form { O,A,X J, where AC X. In 1964, 0. Frolich [9] characterized the dual atoms called ultratopologies as ^)(X \ [ a] ) V on X. Also, 6U, where a E X and 'LA is an ultrafilter E(X) is atomic as well as dually atomic, but not modular if JXJ > 2. In 1966, A.K. Steiner [34] proved that the lattice of topologies on an arbitrary set is complemented. In this chapter we investigate the lattice structure of the set of all fuzzy topologies on a given set X. For a given topology t on X, we study properties of the lattice '_3-t of fuzzy topologies defined by families of lower semicontinuous functions with reference to t on X. 45

2 -46- We prove that the set of all fuzzy topologies forms a complete atomic lattice under the natural order of set inclusion. This lattice is not complemented and has no dual atoms. Also we prove that topologically generated fuzzy topologies and crisp topologies have complements Preliminaries Definition Let (X,T) be a topological space, f:x -^ [0,1] is lower semicontinuous a function if f-1 (a,l] is open in X for 0 < a ^,< 1 [38]. Notations If F is a fuzzy topology on the set X, let Fc denote the 0-1 valued members of F, that is Fc is the set of all characteristic maps in F. Then Fc is a fuzzy topology on X. Throughout this chapter if AC-X, let µa denote the characteristic map of the subset A, that is, µa x) = 1 if x e A and µa(x) = 0 if xe X\ A. Define Fc- ={A C X µa G Fc}. The fuzzy space (X, Fc) is then the "same" as the topological space (X,Fc*).

3 -47- Definition A fuzzy space (X,F) is said to be an induced fuzzy space or an induced space [26], provided that F is the collection of all lower semicontinuous maps from (X,Fc*) into [0,1]. Remark Induced spaces are equivalent to topologically generated spaces of Lowen [1.2.16]. Definition A fuzzy space ( X,F) is said to be a weakly induced fuzzy space (according to H.W. Martin [26]) provided that whenever g E F, then g : (X, Fc*) --^ [ 0,1] is a lower semicontinuous map. It is observed that every induced space is weakly induced and every topological space, regarded as a fuzzy space via the fuzzy topology of all characteristic maps of the open sets, may be thought of as weakly induced space.

4 -48- Theorem (H.W. Martin [26] ) A fuzzy space (X,F) is an induced space iff (X,F) is a weakly induced space such that every constant map from X into [Ot1] belongs to F. Remark For a given topology T on X, the set consisting of all constant functions from (X,T) to [0,1] and characteristic functions corresponding to open sets in T forms a subbasis for W(T) = { f1f is a lower semicontinuous from (X,T to [0,1] Lattice of fuzzy topologies For a given topology t on X, the family '-t of all fuzzy topologies defined by families of lower semicontinuous function from (X,t) to [0,1] forms a lattice under the natural order of set inclusion. The least upper bound of a collection of fuzzy topologies belonging to ^t is the fuzzy topology which is generated by their union and greatest lower bound is that which is their intersection. The smallest and largest elements in at are denoted by 0t and 1t respectively.

5 -49- Theorem t is complete. Let S = [ F, : 7i EA 3 be a subset of and G = fl F,. Then 0,1 E G since 0,1 C- FA, AEA Let a, b E G, then a,b C F^ T E AA. Then a A b E FX T E A. Therefore a A b C. G. Let { a j: j C J be a subset of G. Then V a. E F^ j e; J V a. C G. Therefore and is G E 91t jc-j J Therefore the greatest lower bound of S in Hence t is complete since it has the largest element consisting of all lower semicontinuous functions. Theorem is atomic. Atoms in 9t are of the form T(a) _ {0,1,a^ Let S be any element of 9-t different from Ot. Then S = V T(a). ae S Hence t is atomic.

6 -50- Theorem '31t is not modular. Choose distinct points al, (x2, a3 C (0,1). Then fuzzy topologies t0,1i, t0,1,a1), {Ola21 [o,i,a2a33 1 0,1,a1,a2,a3 forms a sublattice of 3t isomorphic to N5 = Hence the result. Theorem There is no dual atom in '3^t. Let F E Jt such that F L lt; we prove that there is an element G E at such that F < G A lt. By remark F cannot contain at the same time all constant functions as well as all characteristic functions corresponding to open sets in t. Let S be the collection of lower semicontinuous functions which do not belong to F. First we claim that S is infinite. Two cases may arise.

7 -51- Case-l. S contains constant functions. Let constant function a e S. Then there exist a constant function P < a such that F and the elements of the set K = [u p < y < a 3 of constant functions do not belong to F. Then by denseness property of real numbers S is infinite. Case-II. If S contains a characteristic function corresponding to an open set A in t, then there exists a lower semicontinuous function fa G F where fa is defined by fa(x) aifx F A -1 0 otherwise 0 < a < 1 and the elements of the set K = f fa I a<a <l + of lower semicontinuous function do not belong to F.Again by denseness property of real numbers S is infinite. Let g be a lower semicontinuous function belonging to S. Then F < FV1O,l,gJ = G < 1 t where g ^ F.

8 Complementation in the lattice of fuzzy topologies Theorem Atoms in at of the form {O,l,a. where a is a lower semicontinuous function different from characteristic function have no complement in Jt. We divide the proof into two parts. Part-a. Suppose F = { O,l,a is an atom where a is a constant function, a E (0,1). We claim that F has no complement. Clearly It is not a complement of F. Let G be any fuzzy topology in t other than lt. If constant function a E G, then G is not a complement of F since G A F A 0 t. Suppose constant function a does not belong to G. Then there exists a constant function 0< a such that R E G and the set K = YAP < Y < a j of constant functions is not contained in G. Now G = [f A gl f C- F, g c-- G] is a base for H = F VG. Then K is not contained in H. Hence G is not a complement of F.

9 -53- Part-b. Suppose F = 1 0,1,a 3 is an atom where a is a lower semicontinuous function which is neither a constant function nor a characteristic function. We claim that F has no complement. Clearly It is not a complement of F. Let G be any fuzzy topology in 'J t other than lt. Then G cannot contain at the same time all constant functions as well as characteristic functions corresponding to open sets in t since they together generate 1t (Remark 3.1.7). Two cases arise. Case-1. G does not contain all constant functions. If a E G then G is not a complement of F since F/1 G A Ot. Suppose a 4 G. Then there exists a constant function P < a such that REG and the set K = lylp < y < a 3 of constant functions is not contained in G. Then G is not a complement of F., Case-II. G contains all constant functions. Then at least one characteristic function µa corresponding to an open

10 -54- set A in t does not belong to G. If a e G, then G is not a complement of F. Suppose a, G. Then we can find a set K = [ f( I a < 1 of lower semi- A J continuous functions defined by fa(x) = { a oif x C- A therwise such that K is not contained in G. Now consider the base Q3 = {f A g I f C-F, g r. G} of H= F V G. Then K is not contained in H. Thus G is not a complement of F. Remark When t = d, discrete topology on X, ad becomes the lattice of all fuzzy topologies on X. Thus the family of all fuzzy topologies on a set forms a complete atomic lattice under the natural order of set inclusion. This lattice is not even modular. It is not complemented and has no dual atoms. Let E denote the family of fuzzy topologies defined by lower semicontinuous function where each lower semicontinuous function is a characteristic function. Then E is a sublattice of 9d and is lattice isomorphic to the lattice of all topologies on X. The elements of E are the crisp topologies.

11 -55- Theorem Every topologically generated or crisp topology has complement in d. Let F be a topologically generated fuzzy topology. Then there exists a topology T on X such that W(T) = F. Since the lattice of topologies is complemented there exists at least one complement S of T. Then identify S with its characteristic function let it be denoted by FS. Then F and FS are complements of each other. Remark The converse of theorem is not true as shown by the following counter example. Let X be any set and F be the fuzzy topology which consists of constant function a where a C [ 0 71 ] and the constant function 1. Let F' be the fuzzy topology generated by constant functions a where a E ( 19 1 ] together with all characteristic functions. Then F and F' are complements to each other.

12 -56- Remark If we define an equivalance relation by F1 ^- F2 if and only if i(f1) = i(f2). Then ( a^ ',\ ) is lattice isomorphic to ( E,,^<, ). Consider Z = V (t x fit) and define (t,f) N (s,g) t G E if and only if F = G. Then Theorem Zj^- is a complete lattice, isomorphic to the lattice of all fuzzy topologies on X. [ (t,f) ] ^< [ (s,g) ] if and only if F < G [ (t,f) ] A [ (s,g) ] _ [ (d, F A G) ] [ (t,f) ] v [ (s,g) ] _ [ (d, FVG) ] Then (ZjN, < ) is a lattice. If S is any subset of Zjr we can show that S has a greatest lower bound in ZI,- 0 Therefore Z1N is complete.

13 -57- Define 9 C-`1d, < ) ---^ (2. I", `< ) by 9(F) = [( d,f)] Then G is one - one and on to. Fl < F2 in 'j-d e(fl) < (F2) in ZH 9(F1V F2) = L (d, Flu F2) ] L (d,f1) ] V [ (d,f2) ] 9(Fl) V G(F2) 9(F1A F2) = L (d, F1n F2) ] L (d,f1) ] A [ (d,f2) ] A(Fl) A 8(F2) Remark 3,3.7. If we define N in Z by (t,f) N (s,g) if and only if t = s, then ZI,- is a complete lattice isomorphic to the lattice of all topologies on X. i