On some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topological spaces

Fuzzy Sets and Systems 32 (1989) ;03-114 North-Holland

103

ON SOME W E A K E ~ FOPd~S OF FUZZY CONTINUOUS AND F U Z Z Y O P E N MAPPINGS ON F U Z Z Y T O P O L O G I C A L SPACES MN. MUKH~RJEE Deparimen¢ of P~-~reMathematics, Unive~i~, of Calcuuo, 35, Bollygunge Circular l~o~d, Calcutta 700019, India

S.P. SINHA Department of Mathematics, Jogesh Cha~dra Choa,dhury College, 30, Prince Ann,at Shah Road, Calcutta 700 033, India Received July 1987 kevised December 1987

Abstract: The purpose of the present paper is to study some weaker forms of fuzzy continuous and fuzzy open maps between fuz¢~ topologic~i spaces, along with the introduction and investigations of a new class of non-fuzzy t~r~tinuous maps. This latter type of maDpings i.~ s~.,~n to be independent of the other non.continuous functions discussed here ~fi ihe~ : fJl~ ~ioLq ~g characterized and certain properties concerning them ~re prc~ve~ F~,dliy, a compa~dtive study as to their mutual dependence and interr,2a~Jtm eL =~c aifferent conditions is taken up. • . : .~l,~st fuzzy open mapping; fuzzy almost continuous mapping; fuzzy weakly Keywor~s ~ continuous mapping; fuzz' semicontinuous mapping; fuzzy re~,hr ripen set; fuzzy regular closed set; fuzzy semi-open set; fuzzy semi-closed set.

I. Introduction ~ d pre|bnlnaries Since the introduction of fuzzy sets by Zadeh in his classic paper [12] of 1965 and fuzzy tol~'"~gical spaces by Chang i2] in 1968, certain mappings between topological sl;aces, weaker than usual open or continuous ones, have been generalized to and studied in fuzzy topological spaces by different authors. Functions between fuzzy topological spaces under the terminology 'almost fuzzy open' have been introduced by Nanda [9] and also by Ganguly and Saha [3]. We sh~,ll cd|| them a.f.o.N, and a.f.o.G, functions respe,.~tively. These functions were further studied in [8], in which we proved the independence of these definitions, and different characterizations along with other properties of a.f.o.N, functions were also established. However, each of the papers [3] and [8] lacks characterizing theorems for a.f.o.G, functions. We furnish the sought-for theorem in Section 2 of this paper. We also find conditions under which these two types of almost fuzzy open functions can be correlated and this is embodied in the same section. Almost continuous and weakly continuous functi0ns between general topological spaces were introduced and studied respectively by Singal and Singal [10] and Levine [5]. Azad [1] defined the generalized forms of these functions for fuzzy 0165-0114/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

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M.N. Mukherje~~, 5.P. S!n~e

topological spaces by taking some equivalent formulations of the definitions of these functions, but did not investigate whether the extended versions of these equivalences in general topology can be achieved as parallel results in fuzzy setting. In Section 3 of the paper we establish these equivalences along with other results concerning these functions in fuzzy topological spaces. Much earlier than Singal and Singal [10], the definition of another class of functions under t~e same terminology 'almost continuous functions' between topologicR! spaces was given by Husain [4]. As the theory of ~ topological speces contains the theory of topological spaces, it is worth studying aimosl continuity in fuzzy setting along Husain's lic,6 ahd this is done in Section 3. We shall call this new type of functions fuzzy almost continuous fimctions and in order to avoid confusion, Azad's fuzzy almost continuity shall be termed as A-fuzzy almost continuity. It will be seen that our concept of fuzzy almost continuous functions is independent of the notions of A - f u r y almost continuous and fuzzy w~akly continuous functions. In Section 4 our ahn is to study all these [unctions, taken together, to bring forth their mutual interactions under di~ierent c~uditions. In order to make the exposition self-contained as far as practicable, we list some definitions and results that will be fised in the sequel. Let X be a non-empty (ordinary) set and I the unit interval [0, 1]. A fuzzy set A in X is a mapping from X into 1 [12]. A fuzzy point x~ in X is a fuzzy set in X defined by

ll)

fo, yfx,

(y z)

for y ~ x x and o~ are respectively called the support and the value of x,~ [7]. A fuzzy point x~ is said to belong to a fuzzy set A of X iff c~~
[" sup A(z) F(A)(Y)-

othe vise

(y e Y),

L

and f-~(B) is ", fuzzy set in X defined by f-~(B)(x) = B(f(x)) (x ¢ X) [12]. For t fuzzy sets a, B in a space X, A - B iff / (x) - B(x) for all x e ,t", and A ~A, V ' e T} and IntA--Sup{V:V<~A, V E T } . A fuzzy set A in an fts (X, T) is a fuzzy

Weakerforms af fuzzy mappings

105

neighbouthood (f.nbd, in short) of a fuzzy point P,~ iff there exists a fuzzy open set B such that P~ <~B ~A. Definition 1.3 [1]. Let f : (X~ T ) ~ (~; U) be a mapping between two fts's. Then f is called a fuzzy semi-continuous mapping if f-~(A) is a fuzzy semi-open set of X, for each A ,~ U. Rcsu|~ 1.4 [8]. Let A and B be fuzzy sets in X and Y respectively and f : X--, Y be a mapping. Then f-~(B)<.A iff 1 - f ( 1 - A ) ~ > B. Result 1.5 [1]. For a fuzzy set A in an fls (X, T), the foUowing are equivalent: (a) A is a fuzzy semi-closed set. (b) A' is a fuzzy semi-open set. (c) Int Cl A ~~A'.

Resnk 1.6 [2]. l f f : X--> Y, then we have: (a) A <~B implie, f - l ( A ) ~ f - l ( B ) , wi~ereA, B are any two fuzzy sets in Y. (b) f ( f - l ( B ) ) <<-B, where B is any fuzzy set in Y. (c) f-t(f(A))>~A~, for any fuzzy set A in X. (d) A ~ B in X implies f(A)<~f(B) in Y. (e) f ( 1 - A ) ~ 1 - f ( A ) , for any fuzzy set A in X. (f) f-l(1 - B ) = 1 - . f - l ( B ) , for any fuzzy set B in Y. Throughout the paper, by (X, T), (Y, U) or simply by X, Y we shall mean fuzzy topological spaces. 2. Almost fuzzy open functions

Definition 2.1. A functio~ f : ( X , T)--* (Y, U) is said to be (i) almost fuzzy open in the sense of Nanda (a.f.o.N., in short) [9] iff the image of every fuzzy regularly open set of X is fuzzy open in Y, and (ii) almost fuzzy open in the sense of Ganguly and Saha (abbreviated as a.f.o.O.) [3] iff for each A ~ U, f-~(CIA) -<. Cl(f-l(A)).

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M.N. Mukher]ee, $.P. Sinha

We shall now prove the following characterization &eorem for a.f.o.G. functions. Theorem 2.2. For a function f : X--* Y, the following are equivalent: (a) f/s a.f.o.G. (b) For any fuzzy set A of Y, f-l(alIntA)<~ Clf-l(A). (c) For any fuzzy set B of X, f(Int B) ~1 - Int O f ( B ) and hence f(Int B) <~ lnt CIf(B). (c) ~ (d): Straight forward. (d) ~ (e): Let A be a fuzzy set in Y and B be a fuzzy closed set in X such that f-~(A)<~B. Now 1 - B is fuzzy open in X and so by (d), it follows that f(1 - B) <~lnt Clf(l - B) which implies that f(1 - B) is fuzzy preopen in Y. Let C = 1 - f ( 1 - B). Then C is fuzzy preclosed and by Result 1.4, A ~
Weakerforms offuzzy mappings

107

Proof. Let A be a ,fuzzy open set in Y. Then f - l ( A ) is fuzzy semi-open in X and hence f-~(A)<~ CI Intf-~(A) (by Result 1.5). Since O Intf-~(A) is fi~zzy regular closed in X, by "Iheorem 2.3 there exists a fuzzy closed set B in Y containing A such that f - l ( B ) ~ (21 Intf-l(A). How, A ~
f - l ( A ) <<-f-1(el A ) ~ f-'J(Cl B ) = f - l ( B ) <~0 Int f-a(A ) <~CI f - l ( A ). Consequently, f is a.f.o.G.

Theorem 2.$, An a.f.o.O, function f :X--~ Y is a.f.o.N, if the image of each fuzzy semi-closed set is fuzzy semi-closed. Proof, In fact, let A be a fi~zy semi-closed set in X. Since f is a.f.o.G., by Theorem 2.2 we have f(Int A)~< Int Of(A). Since f(A) is fuzzy semi-closed, by Result 1.5, Int C!f (,4) ~
~|mof~t continuous, fuzT,y wes~ly eon~uous and ~

almost

continuous mappings Fuzzy almost continuous mappings were defined and studied by Azad [1]. We shall call it 'A-fuzzy almost contie~ous mapping'. It will now be our endeavour to further characterize and investigate other properties of this mapping. Definition 3.1 [1]. A mapping f : ( X , T)---~(Y, U) is called A-fuzzy almost continuous ifff-l(A) is fuzzy open in X for each fuzzy regular open set A of Y. Theorem 3.2 [1]. Let f : X - , Y be a mapping. Then the foUowing are equivalent: (i) f is A-fuzzy almo,st con~nuo~. (ii) f - l ( B ) is a fuzzy closed set, for each fuzzy regular closed set B of Y. (iii) f - t ( B ) ~ Intf-~(Int CI B), for each fuzzy open set B of Y. ' (iv) Clf-I(CI Int B) ~ f-l(B), for each fuzz), closed set B of Y.

Theorem 3.3. A mapping f :X.--~ Y from an fts X to another fts Y is A-fuzzy almost continuous iff for each fuzzy point x~ of X and each fuzzy open set B of Y containing f(x~), there exists a fuzzy open set C of X containing x~ such that f(C) <<-Int C! B. Proof. Let f be A-fuzzy almost continuo~:g. Let x~, be a fuzzy poh~t in X an~ B be a fuzzy open nbd off(x~) in Y. Then by Theorem 3.2, f - l ( B ) <~Intf-l(Int CI B). Let C =f-~(Int C1 B) and D = I n t o B. Then D contains f(x~). Now 1 - D is fuzzy closed and hence, CI[f-~(CI Int(1 - D))] ~
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M.N. Mukherjee, S.P. Sinlm

(using Theorem 3.2). Since D (=grit CI B) is fuzzy regular open, 1 - D is fuzzy regular closed, we have

f - t ( O ) <~ 1 - Cl[f-'(Cl Int(l - O))] = 1 - Cl[f-~(1 - O)] = 1 - CI[1 - f - ~ ( D ) ] = Intf-t(D), so that f-~(D)ffilntf-t(D). Hence C is fuzzy open. Now, A < ~ f - T ( A ) << f-X(D) ffi C and f(C) ~ff'~(lnt Cl B) ~
Conversely, let B be any fuzzy regular open set in Y arid x,~ be any fuzzy point in X such that x,, ~f-~(B). By hypothesis, there exists a fuzzy open set C in X with xo, ~ C such that f(C) ~ Int CI B -- B. Thus x,~ <~C <<,f-~f(C) <<,f-~(B). Consequently,f-t(B) iS a fuzzy nbd of each of its points and hence is fuzzy open. Thus f is A-fuzzy almost continuous.

Theorem 3.4. A mapping f : X-~, Y is A-fuzzy almost continuous iff Cl(f-t(A )) <~ f-a(C!fA)) for all fuzzy semi-open sets A of Y. Proof. Let f be A-fuzzy almost continuous and A be a fuzzy semi-open set in Y~ Then A ~
Cl f - l ( A ) <. Cl f - l ( B ) = f-~( B) -- f-t(Cl lnt A ) <~f-t(CI A ). Conversely, since every fuzzy regular closed set is fuzzy semi-open, for every fuzzy regular closed set A in Y, we have Clf-l(A) ~
then: (a) the inverse image f - t ( A ) of each fuzzy regular open set A of Y is a fuzzy regular open set in X, (b) the inverse image f - l ( B ) of each fuzzy regular closed set B of Y is a fuzzy regular closed set in X. Proof, (a) Let A be an arbitrary fuzzy regular open set in 1I. Then since f is A-fuzzy almost continuous, f - l ( A ) is fuzzy open and hence we obtain that f-t(A)<-IntCIf"~(A). Since CIA ~s fuzzy regular closed, by Theorem 3.2, f - t ( C I A ) is fuzzy closed and hence

lnt CI f-~(A) ~ CI f-'(A) ~ Cl f - 1 ( O ,4) - f-~(Cl A). As f is a.f.o.N, and I n t C l f - l ( A ) flint CI(f-I(A))] is fuzzy open. Now,

f[Int CI(f-I(A))] ~
is a fuzzy regular open set in X, CI ,~,

Weakerforms of.fuzzy mappings

|09

and thus flint Cl(f-'(A))] = Int f[Int CI(f-~(A ))] ~
f-l(IntA) <~Int f-l(Int Cl IntA) <~ll~tf - l ( O lntA) = Intf"~(A). Now f - l ( A ) being closed, C! Intf-l(A) <~f-i(A). Thus f-~(A) = Of-~(Int A) <~ CI Intf-~(A). Hence f-~(A) = Cl Intf-~(A) and consequently~ f-~(A) is f u r y regular closed in .t". The other part is obvious. Kem~k 3.7. Later on we shall give an improved version of the above theorem (see Remark 4.2). Definition 3.8. A family F of non-zero fuzzy sets in an fts (X, T) will be called a fuzzy directed family iff for all A, B in F, there exists a member C of F such that C<~Af3B. Defin|flon 3.9. Let F be a fuzzy directed family in an fts (X, T) and x~, be a fuzzy point in X. (a) F is said to converge (almost converge) to x~, iff for any fuzzy open nbd B of x~, (fuzzy regular open set B containing x~) there exists a member C of F ~uch that C ~ B.

(b) x~ will be said to be a fuzzy almost cluster point of F in X iff every member of F intersects every fuzzy regular open set containing x~,. (Two fuzzy sets A and B are said to intersect if[ A N B ~ 0 , i.e., there exists x ~ X such that min(A(x), B(x)) > 0 [7].) Resullt 3.10. lf f :X--} Y and F is a fuzzy directed family of non-zero fuzzy sets in X, then f(F) - {f(A):A ~ F} is also a fuzzy directed family in Y.

PmoL A e F :~ A ~ O :~ f(A)~O. Let f(A), f ( B ) e f ( F ) , where A, B e F . Then there exists C ~ F such that C ~ A N B. Hence f(C) <-f(A N B) ~ f ( A ) N

f(z

110

M.N. Mukherjee, $.P. Sinha

"l['heorem 3.H. l f f : X--~ Y is A ~ t z z y almost continuous then for each fuzzy point x. of X and every fuzzy directed family F in X converging to x., f ( x . ) is an almost cluster point of the fuzzy directed family f (F). Proef° Let B be a fuzzy regular open set in Y withf(x~) ~ 0. Then (it 13 C)(y) > 0 such that A(y) > 0 and C(y) > 0. Now~

¢(y)>0 :~ U(y)>0 ~ f(U)[f(y)]>0 => B(f(y))>0. Also,

A(y)>0 => f ( A ) ( f ( y ) ) > O . Thus B N f ( A ) ~ 0 an~ the theorem, ~ pr~ed.

Theorem 3°~° Let f :X.-~ Y, f is A-fuzzy filmost continuous if for each fuzzy directed family F in X converging to some fuzzy point x,~ in X, the.fuzzy directed family f(F) almost converges to f(x,,). ~mefo I f f i s not A-fuzzy almost continuous, there exist a fuzzy point x,, in X and a fuzzy regular open set B in Y w~th f ( x , ) <~B such that for each ftw.zy open nbd U of x~, f(U)~B. Let F denote the family of all fuzzy open nbd's of x,, in X. Then F is a fuzzy directed family in X converging to x,,. By hypothesis f ( F ) must almost converge to f(x,~). But B is a fuzzy regular open set containing f(x,,) such that for every member f(U) o f f ( F ) (where U ~ F), f ( F ) ~ B . Hence we arrive at a contradiction. 'Re concept of fuzzy weakly continuous mapping was defined by Azad [1], where he studied the concept with regard to product spaces and in relation to fuzzy semi-continuous and graph functions. We propose to characterize such a mapping here. D e ~ f i e n 3°13 [1]. A mapping f :X--~ Y from an fts X to another fts Y is called fuzzy weakly continuous iff for each fuzzy open set B of Y, f-~(B)~< Intf-t(Cl B).

[,emma $o14. Let A and B be two fuzzy sets in a space X. Then A <~B iff for any fuzzy point x . of X, x. <~A ~ x~ ~ B. F~oof. Let the condition hold. If A ~ B , then there is x ~ X such that A(x)> B(x). Choose p ¢ (0, 1] such that A(x) > P > B(x). Then xp is a fuzzy point in X such that xp <. A but xp ~ B, going against the hypothesis. Conversely, if A <~B, then for any fuzzy set C of X, C ~ C <~B.

" l ~ e o ~ 3o15. A mapping f :X--~ Y is fuzzy weakly continuous iff for each fuzzy point xp of X and each fuzzy open set B of Y containing f(Xp), there exists a fuzzy open ~e, A of X containing xp such that f (A ) <<.CI B.

Weakerforms of fuzzy nmppings

Ili

Proof. Suppose f is fuzzy weakly continuous. Let Xp be a fuzzy point in X and B be a fuzzy open set in Y such that f(Xp) <~B. Then

xp ~ f - l f (xp) <~f-t(B ) <, Int f-x(Cl B). Let A ffi Intf-t(Cl B). Then f(A) = f ( l n t f-~(Cl B)) <~ff-t(Cl B) ~
Theorem 3.17. A mapping f :X--. Y is fuzzy almost continuous iff for each fuzzy point xp of X and every fuzzy open set B containing f(xp), there exists a fuzzy open set A containing Xp such that f - l ( B ) is fuzzy de~e in A. Theorem 3.18. A mapping f :X-~ Y from an fls X to another fts Y is fuzzy almost continuous iff f - i ( B ) <~Int O F t ( B ) for every fuzzy open set B of Y. Proof. Let f be fuzzy almost continuous and x, be a fuzzy point in X such that x,, <~f-l(B) where B is fuzzy open in If. By hypothesis, CIf-I(B) is a fuzzy nbd of x,~. Then x,~ <~Im CIf-I(B). Hence x,, <.f-l(B) =~ x, <<-Int Clf-1(B). Then by Lcmma 3.14, f - l ( B ) <~Int Clf-t(B). Conversely, let x~, be any fuzzy point in X and B be any fuzzy open nbd of f(x~). Then x~, <~f-t(B)<-IntClf-~(B)<.Clf-l(B). Thus Clf-l(B) is a fuzzy nbd of x,~ and consequently f is a fuzzy almost continuous mapping. 4. Mural re|aiiousWps

It is clear that a mapping which is A-fuzzy almost continuous, is obviously fuzzy weakly continuous, though the converse is false as shown by Azad [1]. However, we have:

Theorem 4.1. If f :X-.> Y is a.f.o.G, and .fuzzy weakly continuous then f is A-fuzzy almost continuous.

M.N. Mukherjee, S.P. Sitrha

112

I~eof. Let Xp be a fuzzy point in X and B be an open set in Y containing f(Xp). Then by Theorem 3.15, there exists a fuzzy open nbdA of Xp such that f(A) <<.0 B. Now since f is a.f.o.G., f(A) <. Int O f ( A ) ~ lnt CI(B) and consequently by Theorem 3.3, f is A-fuzzy almost continuous. R e m ~ 4.2. In view of the above theorem and Theorem 3.6, it now follows that for a mapping f : X - - , Y which is fuzzy weakly continuous and a.f.o.G., the inverse, image of a fuzzy regular o ~ n (closed) set of Y is fuzzy regular open (dosed) in X.

The fo|lowing two examples show that fuzzy almost continuity and fuzzy weak continuity ate independent notions.

gnmple 4.3. Let X-- (a, b, c}, Y~ = {0, I,A}, ~ = {o, I, B} where A(a)---O, A(b)=~, A(c)-~i and

B(a)=l, B(b)=0, ~(c)--0. Consider the identity mapping f : ( X , T1)~(X, 7"2). The only non-zero fuzzy regular open set in (X, T2) is I and hence f is A-flnT~almost continuous and consequently a fuzzy weakly continuous mapping. Now B ~ T2 and f-t(B)= B.

Also, Tt-CIf-t(B)-A' and T~-Int[T~-Cl(f-a(B))]--0, so that f-'(B)ffi

B ~IntC|f-I(B). Hence by Theorem 3.18, f is not a fuzzy almost continuous mapping.

E~l~pte 4.4. Let X = {a, b, c}, T~-- {0, 1, A, B}, T2= {0, 1, D}, where D(a)=~,

D(b)=~,

D(c)-L

A(a)-- ~, A ( b ) - 1, A(c) = ~, and

8(a)fL B(b)=~, B(c)ffi~.

Consider the mapping f : X--~ X defined by

f(a)-b,

f(b)--c

and f(c)ffia.

Now, D is a fuzzy open set in (X, 7"2)and T2 - Ci D = D'. E =y-I(D) is the fuzzy set given by E(a)ffi ~, E(b)--~ and E ( c ) - ½. Since there is no Tt-closed set, other than 1, which contains f - | ( D ) , we have TI-CIf-t(D)---1, so that f-l(D) <~7"1- Int Tl - CIf-I(D) = 1. Hence j'is fuz~ almost continuous. Again, f - t ( T 2 - C I D ) = F (say) is given by F(o)-23, F(b)--~, F(c)ffi12 and since none of A, B, 1 is contained in f - ~ ( T 2 - C I D ) it follows that T ~ - I n t f - ~ ( ~ - C I D ) - - 0 . Hence f - t ( D ) ~ T ~ - I n t f - ~ ( T 2 - C | D ) . Thus f is not fuzzy weakly continuous. In the next two theorems we investigate under which conditions these two independent mappings, viz. fuzzy almost continuous and fuzzy weakly continuous mappings, can be correlated.

Weaker forms of fuzzy mappings

!13

Thcoren~ 4.5. If a mapping f :X--, Y is f l : z y weakly continuous and a.f.o.G. then it ~ fuzzy almost continuous. Proof. Let f : X--* Y be fuzzy weakly continuous and a.f.o.G, and B be any fuzzy open set in Y. As f is a.f.o.G., f-I(CIB)<.CIf-~(B). Again~ since f is fuzzy weakly continuous, f-t(B) <<.Intf-l(Ci B). Hence f-a(B) <. Intf-l(C! B)<~ lnt O F t ( B ) and consequently by Theorem 3.18, f is fuzzy almost continuous.

Theorem 4°6. A fuzzy almost continuous mapping f :X--, Y is fuzzy weaRly conffnuous if C| f-I(B) <~f-l(Cl B), for eve~ fuzzy open set B in Y. ~f. Let ap be a f u r y point in X and B be a fuzzy open nbd off(xp). By fuzzy almost continuity of f and the given condition, there exists a fuzzy open set A in suctt that xp ~ A <<-OFt(B)<~f-l(Ci B). Hence f(A) <~CI B and consequently by virtue of Theorem 3.15, f is fuzzy weakly continuous. Finally, from what we have already deduced, we ca~Jmake a comparative study of fuzzy almost continuous and A-fuzzy almost continuous functions. Example 4.3 presents an A-fuzzy almost continuous function which is not fuzzy almost continuous, whereas in Example 4.4 a fuzzy almost continuous function which is not A-fuzzy almost continuous, is exhibited. Thus: Reset 4.7. Fuzzy almost continuity and A-fuzzy almost continuity are independent notions. Since an A-fuzzy almost continuous mapping is fuzzy weakly continuous, we have: ~eorem 4.8. A mappi;~g f : X - . * Y which is fuzzy almost continuous and

a.f.o.G.,/s A-fuzzy almost continuous if Cif-l(B) ~ f - l ( C l B), for every fuzzy open set B of Y. Proof. Follows from Theorem 4.1 and Theorem 4.6. ~eorem 4.9. An A-fuzzy almost continuous mapping f is fuzzy almost continuous if f is a.f.o.G. Preof, Follows from Theorem 4.5. Refereaces [I] K.K. Azad, On fuzzy semicontinuity, [uzzy a|most coutinuity and fuzzy weakly continuity, Z Math. Anal. Appl. 82 (1981) 14--32. [2] C.L. Chang, Fuzzy topologica| spaces, J. Math. Anal AppL 24 (1968) I~2.-1~. [3] S. Ganguly and S. Saha, A note on semi-open sets in fuzzy topological spaces, Fuzzy Sefs and Systems 18 (1986) 83-96. [4] T. Husain, Almost continuous mappings, Prace Mat. 10 (1966) 1-7.

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M.N. Mukher]ee, S.P. $inha

[5] N. Levine, A decomposition of continuity in topological spaces, Amer. Math. Monthly 68 (1961) 44--46. [6] A.5. Mashour, M.H. Ghanim and M.A. Fath Alia, On fuzzy non-continuous mappings, Bull. Calcutt~.~Math. $oc. 78 (1986) 57-69. [7] Pu Pao Ming and Lie Ying Ming, Fuzzy topology, I, Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. M'~h. Anal. AppL 76 (!980) 571-599. [8] M.N. Mukherjee and S.P. Sinha, Irresolute and almost open functions between fuzzy topological spaces, Fuzzy ~'ts and Systems 29 (1989) 381-388. [9] S. Nanda, On fuzzy topological spaces, Fuzzy Sets and Systems 19 (1986) 193-197. ii~] ~s,.K. Singal and A.R. Singal, Almost continuous mappings, Yokohama Math. J. 16 (1968) 6~-73. [11] C.K. Wong, Fuzzy points and local properties of fu2~ topology, J. Math. Anal. Appl. 46 (1974) 316-328. [12] L.A. Zadeh, Fuzzy sets, Inform, and Control 8 (1965) 338-353.