On fuzzy topological spaces

Fuzzy Sets and Systems 19 (1986) 193-197 North-Holland

193

SHORT COMMUNICATION

ON FUZZY

TOPOLOGICAL

SPACES

Sudarsan N A N D A Department of Mathematics, Indian Institute of Technology, Kharagpur, India

Received December 1983 Revised January 1985 In this note the concept of almost fuzzy continuous mapping from a fuzzy topological space into another is introduced and discussed. Keywords: Fuzzy sets, Fuzzy topological spaces, Almost continuous mapping.

1. In~oductlon The concept of fuzzy sets and fuzzy set operations was first introduced by Zadeh in his classic paper [10]. Subsequently, several authors including Zadeh [10] have discussed various aspects of the theory and applications of fuzzy sets. Chang [2], Wong [7, 8, 9], Goguen [4] and others applied some basic concepts from general topology to fuzzy sets and developed a theory of fuzzy topological spaces. Singal and Singal [6] introduced the notion of almost continuous mappings in topological spaces. The purpose of this paper is to introduce and study almost fuzzy continuous mappings from a fuzzy topological space into another and this will fill up a gap in the existing literature on fuzzy topological spaces.

2. Preliminaries Let X be any set and let L be a lattice. In particular L could be [0, 1]. A fuzzy set A in X is a mapping tzA : X ~ L which assigns to each element x e X a degree of membership, ~A(x)~ L. Thus an ordinary set is a special type of a fuzzy set in which the range of the mapping is the lattice {0, 1}. W e first quote some definitions which will be needed in the sequel. Let A and B be any two fuzzy sets 0165-0114/86/$3.50 ©[ 1986, Elsevier Science Publishers B.V. (North-Holland)

194

S. Nanda

in X. Then we define A =B

iff

gA(x) = t~B(x) for all x ~X,

A ~B

iff

/~A(x) ~< tLs(x) for all x ~ X,

C=AOB D=ANB E = A'

gc(X) = max[irA(x), t~s(x)] for all x ~ X, iff No(x) = min[t~A(x), t~s(x)] for all x ~ X,

iff iff

gE(x) = 1-t~A(x) for all x ~X,

More generally, if L is a complete lattice, for a family of fuzzy sets A = {A~ : i ~/},

C=U

A~

iff

tXc(X)=sup{tza,(x)} for all x ~ X

D = 0 Ai

iff

tLD(X) = inf{tt~(x)} for all x ~ X

and

The symbol O is used to denote the empty fuzzy set (t~0(x)= 0 for all x ~ X) and for X we have the definition ~Zx(X)= 1 for all x ~ X. We shall throughout assume that L is a complete lattice. A fuzzy topology is a family T of fuzzy sets in X which satisfies the following conditions: (j) ~J,X~T, (ii) A, B ~ T implies that A f3 B ~ T, (iii) Ai ~ T for each i ~ I implies that I J~Ai ~ T. T is called a fuzzy topology for X and the pair (X, T) is a fuzzy topological space (fts in short). Ever~ m e m b e r of T is called a T-open fuzzy set. A fuzzy set is T-closed if its complement is T-open. The indiscrete fuzzy topology contains only and X while the discrete fuzzy topology contains all fuzzy sets in X. Let A be a fuzzy set in a fts(X, T). The largest T - o p e n fuzzy set contained in A is called the interior of A and is denoted A °. The smallest T-closed fuzzy set containing A is called the closure of A and is denoted by A. Let f be a mapping from a set X into a set Y. Let B be a fuzzy set in Y with membership function t~B. The inverse of B, written as f-l[B], is a fuzzy set in X whose membership function is defined by ~:-,[~,](x) = ~ B

([(x))

for all x ~ X. Let A be a fuzzy set in X with membership function t~A. The image of A, written as f[A], is a fuzzy set in Y whose membership function is given by sup t~A ( z ) i f txnA)(y) = ` [ O~,_q,j

/ - l [ y ] # O, otherwise.

A function f from a fts (X, T) into fts (Y, U) is said to be F-continuous (fuzzy continuous) if the inverse of every U - o p e n fuzzy set is T-open. f is said to be F-open (F-closed) if it maps a T - o p e n (T-closed) fuzzy set onto an U - o p e n (U-closed) fuzzy set. We now introduce our definitions.

On fuzzy topologicalspaces

195

Definition L A fuzzy set A in a fts (X, T) is called regularly T-open if it is the interior of its own closure, that is, A = (A)° or equivalently, if it is the interior of some T-closed fuzzy set. Definition 2. A fuzzy set A in (X, T) is called regularly T-closed if it is the closure of its own interior, that is, A = (A°), or equivalently, if it is the closure of some T-open fuzzy set. Definition 3. A function f : (X, T)--~ (Y, U) is said to be almost F-continuous if the inverse of every regularly U-open fuzzy set is T-open. Dethaition 4. f:(X, T)---~ (Y, U) is called almost F-open if the image of every regularly T-open fuzzy set is U-open. Definition 5. f:(X, T)--* (Y, U) is called almost F-closed if the image of every regularly T-closed fuzzy set is U-closed. Definition 6. f : (X, T)--* (Y, U) is said to be almost F-quasi-compact if it is onto and if A is U-open whenever f - l ( A ) is regularly T-open.

3. T h e results T h e o r e m 1. f :(X, T)---~(Y, U) is almost F-continuous if and only if the inverse of

every regularly U-closed fuzzy set is T-closed. Proof. Let A be any regularly U-closed fuzzy set. Then A ' is regularly U-open and therefore f-l[A'] is T-open. But f-l[A'] = (f-l[A])'. Thus (f-~[A ])' is T-open and hence f-~[A] is T-closed. Conversely, let A be any regularly U-open fuzzy set. Therefore A ' is regularly U-closed. Now f-l[A'] is T-closed. Thus (f-l[A])' is T-closed and hence f-~[A] is T-open. Thus f is almost F-continuous and this completes the proof. T h e o r e m 2. Let (X, T), (Y, U) and (Z, V) be fuzzy topological spaces. Let f be an

F-open and F-continuous map of (X, T) onto (Y, U) and let g be a map of (Y, U) into (Z, V). Then g of is almost F-continuous if and only if g is almost Fcontinuous. Proof. Let g of be almost F-continuous. Let A be any regularly V-open fuzzy subset of Z. Therefore (g o f ) - l ( A ) is T-open and so f - l ( g - l ( A ) ) is T-open. Since f is F-open we have f ( f - l ( g - l ( A ) ) ) is U-open. Thus g - l ( A ) is U-open and hence g is almost F-continuous. Conversely, let g be almost F-continuous. Let A be any regularly V-open fuzzy set. Then g - l ( A ) is U-open. Since f is F-continuous, f - l ( g - l ( A ) ) is T-open. Thus (g o f ) - l ( A ) is T-open and hence g o f is almost F-continuous.

S. Nanda

196

Theorem 3. Let f be a 1-1 mapping of a fls (X, T) onto a f t s (Y, U). Then the

following conditions are equivalent: (i) f is almost F-open, (if) f is almost F-closed, (iii) f is almost F-quasi-compact, (iv) f-1 is almost F-continuous. Proof. Assume that (i) holds. Let A be a regularly T-closed fuzzy set. Then A ' is regularly T-open and hence f ( A ' ) = (f(A))' is U-open so that f ( A ) is U-closed and therefore (if) holds. Now assume that (if) holds and let f - I ( A ) be a regularly T-closed fuzzy set. This implies that f(f-~(A)) is U-closed, that is, A is U-closed and hence (iii) holds. Let (iii) hold and let A be a regularly T-open fuzzy set. Then f-x(f(A) is regularly T-open. This implies that f ( A ) is U-open, that is, (f-~)-~(A) is U-open and therefore (iv) holds. If (iv) holds and if A is a regularly T-open fuzzy set, then (/-x)-~(A) is U-open and hence (i) holds. This completes the proof. Theorem 4. Let f be a map from (X, T) onto (Y, U) and g be a map from (Y, U)

onto (Z, V). If f is almost F-continuous and if g ° f is F-open (F-closed), then g is almost F-open (almost F-closed). Proof. Suppose that f is almost F-continuous and g of is F-open (F-closed). Let A be any regularly U-open fuzzy subset. Then f - l ( A ) is a T-open (T-dosed) fuzzy set. Since (g o f) is F-open (F-closed), (g o f)(f-~(A)) is V-open (V-closed) fuzzy set. But (g o f ) ( f - l ( A ) ) = g(A). Thus g(A) is V-open (V-closed). Therefore g is almost F-open (almost F-closed) and this completes the proof of the theorem.

4. Remark

It may be noted that fuzzy topologies can also be introduced from a definition like the Kuratowsky closure and in particular, from fuzzy neighbourhood systems (see [1, 4]). For instance, a closure map is a function a:LX---*L x with the following axioms: a ( ¢ ) = ¢,

a(A)=A,

a(X) = x,

a(AUB)=a(A)Ua(B),

a2(A)=a(A).

From the closure we define the interior map i : L × ~ L x as

i(A) = (a(AC)) c. A fuzzy set A is closed iff a ( A ) = A and is open iiI i(A)= A. If a and b are closure maps, a map f : ( X a ) ~ (Yb) is continuous iff A c X implies f ( a ( A ) ) c b(f(A)) (a definition which is correct even for pretopologies). For fuzzy topologies it is equivalent to say that the inverse image of every closed fuzzy set is closed.

On fuzzy topological spaces

197

Almost continuity is then obtained by considering a subfamily of the family of all closed sets of Y namely, the family of all regularly closed sets (see Theorem 1). If we respectively denote R °, R~ the family of the regularly open, regularly closed fuzzy sets of (Xa), then we have the following properties:

X, OeR°,R~,, a~R~ 4=~ ACeR °, A, B e R ~ ~ AUBeR~,, A, B e R ~ F/~ A N B e R ~ .

5. C o n d u ~ o n

The concept of almost continuous mapping of general topology is intimately connected with almost compactness in the same manner as continuity is related to compactness. In this paper we have introduced the concept of almost fuzzy continuous mapping and this fills up a gap in the existing literature on fuzzy topological spaces. Since fuzzy continuity and fuzzy compactness are also related just as in general topology it is hoped that the concept of almost fuzzy continuity will give rise to some related concept of almost fuzzy compactness. Finally the author wishes to thank the referees for several valuable suggestions which improved the presentation of the paper; particularly, for bringing the works of Badard [1] and Lowen [5] to his notice and for suggesting the remark of Section 4.

References [ 1] R. Badard, Fuzzy pretopological spaces and their representation, J. Math. Anal. Appl. 81 (1981) 378-390. [2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [3] F. Conrad, Fuzzy topological concepts, J. Math. Anal. Appl. 74 (1980) 432--440. [4] J.A. Goguen, The fuzzy Tychonoff theorem, J. Math. Anal. Appl. 43 (1973) 734-742. [5] R. Lowen, Fuzzy neighbourhood spaces, Fuzzy Sets and Systems 7 (1982) 165-169. [6] M.K. Singal and A.R. Singal, Almost continuous mapping, Yokohama Math. J. 16 (2) (1968) 63-73. [7] C.K. Wong, Covering properties of fuzzy topological spaces, J. Math. Anal. Appl. 43 (1973). [8] C.K. Wong, Fuzzy topology, Product and quotient theorems, J. Math. Anal. Appl. 45 (1974) 512-521. [9] C.K. Wong, Fuzzy points and local properties of fuzzy topology, J. Math. Anal Appl. 46 (1974) 316-328. [10] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.