E L S E V I E R Fuzzy Sets and Systems 107 (1999) 227~30

FUZ2Y sets and systems

www.elsevier.com/locate/fss

Fuzzy totally continuous and totally semi-continuous functions Anjan Mukherj ee*

Department of Mathematics, Government Degree College, Khowai, West Tripura, India

Received January 1997; received in revised form October 1997

Abstract

The aim of this paper is to introduce two new classes of functions, called fuzzy totally continuous and fuzzy totally semi- continuous functions. Their characterizations, examples, composition of these functions, their relationships with other fuzzy functions and preservation of some fuzzy spaces under these functions are studied. ~) 1999 Elsevier Science B.V. All rights reserved.

Keywords: Fuzzy topology; Fuzzy clopen; Fuzzy semi-clopen; Fuzzy totally continuous and fuzzy totally semi-continuous functions

AMS classification." 54A40

1. Introduction and preliminaries

Semi-open (closed) sets play an important role in general topology. Azad [2] generalized these notions to fuzzy topological spaces (fts). A fuzzy subset in a fts X is said to be fuzzy semi-clopen [8], if and only if it is both fuzzy semi-open and fuzzy semi-closed. This set plays an important tool in the study of fuzzy s-closed [8] spaces. In this paper, we introduce two new classes of functions called fuzzy totally continuous and fuzzy totally semi-continuous functions. In Section 2, some ex- amples and relationships between these new classes and other classes of fuzzy functions are obtained. In Section 3, the preservation of some fuzzy topo- logical structures are examined under these func- tions. Several properties of these functions are also established.

*Correspondence address: B-31, O.N.G.C colony, Agartala 799014 (W) Yripura, India.

We would like to mention the following definitions and results:

Definition 1.1. Let 2 be a fuzzy subset of a fls(X, F), then

(i) 2 is called fuzzy semi-clopen [8] (resp. fuzzy semi-open [2]) if and only if there exists a fuzzy regular open (resp. fuzzy open) subset t/ of X such that r/c_ 2 c_ cl(r/).

(ii) 2 is called fuzzy semi-closed [2] if int(C12) c_2.

(iii) 2 is called fuzzy regular open [2] if 2 = int (C12).

Theorem 1.2 (Allam and Zahran [1, Theorem 4.4]). 2 is fuzzy regular semi-open if and only i f2 is fuzzy semi-open and 2 is fuzzy semi-closed subset of X.

Theorem 1.3 (Sinha and Malakar [8, Theorem 3.3(b)]). A fuzzy topological space (X,F) is called fuzzy s-closed if and only if every fuzzy cover of X by fuzzy semi-clopen subsets has a finite sub cover.

0165-0114/99/$ see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(97)00320-5

228 A. Mukherjee / Fuzzy Sets and Systems 107 (1999) 227-230

Definition 1.4. A fuzzy topological space (X,F) is called fuzzy almost compact [4] if and only if every fuzzy open cover of X has a finite subcover whose closures covers X.

Definition 2.3. A function f : X ~ Y from a fts(X, Fl ) to another fls(Y, F2) is said to be fuzzy totally semi- continuous if the inverse image of every fuzzy open subset of Y is fuzzy semi-clopen subset of X.

Definition 1.5. Let f : (X, FI ) -~ ( Y, Y2 ) be a function from a f t s (X,F 1 ) to another fts(Y, F2), then:

(i) f is called fuzzy semi-continuous [2] if the in- verse image of every fuzzy open subset of Y is fuzzy semi-open subset of X.

(ii) f is called fuzzy completely continuous [3, 6] if the inverse image of every fuzzy open subset of Y is fuzzy regular open subset of X.

(iii) f is called fuzzy semi-irresolute [5], if the in- verse image of every fuzzy semi-clopen subset of Y is fuzzy semi-clopen subset of X.

It is evident that every fuzzy totally continuous function is fuzzy totally semi-continuous, but the con- verse need not be true as can be seen from the follow- ing example:

Example 2.4. Consider the Example 4.5 of [2], the fuzzy subsets kq and #2 of a closed unit interval I are defined as follows:

0, O~<x~< 1,

#l(X)= 2 x - l , ½~<x~<l

2. Fuzzy totally continuous and fuzzy totally semi-continuous functions

and

In this section, the two new classes of function are introduced. Their characterization examples, compo- sitions of these functions and their relationships with other fuzzy functions are established.

1, O~x<¼,

#2(x)= - 4 x + 2 , ¼~<x~<½,

0, 1 ~<x~<l.

Definition 2.1. A function f :X ~ Y from a fls(X, F1 ) to another Its(Y, F2) is said to be fuzzy totally contin- uous if the inverse image of every fuzzy open subset of Y is a fuzzy clopen subset of X.

Example 2.2. Let 21 and 2 2 be two fuzzy subsets of I = [0, 1] defined as follows: For eachx EL 21(x) = x and 22(x) = 1 - x.

We consider the fuzzy topology F = {0,21,22, 21 [.-J 22, 21 A 22, 1 } and the mapping f : (L F ) ~ (L F) defined by f ( x ) = x for each x E I. It is clear that the fuzzy subsets 21,22, 21 U 22, 21 V) 22, are both fuzzy open and fuzzy closed and hence fuzzy clopen in (LF). Since the inverse image of every fuzzy open subset is fuzzy clopen in (I,F), f is fuzzy totally continuous.

It is seen that fuzzy totally continuous function im- plies fuzzy completely continuous function. In [3], it is also seen that fuzzy completely continuous function implies fuzzy continuity, but the converse is not true [3, Example 2].

Clearly, F1 = {0 ,# l ,~2 , / t lUk t2 ,1 ) and F2 = {0, btl,1) are two fuzzy topologies on the set I. Let f : (LF1) -+(LF2) be a function defined by f ( x ) = min(2x, 1 ) for each x E I. Here f - l (0) = 0, f - l ( 1 ) = l , f - l ( / t l ) = # ~ ; also Cl#1=#~. Thus, /z~ is fuzzy semi-open in (LF1) and it is also fuzzy semi-closed in (LF1). Hence, f is fuzzy to- tally semi-continuous. Now, #I ~ FI; thus, /4 is not fuzzy clopen in (L F1 ). Hence, f is not fuzzy totally continuous.

Every fuzzy totally semi-continuous function is fuzzy semi-continuous, but the converse is not true which can be seen from the following example:

Example 2.5. Considering Example 2.4 and taking F2={0,kq,//1,1}, we observe that the function g : ( L F 1 ) ~ ( L F 2 ) defined by g(x)=min(2x, 1) is fuzzy semi-continuous. Here g- 1 (0) = 0, g-1 (1) = 1, g-l(/tl)----/l~, g-l(# ' l )=/t2. Here 0-1(/2'1)----#2 EEl which is not fuzzy semi-clopen in (LF1) shows that f is not fuzzy totally semi-continuous.

A. Mukherjee/Fuzzy Sets and Systems 107 (1999) 227 230 229

Now, we have the following implications:

Fuzzy totally ~ Fuzzy completely =~ Fuzzy continuity continuity continuity

Fuzzy totally ~ Fuzzy semi-continuity semi-continuity

A fts(X,F) is said to be fuzzy extremally discon- nected (FED) if and only if the closures of every fuzzy open subset is fuzzy open in X. By using Lemma 3.3 of [8] if (X,F) is a FED space then Sc1(2)= c1(2) for every fuzzy semi-open 2 in X. We have the following obvious theorem:

Definition 2.10. A function f : (X,F) ~ (Y, FI ) from fts(X, F ) to another fts( Y, F1 ) is said to be fuzzy almost semi-open if the image of every fuzzy semi-clopen subset is fuzzy open.

Theorem 2.11. I f f : (X,F) ~ (Y, F1 ) is a onto fuzzy almost semi-open and fuzzy totally semi-continuous function and g : (Y, F1 ) ~ (Z, F2 ) is a function such that g o f is fuzzy totally semi-continuous, then g is fuzzy continuous.

Proof. Let 2 be fuzzy open subset of Z. Then (g o f ) - 1 ( 2 ) is a fuzzy semi-clopen subset of X. But f ( g o f ) - l ( 2 ) = g - l ( 2 ) is a fuzzy open subset of Y. Hence, g is fuzzy continuous. []

Theorem2.6. Let (X,F) be a FED space. I f f : (X ,F)-+(Y, F1) is a fuzzy continuous (resp. fuzzy semi-continuous)function then it is also a fuzz), totally continuous (resp. fuzzy totally semi- continuous) function.

Proof. Straightforward. []

Example 2.7. Consider Example 2.4; FI = {0, #l, #2, plU]22,1} and F = {0, ]21, #'1,1} are two fuzzy topologies on 1. Here (LF) is a FED space. Let f : ( L F ) - - + ( L F I ) be a function defined by f ( x ) = x / 2 . Then f - l ( 0 ) = 0 , f - I ( 1 ) = 1, f - l ( # l ) = 0, f 1(]22) = ]2t 1 ~---- f- l ( ]2t O]22). Thus, f is fuzzy continuous. Here #'l is both fuzzy open and fuzzy closed on (/ ,F); thus f is also fuzzy totally continuous.

Next we prove some results on the composition of functions:

Theorem 2.8. I f f : (X, F) --* (Y, F1 ) is fuzzy totally semi-continuous and g : (Y, FI )--+ (Z, F2) is fuzzy continuous, then g o f :(X,F)--+(Z, F2) is a fuzzy totally semi-continuous function.

Proof. Obvious. []

Theorem 2.9. I f f : (X,F) --+ (Y, F1 ) is fuzzy semi- irresolute and o:(Y, F1)--*(Z, F2) is fuzzy totally semi-continuous function, then g o f : (X, F) --+ ( Z, F2 ) is fitzzy totally semi-continuous.

Proof. Straightforward. []

3. Preservation of some fuzzy spaces under fuzzy totally semi-continuous functions

It was proved in Lemma 3.9 of [9] that if 2 is fuzzy semi-clopen and A is a fuzzy open crisp subset of a fts(X,F), then 2NA is fuzzy semi-clopen in (X,F). We have the following theorem:

Theorem 3.1. I f f : X ~ Y is fuzzy totally semi- continuous function and A is fuzzy open crisp subset o f X, then fA :A ~ Y is also fuzzy totally semi- continuous.

Proof. Let 2 be a fuzzy open subset of Y; then f - 1(2) is fuzzy semi-clopen in X. Now, by Lemma 3.9 of [9], f l(2)AA = fA-l(,~) is fuzzy semi-clopen in X. Hence, the theorem. []

Theorem 3.2. I f f :X ---* Y is a fuzzy totally semi- continuous onto function and X is fuzzy s-closed, then Y is fuzzy compact.

Proof. Let {2a: a E A}; A being the index set be a fuzzy open cover of Y. Then {f - l (2a) : a E A} is a fuzzy semi-clopen cover of X. By fuzzy s-closedness of X, there exists a finite subfamily of { f - l ()~a)} such that

- - 1 ~ f ( a , ) = l , . i l

230 A. Mukherjee/Fuzzy Sets and Systems 107 (1999) 227-230

Now

which implies that Y is fuzzy compact. []

Theorem 3.3. I f f : X ~ Y is a fu z zy semi-con- tinuous onto function and X is f u z zy s-closed, then Y is fuzzy almost compact.

Proof. Let {t/a: a E A} be a fuzzy open cover o f Y; then { f - l ( t / a ) : a C A} is a fuzzy semi-open cover o f X . By fuzzy s-closedness o f X , there is a finite sub-

n family o f f - l ( t / a ) such that [-Ji=l S c l f - l ( t / a ~ ) = lx . Now

ly = f ( l x ) = f S c l f - l ( r / a i )

i = ,

References

[1] A.A. Allam, A.A. Zahran, On fuzzy f-continuity and ~-near compactness in fuzzy topological spaces, Fuzzy Sets and Systems 50 (1992) 103-112.

[2] K.K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82 (1981) 14-32.

[3] R.N. Bhaumik, A. Mukherjee, Fuzzy completely continuous mappings, Fuzzy Sets and Systems 56 (1993) 243-246.

[4] A. Di Concilio, G. Geeta, Almost compactness in fuzzy topological spaces, Fuzzy Sets and Systems 13 (1984) 187-192.

[5] S. Malakar, On fuzzy-irresolute and strongly irresolute functions, Fuzzy Sets and Systems 45 (1992) 239-244.

[6] M.N. Mukherjee, B. Ghosh, Some stronger forms of fuzzy continuous mappings on fuzzy topological spaces, Fuzzy Sets and Systems 38 (1990) 375-387.

[7] T.M. Nour, Totally semi-continuous functions, Indian J. Pure Appl. Math. 26 (7) (1995) 675-678.

[8] S.P. Sinha, S. Malakar, On s-closed fuzzy topological spaces, J. Fuzzy Math. 2 (1) (1994) 95-103.

[9] A.M. Zahron, Fuzzy regular semi-open sets and as-closed spaces, J. Fuzzy Math. 2 (3) (1994) 579-585.

which implies Y is fuzzy almost compact. []