Ultra L-Topologies in the Lattice of L-Topologies

Ultra L-Topologies in the Lattice of L-Topologies

L-Topologies

Raji George

Department of Mathematics, St.Peters College, Kolenchery -682311,Ernakulam Dt ,Kerala State,India.

T. P. Johnson

Applied Sciences and Humanities Division,School of Engineering, Cochin University of Science and Technology, Cochin-22,

Kerala State, India.

Abstract

| | | |

| | | |

We study the principal and nonprincipal ultra L-filters on a nonempty set X,where L is a completely distributive lattice with order reversing involution. Using this notion we study the topological properties of principal and nonprincipal ultra L- topologies. If X has n elements and L is a finite pseudo complemented lattice or a Boolean lattice, there are n(n 1)mk principal ultra L-topoogies, where m is the number of dual atoms and k is the number of atoms. If X is infinite, there are X principal ultra L-topologies and X nonprincipal ultra

L-topologies.

Keywords. L – filter, Principal and Nonprincipal Ultra L-Topologies, Simple extension.

AMS Subject Classification. 54A40

1. Introduction

   The Purpose of this paper is to identify the Ultra L-Topologies in the lattice of L-Topologies. For a given topology on X , T. P. Johnson [5] studied the properties of the lattice S,L of L-Topologies defined by families of scott

   | |

   | |

   | |

   | |

   continuous functions with reference to on X. In [5] Johnson has proved that S,L is complete, atomic and not complemented. Also he has showed thatS,L is neither modular nor dually atomic in general. In [3] Frolich proved that if X = n there are n(n 1) Principal ultra topologies in the lattice of topologies. In [8] A. K. Steiner studied some topological properties of the ultraspaces. In this paper we showed that if X = n and L is a finite pseudocompleted chain or a Boolean lattice, there are n(n 1)mk Principle Ultra L- Topologies, where m and k are the number of dual atoms and atoms

   in L respectively. If X is infinite, there are |X| Principal Ultra L- Topologies and |X| Nonprincipal Ultra L-topologies. Also we studied some topological properties of the Ultra L Topologies and characterise the T1, T2 L-topologies.

2. Preliminaries

   /

   /

   Let X be a non empty ordinary set and L = L( , , ,/ ) be a completely distributive lattice with the smallest element 0 and largest element 1((0 = 1) and with an order reversing involution a a/ . We identify the constant function with value by . The fundamental definition of L-fuzzy set theory and L-fuzzy topoogy are assumed to be familiar to the reader (in the sense of Chang [2] and Goguen [4]). Here we call L-fuzzy subsets as L-subsets and L- fuzzy topology as L-topology. For a given topology on X, the family S,L of all L-topologies defined by families of Scott continuous functions from (X, ) to L forms a lattice under the natural order of set inclusion. The least upper bound of a collection of L-topologies belonging to S,L is the L-topology generated by their union and the greatest lower bound is their intersection. The smallest and largest elements in S,L are denoted by 0s, and 1s, respectively.

   In this paper, L-filter on X are defined according to the definition given by A. K. Katsaras [6] and P. Srivastava and R. L. Gupta[7], by taking L to be the membership lattice, instead of the closed unit interval[0, 1].

   Definition 2.1. A non empty subset U of LX is said to be an L-filter

   if

   i.0 / U

   ii.f, g U implies f g U and

   iii.f U , g LX and g f implies g U .

   An L-filter is said to be an ultra L-filter if it is not properly contained in any other L-filter.

   Definition 2.2. Let x X, L An L-point x is defined by x(y) =

   ( if y = x

   where 0 < 1

   0 if y x

   Definition 2.3. In a filter U , if there is an L subset with finite support, then U is called a principal L-filter.

   Example 1: {f LX |f x, where x is an L-point}.

   Definition 2.4. In a filter U , if there is no L subset with finite support, then U is called a nonprincipal L-filter.

   { | }

   { | }

   { | }

   { | }

   Example 2: f LX f > 0 for all but finite number of points . Let f be a nonzero L-subset with finite support. Then U (f ) LX defined by U (f ) = g LX g f is an L-filter on X, called the Principal L-filter at f .Every L-filter is contained in an ultra L-filter. From the definition it follows that on a finite set X, there are only Principal ultra L-filters.

3. Ultra L-topologies

   An L-topology F on X is an ultra L-topology if the only L-topology on X

   strictly finer than F is the discrete L-topology.

   { | }

   { | }

   Definition 3.1. [9] Let (X, F ) be an L-topological space and suppose that g LX and g / F . Then the collection F (g) = g1 (g2 g) g1, g2 F is called the simple extension of F determined by g.

   Theorem 3.1. [9] Let (X, F ) be an L-topological space and suppose that F (g) be the simple extension of F determined by g. Then F (g) is an L-topology on X.

   Theorem 3.2. [9] Let F and G be two L-topologies on a set X such that G is a cover of F . Then G is a simple extension of F .

   Theorem 3.3. [3] The ultraspaces on a set E are exactly the topologies of the form S(x, U ) = P (E {x}) U where x E and U is an ultrafilter on E not containing {x}.

   Analogously we can define ultra L-topologies in the lattice of L-topologies according to the nature of Lattices. If it contains Principal ultra L-filter, then it is called Principal ultra L-topology and if it contains nonprincipal ultrafilter, it is called nonprincipal ultra L-topology.

   Theorem 3.4. [1] A principal L-filter at x on X is an ultra L-filter iff

   is an atom in L.

   Theorem 3.5. Let a be a fixed point in X and U be an ultra L-filter

   not containing a, 0 L. Define Fa = {f LX |f (a) = 0}. Then

   S = S(a, U ) = Fa U is an L-topology.

   Proof. Can be easily proved.

   Theorem 3.6. If X is a finite set having n elements and L is a finite pseudo complemented chain or a Boolean lattice, there are n(n 1)mk Prin- cipal ultra L-topologies, where m and k are the number of dual atoms and atoms in L respectively. If k = m there are n(n 1)m2 ultra L-topologies.

   Illustration :

   { } { }

   { } { }

   1. Let X = a, b, c , L = 0, , , 1 , a pseudo complemented chain. Here

      is the atom and is the dual atom.

      1

      0

      Figure 1:

      { | }

      { | }

      Let S = S(a, U (b)) = {f |f (a) = 0} {f |f b}, S does not contain the L-points a, a , a1 Then S(a, U (b), a ) = S(a ) = simple extension of S by a = f (g a ) f, g S, a / S is an ultra L-topology, sinceS(a1) is the discrete L-topology. similarly

      if S = S(a, U (c)), thenS(a ) is an ultra L-topology. if S = S(b, U (a)), thenS(b )

      if S = S(b, U (c)), thenS(b )

      if S = S(c, U (a)), thenS(c )

      if S = S(c, U (b)), thenS(c )

      Number of ultra L-topologies= 6 = 3 2 1 1 = n(n 1)m2, where n =

      3, k = m = 1

      { }

      { }

   2. Let X = a, b, c , L = [0, 1], then there is no ultra L-topology, since [0, 1] has no dual atom and atom.

   3. Let X = {a, b, c}, L =Diamond lattice {0, , , 1}

   1

   0

   Figure2:

   Here , are the atoms as well as the dual atoms. Let S = S(a, U (b)) =

   {f |f (a) = 0} {f |f b} , does not contain the L-points a, a , a1. Then the simple extension S(a) contains the fuzzy point a also. Let S1 = S(a). Then the simple extension S1(a ) contains all L-points and hence it is discrete. So S(a) = S(a, U (b), a) is an ultra L-toology. Sim- ilarly the simple extension S(a ) = S(a, U (b), a ) is an ultra L-topology. If S = S(a, U (b ) = {f |f (a) = 0} {f |f b } , Then the simple exten- sions S(a) and S(a ) are ultra L-topologies.That is corresponding to the

   elements a and b there are 4 ultra L-topologies. Similarly corresponding to the elements a and c , there are 4 ultra L-topologies. So there are 8 ultra L-topologies corresponding to a. Similaly there are 8 ultra L-topologies cor- responding to b and 8 ultra L- topologies corresponding to c . Hence total ultra L-topologies = 8 + 8 + 8 = 24 = 3 2 2 2 = n(n 1)m2, where n = 3, k = m = 2

   4.Let X = {a, b, c}, L = P (X) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}.1 =

   {a}, 2 = {b}, 3 = {c}, 1 = {a, b}, 2 = {a, c}, 3 = {b, c}. Atoms are

   1, 2, 3and dual atoms are 1, 2, 3

   Let S = S(a, U (b1)) = {f |f (a) = 0} {f |f b1}, does not con- tain the L-points a1, a2, a3, a1, a2, a3, a1. Let S1 = Simple extension of S by a1 denoted by S(a1).Then S1 contains more L-subsets than S,but

   1

   1 2 3

   1 2 3

   0

   Figure 3:

   not discrete L-topology. Let S2 = S1(a2) , Simple extension of S1 by a2. Then S2 contain more L subsets than S1 but not discrete L-topology. Let S3 = S2(a3) , Simple extension of S2 by a3, which is a discrete L-topology. Hence S2 = S1(a2) is an ultra L-topology, which is the L- topology generated by S(a1) and S(a2). Also L-topology generated by S(a1) and S(a3) and L-topology generated by S(a2) and S(a3) are ultra L-topologies. That is if S = S(a, U (b1)), there are 3 ultra L-topologies. Similarly if S = S(a, U (b2)), there are 3 ultra L-topologies and if S = S(a, U (b3)), there are 3 ultra L-topologies. So corresponding to the el- ements a, b there are 9 ultra L-topologies. Similarly corresponding to the elements a, c there are 9 ultra L-topologies. Hence there are 18 ultraL- topologies corresponding to the element a. Similarly corresponding to each element b and c there are 18 ultra L-topologies. So total number of ultra L-topologies = 54 = 3 2 3 3 = n(n 1)m2, n = 3, k = m = 3.

   5.Let X = {a, b, c, d}, L = P (X) = {, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d},

   {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}, {c, d, a}, X} . Let {a} = 1, {b} =

   2, {c} = 3, {d} = 4, {a, b} = 1, {a, c} = 2, {a, d} = 3, {b, c} = 4, {b, d} =

   5, {c, d} = 6, {a, b, c} = 1, {a, b, d} = 2, {b, c, d} = 3, {c, d, a} = 4 If

   S = S(a, U (b1)), there are 4 ultra L -topologies. If S = S(a, U (b2))

   If S = S(a, U (b3))

   Figure 4:

   If S = S(a, U (b4))

   | |

   | |

   So corresponding to the elements a, b,there are 16 ultra L-topologies. Simi- larly corresponding to the elements a, c,there are 16 ultra L-topologies and corresponding to the elements a, d, there are 16 ultraL-topologies. Hence there are 48 ultra L-topologies corresponding to the element a. Similarly corresponding to each elements b, c and d, there are 48 ultra L-topologies. So total number of ultra L-topologies = 48 4 = 192 = 4 3 4 4 = n(n 1)m2, n = 4, k = m = 4. In general if X = n and L is a finite pseudo complemented chain or a Boolean lattice, there are n(n 1)mk ultra

   L-topologies where m and k are the number of dual atoms and number of atoms respectively. If k = m, it is equal to n(n 1)m2.

   Remark 3.1. If L is neither a finite pseudo complemented Lattice nor a Boolean Lattice, we cannot identify the ultra L-topologies in this way.

   Example:

   Let X = {a, b, c}, L = D12 = {1, 2, 3, 4, 6, 12}

   1

   2

   1

   1 2

   0

   Figure 5:

   Here the atoms are 1 = 2, 2 = 3 and dual atoms are 1 = 4, 2 = 6 . If S = S(a, U (b1)) = {f |f (a) = 0} {f |f b1} ,L-topology generated by S(a1) and S(a2) does not contain the L-pointa2. It is not a discrete L-topology.So we cannot say that S(a1) is an ultra L-topology.

   | | | |

   | | | |

   Theorem 3.7. If X is infinite and L is a finite pseudo complemented chain or a Boolean lattice, there are X principal ultra L-topologies and X non principal ultra L-topologies.

   Illustration:

   If X is countably infinite, cardinality of X, |X| = 0. If X is uncountable,

   | |

   | |

   X > 0

   case 1: X is infinite and L is finite

   Let X = {a, b, …..}, L = {0, , , 1} a pseudo complemented chain.LetS = S(a, U (b)) = {f |f (a) = 0} {f |f b}. S does not contain the fuzzy points a, a , a1. Here S(a ) = S(a, U (b), a ) is a principal ultra L- topology since S(a1) is the discrete L-topology, where S(a ) is the simple

   | |

   | |

   | | | |

   | | | |

   extension of S by a . Similarly we can identify other L-topologies. Hence corresponding to the element a, there are X 1 = X principal ultra L- topologies. Similarly corresponding to each element b, c, d, …… there are X principal ultra Ltopologies So total number of principal ultra L-topologies

   | |

   | |

   = |X||X| = |X|. If S = S(a, U ) = {f |f (a) = 0} U , where U is a non- principal ultra filter not containing a, 0 /= L. Then the simple extension of S by a = S(a ) = S(a, U , a ) is a nonprincipal ultra L-topology since S(a1) is discrete L-topology. So there are X non principal ultra L topolo-

   gies.

   case 2: X and L are infinite

   LetX = {a, b, c, ….}, L = P (X). There are |X| atoms and |X| dual atoms.

   Number of Principal ultra L-topologies corresponding to one element= |X||X|(|X|

   1) = |X| Hence total number of principal ultra L-topologies= |X||X| = |X|. Let S = S(a, U ) = {f |f (a) = 0} U , where U is a nonprincipal ultra filter not containing a, 0 /= L . There are |X| nonprincipal ultra L-filters . Corresponding to a there are |X||X| = |X| nonprincipal ultra L-topologies. So total number of nonprincipal ultra L-topologies = |X||X| = |X|.

4. Topological Properties

   PRINCIPAL ULTRA L-TOPOLOGIES

   Let X be a nonempty set and L is a finite pseudo complemented chain. If S = S(a, U (b) = {f |f (a) = 0} {f |f y}, then a principal ultra L-topology = S(a, U (b), a ) = S(a ) , which is the simple extension of S

   by a = {f (g a ), f, g, S, a / S}, where a, b X, and are the

   atom and dual atom in L respectively.

   Let X be a nonempty set and L is a Boolean lattice. If S = S(a, U (b)) =

   {f |f (a) = 0} {f |f b} where a, b X, is an atom, then a Principal

   ultra L-topology = S(a, U (b), aj ) = L-topology generated by any(m 1),

   S(ai) among m, S(ai), i = 1, 2, …, m, j = 1, 2, …, m, i j if there are m

   dual atoms 1, 2, …m, where S(ai) = S(a, U (b), ai)

   Theorem 4.1. Let X be a nonempty set and L is a finite pseudo complemented chain or a Boolean Lattice. Then every principal ultra L- topology is L T0 but not L T1.

   Example:

   Let X is a non empty set

   Let L is a finite pseudo complemented chain and a, b X, , are atom and dual atom in L respectively. Take two distinct L points a1, b. b is an open L subset contain b but not a1. SinceU (b) = {f |f b}, any open set contain a1 must contain b. So S(a, U (b), a ) is L T0 but not L T1. Let L is a Boolean lattice and a, b X, is an atom and 1, 2, ….. are dual atoms in L. Take two distinct L-ponints a1, b.b is an open L subset contain b but not a1 . Since U (b) = {f |f b}, any open set contain a1

   must contain b. So S(a, U (b), aj ) is L T0 but not L T1.

   p/>

   Definition 4.1. An L-topological space (X, F ), F LX is called door

   L-space if every L-subset g of X is either L-open or L-closed in F .

   Example : Let X = {a, b} and L = {o, .5, 1}.Define f1(a) = 0, f1(b) = 0, f2(a) = 0, f2(b) = .5, f3(a) = 0, f3(b) = 1, f4(a) = .5, f4(b) = 0, f5(a) =

   .5, f5(b) = .5, f6(a) = .5, f6(b) = 1, f7(a) = 1, f7(b) = 0, f8(a) = 1, f8(b) =

   .5, f9(a) = 1, f9(b) = 1 . Let F = {f1, f9, f2, f3, f4, f5, f6}. Thenf7 and f8 are closed L-subsets. So (X, F ) is a door L-space. Let X = {a, b, c}, L = [0, 1] and the the L-topology F = {0, µ{a}, µ{b,c}, 1}.Then (X, F ) is not a door L- space sinceµ{b} is neither an L open set nor an L-closed set. In a principal ultra L-topology S(a, U (b), aj ) every L subset of X is either open or closed

   if L is a finite pseudocomplemented chain or a Boolean lattice . So the principal ultra L-topological space is a door L space.

   Definition 4.2. An L-topological space (X, F ) is said to be regular at an L-point a if for every closed L subset h of X not containing a, there exists disjoint open sets f, g such that a f and h g.(X, F ) is said to be regular L-topology if it is regular at each of its L-points.

   Theorem 4.2. Let X be a non empty set and L = P (X). Then the principal ultra L-topology S(a, U (b), aj ) is not regular if |X| 3.

   Example: let X = {a, b, c}, L = P (X) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}.1 =

   {a}, 2 = {b}, 3 = {c}, 1 = {a, b}, 2 = {a, c}, 3 = {b, c}. Atoms are

   1, 2, 3 and dual atoms are 1, 2, 3 . Take = 1 in the principal ultra L-topology S(a, U (b), a3). Consider a1. a3 is a closed L subset not con- taining a1.Consider the open sets f, g such that f (a) = 1, f (b) = 1, f (c) = 1, g(a) = 2, g(b) = 0, g(c) = 0.f is an open set containing a1 andg is an open set containing a3 but f g /= 0. That is f and g are not disjoint.

   Definition 4.3. An L topological space (X, F ) is said to be Normal if for every two disjoint closed L subsets h and k , there exists two disjoint open L subsets f, g such that h f and k g.

   Theorem 4.3. Let X be a nonempty set and L = P (X). Then the principal ultra L-topology S(a, U (b), aj ) is not a normal L-topology if

   |X| 3.

   Example: Let X = {a, b, c}, L = P (X) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}.1 =

   {a}, 2 = {b}, 3 = {c}, 1 = {a, b}, 2 = {a, c}, 3 = {b, c}. Atoms are

   1, 2, 3 and dual atoms are 1, 2, 3.Take = 1 in the principal ultra L-topology S(a, U (b), a3).a2 and a3 are disjoint closed L subsets. There are no disjoint open L subsets containing a2 and a3.

   NON PRINCIPAL ULTRA L- TOPOLOGY

   Let X be an infinite set and L is a finite pseudocomplemented chain. If

   S = S(a, U ) = {f |f (a) = 0} U where U is a non principal ultra L-filter

   not containing a, 0 L .Then the non principal ultra L-topology=

   S(a, U , a ) = S(a ),is the simple extension of S by a = {f (g a ), f, g,

   S, a / S},where a, b X, is the dual atom in L.

   Let X be an infinite set and L is a Boolean lattice. If S = S(a, U ),

   Then a non principal ultra L-topology S(a, U , aj ) = L-topology generated by any(m 1), S(ai) among m, S(ai), i = 1, 2, …, m, j = 1, 2, …, m, i /= j if there are m dual atoms 1, 2, …m where S(ai) = S(a, U , ai). Here m can be assumed infinite values.

   Theorem 4.4. Every non principal ultra L-topology S(a, U , aj ) is

   L T1.

   Proof. Let a, b be any two distinct L-points. Since U is a nonprincipal ultra L-filter,there exists L open sets containing each L-points but not the other.

   Theorem 4.5. Every non principal ultra L topology S(a, U , aj ) is

   L T2.

   Proof. Let a, b be any two distinct L-points. Since U is a nonprincipal ultra L-filter, we can find disjoint open sets f and g such that a f, b / f and b g, a / g

   Theorem 4.6. Every non principal ultra L-topology is a door L-space. Proof. In S(a, U , a ) every L subset of X is either L closed or L open,

   if L is a finite pseudo complemented chain . If L is a Boolean Lattice,in

   S(a, U , aj ) every L subset is either L cosed or L open

   Theorem 4.7. If X is an infinite set and L is a finite pseudo comple- mented chain or a diamond lattice, then the non principal ultra L-topology S(a, U , a ) is a regular L-topology.

   Proof. It is trival.

   Theorem 4.8. Let X is an infinite set and L = P (X).Then the non principal ultra L-topology S(a, U , aj ) is not a regular L-topology.

   Proof. Let X = {a, b, c….}, L = P (X). Let 1, 2, …. are atoms and 1, 2, … are dual atoms in L. Consider a1. Then there exists a closed L subset ai for some i not containing a1. But we cannot find disjoint open L subsets f and g such that f contains a1 and g contains ai.

   Theorem 4.9. If X is infinite set and L is a finite pseudo complemented chain or a diamond lattice, the non principal ultra L topology S(a, U , a ) is a normal L topology.

   Proof. It is trivial

   Theorem 4.10. If X is an infinite set and L = P (X), then the non principal ultra L topology S(a, U , aj ) is not a normal L-topology.

   Proof. Let X = {a, b, c, ….}, L = P (X). Let 1, 2, … are atoms and 1, 2, ….. are dual atoms in L. Then there exists two closed L subsets ai and aj for some i and j . But there doesnot exists disjoint open L subsets f and g such that f contains ai and g contains aj .

   Theorem 4.11. Let X is an infinite set and L is a finite pseudo com- plemented chain or a Boolean lattice. An ultra L-topology F is a T1 L topology if and only if it is a non principal ultra L-topology.

   Proof. Suppose that the ultra L-topology F is a T1 L topology. We have to show that F is a non principal ultra L-topology. F is a principal

   ultra L-topology implies F is not a T1 L topology. So we can say that F

   is a T1 L topology implies F is a non principal ultra L-topology.

   Next assume that F is a non principal ultra L-topology. Then by theorem

   4.4 F is a T1 L topology.

   Theorem 4.12. An L-topology F on X is a T1 L topology if and only if it is the infimum of non principal ultra L-toologies.

   Proof. Necessary part

   Any L-topology finer than a T1 L topology must also be a T1 L topology. So a T1 L topology can be the infimum of only non principal ultra L topologies.

   Sufficient part

   Each non principal ultra L-topology on X contains non principal ultra L- filter. So there exists distinct L points a, b where a, b X; , Land L opensets f, g such that a f, b / f and a /, b g. This is also true in the infimum of any family of non principal ultra L-topologies since every L points are closed in non principal ultra L-filters. So infimum of any family of non principal ultra L-topologies is a T1 L topology.

   Theorem 4.13. Let X is an infinite set and L is a finite pseudo com- plemented chain or a Boolean lattice . Then an ultra L-topology is a T2 L topology if and only if it is a non principal ultra L-topology.

   Proof. Suppose that an ultra L-topology is a T2L topology. This implies that the ultra L-topology is a T1 L topology. Hence it is a non principal ultra L-topology.

   Conversely suppose that the ultra L-topology is a non principal ultra L- topology. Since a non principal ultra L-topology contains a non principal ultra L-filter, for any two distinct L points in the non principal ultra L- topology there exists disjoint L open sets contains each L point but not the other. So it is a T2 Ltopology.

5. Mixed L- topologies

   In[8] Steiner studied the mixed topologies. Analogously we ca say that a mixed L-topology on X is not a T1 L topology and does not have a principal representation. Thus a mixed L-topology is the intersectionof a T1 L topology and a principal L-topology.

   The representation of a mixed L-topology as the infimum of a T1 L

   topology and a principal L topology need not be unique.

   Example:

   Let C = {µA|X A is finite} 0 , is a T1 L topology and and / be the principal L-topologies given by = aX{b,c}S(a, U (b), aj ) ,

   / = aX{b}S(a, U (b), aj ) , is an atom and j s are dual atoms in L.

   C = {µA|b A, X A is finite or f = 0} = C / is a mixed L-topology.

   Here c andc / /. That is the representation of a mixed topology as

   the infimum of T1 L topology and principal L-topology need not be unique.

6. Conclusion

   In this paper we identified the principal and non principal ultra L-topologies and studied some topological properties. Also we introduced mixed L-topologies.

7. Future scope

   Study some properties of mixed L-topologies.

8. Aknowledgement

   We would like to thank Dr. T. Thrivikraman, Former Head, Dept. of Math- ematics, Cochin University of Science and Technology, Cochin, Kerala State, India for discssions and suggestions. The first author wishes to thank the University Grant Commission, India for giving financial support.

   References

   1. S. Babu Sundar: Some Lattice Problems in Fuzzy Set Theory and Fuzzy Topology, Thesis for Ph.D.Degree, Cochin University of Science and Technology,1989.

   2. C. L. Chang: Fuzzy topological spaces, J. Math. Anal. Appl.24(1968)191-201.

   3. O. Frohlich: Das Halbordnungssystem der topologischenRaume auf einer Menge, Math. Annnalen 156(1964), 79-95

   4. J. A. Goguen: L-fuzzy sets, J. Math.Anal.Appl.18(19670 145-174

   5. Johnson T.P.: On Lattices Of L-topologies, Indian Journal Of Mathe- matics, Vol.46 No.1 (2004) 21-26.

   6. Katsaras A. K.: On fuzzy proximity spaces, J. Math Ana.and Appl. 75(1980) 571-583.

   7. Srivastava and Gupta R. L.: Fuzzy proximity structure and fuzzy ultra filters, J. Math.Ana. and appl.94(1983) 297-311.

   8. A. K. Steiner: Ultra spaces and the lattice of topologies.Technical Re- port No. 84 June 1965.

   9. Sunil C. Mathew: A study on covers in the lattices of Fuzzy topolo- gies,Thisis for Ph.D Degree. M. G. University 2002

(On FIP)

Department of Mathematics,

Cochin University of Science and Technology, Cochin-27, India.