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Fuzzy non-continuous mappings and fuzzy pre-semi-separation axioms
FU2ZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 94 (1998) 261-268
Fuzzy non-continuous mappings and fuzzy pre-semi-separation axioms B a i S h i - Z h o n g a, W a n g W a n - L i a n g b a Department of Mathematics, Xiangtan University, Xiangtan Hunan, China b The Capital Normal University, Beijing, 100037 China Received February 1993; revised July 1996
Abstract The aim of the paper is mainly to introduce and study fuzzy pre-semi-open mapping, fuzzy pre-semi-irresolute mapping, fuzzy pro-semi-separation axioms and fuzzy pre-semi-connectedness in fuzzy topological spaces. © 1998 Elsevier Science B.V.
Keywords. Fuzzy topological space; Pre-semi-open set; Pre-semi-q-neighbourhood; Pre-semi-irresolute mapping; Pre-semi-Ti axioms; Pre-semi-connected
1. Introduction and preliminaries In 1980, Pu and Liu introduced a new concept of the so-called q-neighbourhood, which could reflect the features of neighbourhood structure in a fuzzy topological space (f.t.s., for short), and by this q-neighbourhood the adherent or cluster points of a fuzzy set were characterized [9]. This gives us a v e r y convenient and much sought for means for formulating closure of a fuzzy set, since such cluster points cannot be characterized in terms of fuzzy open neighbourhoods as is done in general topology. So it is natural to define pre-semi-closure of a fuzzy set in terms of pre-semi-q-neighbourhoods, and this we do in Section 2 and ultimately show, as one should expect, that pre-semi-closure of a fuzzy set A is the intersection of all fuzzy pre-semi-closed sets containing A. Fuzzy topological spaces and fuzzy continuity were introduced in Chang [5] early in 1968. The notion of fuzzy continuity has been proved to be of fundamental importance in the realm 0165-0114/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PH SO 165-01 14( 96 )00225-4
of fuzzy topology. Along with this, many workers [ 14 , 7, 8, 11 ] have studied fuzzy non-continuity. One of them [4] introduced and studied fuzzy pre-semiopen sets and fuzzy pre-semi-continuous mappings in fuzzy topological spaces. Here in Section 2 of this paper we establish some new characteristic properties of fuzzy pre-semi-continuous mapping by the concepts of fuzzy pre-semi-open q-neighbourhoods. And we introduce and study fuzzy pre-semi-open and fuzzy pre-semi-closed mappings. In Section 3 we introduce the fuzzy pre-semi-irresolute mapping and establish some of its characteristic properties and also discuss the relations between it and fuzzy continuous or fuzzy pre-semi-continuous mappings. In Section 4 we follow the definitions of [6], with the help of fuzzy pre-semi-q-neighbourhoods, to introduce fuzzy pre-semi-separation axioms and establish some of their properties. As a natural follow-up of the study of the fuzzy pre-semi-open sets, in Section 5 we introduce and study fuzzy pre-semi-connectedness of a fuzzy set.
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Bai Shi-Zhon9, Wang Wan-Liany/ Fuzzy Sets and Systems 94 (1998) 261-268
Throughout the paper by (X, 6) and (Y, r) or simply by X and Y we mean fuzzy topological spaces. A fuzzy point in X, whose value in e (0 < ~ ~< 1) at the support x E X, is denoted by x~. A fuzzy point x~ EA, where A is a fuzzy set in X, iff ~ ~< A(x). For two fuzzy sets A and B in X, we write A qB to mean that A is quasi-coincident (q-coincident, for short) with B, i.e., there exists at least one point x E X such that A(x) + B ( x ) > l [9]. Negation of such a statement is denoted as A ~B. B is said to be a quasi-neighbourhood (q-nbd, for short) of A iff there exists a fuzzy open set U such that A q U ~< B [9]. B is said to be a semi-q-nbd of a fuzzy point x~ in X iff there exists a fuzzy semi-open set V such that X~ q V ~< B [7]. The constant fuzzy sets taking on the values 0 and l on X are designated by 0x and lx, respectively. The notations A ° , A - , A o , A _ , A ~ and supp A will, respectively, stand for the interior, closure, semi-interior, semi-closure, complement and support of the fuzzy set A.
2. Fuzzy pre-semi-eontinuous and pre-semi-open mappings Definition 2.1 (Bai [4]). A fuzzy set A in X is said to be (1) fuzzy pre-semi-open iff A ~< (A-)o; (2) fuzzy pre-semi-closed iffA /> (A°)_. Obviously, A fuzzy set A is fuzzy pre-semi-open iff A ~is fuzzy pre-semi-closed. It is known that any union of fuzzy pre-semi-open sets is fuzzy pre-semiopen but the intersection of any two fuzzy presemi-open sets or even the intersection of a fuzzy pre-semi-open set with a fuzzy open set may not be fuzzy pre-semi-open [4]. Every fuzzy semi-open set [ l] is fuzzy pre-semi-open and every fuzzy pre-open set [3] is fuzzy pre-semi-open, and that none of the converses need be true [4].
Every fuzzy q-nbd of a fuzzy point is always a fuzzy semi-q-nbd of the fuzzy point [7]. Clearly every fuzzy semi-q-nbd of a fuzzy point is always a fuzzy pre-semi-q-nbd of the fuzzy point, but not conversely.
Definition 2.4. A fuzzy point x~ in X is called a fuzzy pre-semi-cluster point of a fuzzy set A in X iff every fuzzy pre-semi-q-nbd of x~ is q-coincident with A. The set of all fuzzy pre-semi-cluster points of A will be called the fuzzy pre-semi-closure of A and will be denoted by A~. Theorem 2.5. For a f u z z y set A in X, A~ is the intersection o f all f u z z y pre-semi-closed sets, each containing A. Proof. Let B denote the intersection of all fuzzy presemi-closed sets containing A. Suppose x~ E B and if possible, let there exist a fuzzy pre-semi-q-nbd N of x~ such that N 9[ A. Then there exists a fuzzy pre-semiopen set V in X such that x~ q V ~< N, which shows that V 9(A and hence A ~< Vt. As V~is fuzzy pre-semiclosed, B ~< VC Since x~ ¢ V~, we obtain x~ ~B which is a contradiction. Conversely, suppose x~ ~ B. Then there exists a fuzzy pre-semi-closed set D >~ A such that x~ ~D. We then have x~ q D ~ and Ag(D ~, where D ~ is fuzzy presemi-open. Thus x~ ~ A~. [] Corollary 2.6. A f u z z y set A in X is f u z z y pre-semiclosed iff A = A ~. It is clear that A~ ~< A_ ~< A - for each fuzzy set A.
Definition 2.7 (Bai [4]). A mapping f:X---~ Y is said to be fuzzy pre-semi-eontinuous if f - l ( B ) is fuzzy pre-semi-open in X, for each fuzzy open set B in Y; or, equivalently f - l ( D ) is fuzzy pre-semi-closed in X for each fuzzy closed set D in Y.
Definition 2.2. A fuzzy set A in X is called a fuzzy pre-semi-nbd of a fuzzy point x~ in X iff there exists a fuzzy pre-semi-open set B in X such that x~ E B ~< A.
Theorem 2.8 (Bai [4]). A mapping f : X---+ Y isf u z z y pre-semi-continuous iff for any f u z z y point x~ in X and any f u z z y open set V in Y with f ( x ~ ) E V, there exists a f u z z y pre-semi-open set U in X such that x~ E U and f ( U ) <~ V.
Definition 2.3. A fuzzy set A in X is called a fuzzy pre-semi-q-nbd of a fuzzy point x~ in X iffthere exists a fuzzy pre-semi-open set B in X such that x~ qB ~< A.
Theorem 2.9. A mapping f : X ~ Y is Juzzy presemi-continuous iff corresponding to any f u z z y open q-nbd V o f y~ in Y, there exists a f u z z y pre-semi-
Bai Shi-Zhong, Wan9 Wan-Liany/ Fuzzy Sets and Systems 94 (1998) 261-268 open q-nbd U of x~ in X such that f ( U ) <~ V, where y = f(x).
Proof. Let f be fuzzy pre-semi-continuous and let V be a fuzzy open q-nbd of y~ in Y. Then V(y) 4- ~ > 1 and hence there exists a positive real number 3 such that V(y) > 3 > 1 - c~, so that V is a fuzzy open nbd of Yl~" By Theorem 2.8 there exists a fuzzy pre-semi-nbd W ofx/~ such that f ( W ) <~ V. Now, m(x)
>>. ~ ~
W(x)>
l - ~ ~
W ( x ) + ~ > l.
Thus W is a fuzzy pre-semi-open q-nbd ofx~ in X. Conversely, let the given condition hold. Let V be a fuzzy open set in Y and W = f - l ( v ) . If W = 0x then it is fuzzy pre-semi-open in X. If supp W ¢ 0, then for any x Esupp W, put f ( x ) = y so that W(x) = V(y). There exists a positive integer m such that 1/m<~ W(x). We put ~ = 1 - W ( x ) + 1In for n~>m. Then 0<~n ~< 1, for all n~>m. This V(y) 4- ~ = 1 + 1/n > 1, for each n >~ m, and by given condition there exists a fuzzy pre-semi-open q-nbd U,, ofx~,, i n X such that f ( U , ) <~ V, for all n ~> m. Let us put u = U { U ~ : n ~> m}, then f ( U ) <~ V. Also, n ~> m implies U , ( X ) + ~ n > l so that U , ( x ) > W ( x ) - 1In and hence U ( x ) > W(x) - l/n. Thus U(x) >~ W(x) which in turn implies that W ~< U. Again from f ( U ) <~ V, U <~f - l ( V ) = W. Hence U = W and W becomes fuzzy pre-semi-open in X. Thus f is fuzzy pre-semi-continuous. Theorem 2.10. For a mapping f : X ~ Y the fol-
lowing are equivalent: (1) f is Juzzy pre-semi-continuous. (2) f ( A ~ ) <~( f ( A ) ) - for each fuzzy set A in X. (3) ( f - l ( B ) ) ~ <~f I(B ) f o r each Juzzy set B inY. Proof. (1) :::> (2): Let A be a fuzzy set in X. Then ( f ( A ) ) - is fuzzy closed in Y, and by Definition 2.7 f - J ( ( f ( A ) ) - ) is fuzzy pre-semi-closed in X. Furthermore,
A~<~ ( f - ~ f ( A ) ) ~ <~( f
J((f(A))-))~
=f-I((f(A))-). Thus f ( A ~ ) <~f f - l ( ( f ( A ) ) - ) <~( f ( A ) ) - . (2) :::> (3): Let B be a fuzzy set B in Y. By (2),
f((f-l(B))~)
<~( f f - l ( B ) ) -
<~B - .
Thus ( f - l ( B ) ) ~ <~. f - l f ( ( f - I ( B ) ) ~ )
<~f - l ( B - ) .
263
(3) ::~ (1): Let B be a fuzzy closed set in Y. By (3), f - l ( B ) <~( f - l ( B ) ) ~ <. f - l ( B - ) = f - l ( B ) i.e., f - ~ ( B ) = ( f - 1 ( B ) ) ~ . Hence f - l ( B ) is fuzzy presemi-closed in X and consequently, f is fuzzy presemi-continuous. Definition 2.11. A mapping f : X - ~ Y is said to be (1) fuzzy pre-semi-open if f ( A ) is fuzzy pre-semiopen in Y for each fuzzy open set A in X; (2) fuzzy pre-semi-elosed if f ( A ) is fuzzy presemi-closed in Y for each fuzzy closed set A in X. Theorem 2.12. For a mappinq f :X--+ Y the Jollow-
in9 are equivalent: (1) f is fuzzy pre-semi-open. (2) f ( A °) <~( ( f ( A ) ) - ) o for each fuzzy set A inX. (3) ( f I(B))° ~< f - l ( ( B - ) o ) for each fuzzy set BinY. Proof. This is analogous to the proof of Theorem 2.10. [] Theorem 2.13. For a mapping f : X --+ Y the follow-
in9 are equivalent: (1) f is fuzzy pre-semi-closed. (2) ( ( f ( A ) ) ° ) _ ~< f ( A - ) for each fuzzy set A inX. (3) ( f (A ) )~ <~f (A- ) for each fuzzy set A in X. Proof. This is analogous to the proof of Theorem 2.10. [] Theorem 2.14. Let f : X --+ Y be one-to-one and
onto. f is a fuzzy pre-semi-closed mapping iff f I ( B ~ ) < . ( f - I ( B ) ) - for each .fuzzy set B inY. Proof. Let f be fuzzy pre-semi-closed and B be any fuzzy set in Y. Then f ( ( f I ( B ) ) - ) is fuzzy pre-semi-closed in Y. By Corollary 2.6 and since f is onto we have f ( ( f - l ( B ) ) - ) = ( f ( ( f - J ( B ) ) - ) ) ~ >>, ( f f - l ( B ) ) ~ = B~. Again since f is one-to-one we have f - l ( B ~ ) <~f If ( ( , f - I ( B ) ) - ) = ( f - J ( B ) ) - . Conversely, let A be fuzzy closed set in X. By hypothesis f - l ( ( f ( A ) ) ~ ) <<,( f - i f ( A ) ) = A- = A and ( f ( A ) ) ~ = f f - l ( ( f ( A ))~ ) ~ f ( A ).
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Bai Shi-Zhong, Wang Wan-Liang/Fuzzy Sets and Systems 94 (1998) 261~68
Hence f ( A ) = ( f ( A ) ) ~ , i.e., f ( A ) is fuzzy presemi-closed in Y. Thus f is fuzzy pre-semi-closed. []
3. Fuzzy pre-semi-irresolute mappings Definition 3.1. A mapping f : X --~ Y is said to be
Remark 3.3. For the mapping f : X ---, Y, the following statements are valid: (1) f is fuzzy pre-semi-irresolute ~ f is fuzzy pre-semi-continuous. (2) f is fuzzy continuous 5+ f is fuzzy pre-semiirresolute.
fuzzy pre-semi-irresolute if f - l (B) is fuzzy pre-semiopen in X for each fuzzy pre-semi-open set B in Y.
Example 3.4. Let X = { a , b , c } , 6={Ox,A,B, Ix} and
Theorem 3.2. For a mapping f : X --+ Y the following are equivalent: (1) f is fuzzy pre-semi-irresolute. (2) f - ~(B) is fuzzy pre-semi-closed in X for each Juzzy pre-semi-closed set B in Y. (3) f ( A ~ ) <<.( f ( A ) ) ~ for each fuzzy set A in X. (4) ( f - l ( B ) ) ~ <. f - l ( B ~ ) Jor each fuzz), set B inY. (5) For each fuzzy point x~ in X and each Juzzy pre-semi-open set V in Y with f(x~) E V, there exists a fuzzy pre-semi-open set U in X such that x~ E U and f ( U ) <~ V. (6) For each fuzzy point x~ in X and each Juzzy pre-semi-open set V in Y satisfying f ( x ~ ) q V there exists a fuzzy pre-semi-open set U in X such that x~qU and f ( U ) <~ V.
A(a) = 0.2,
A(b) = 0.4,
A(c) = 0.5;
B(a) =0.8,
B ( b ) = 0.8,
B ( c ) = 0.6;
C(a) =0.2,
C(b) =0.1,
C(c) =0.3.
Proof. (1) <==>(2): Straightforward and hence omitted. ( 1) =~ (3) =~ (4) =~ ( 1 ) is analogous to the proof of Theorem 2.10. (1) =~ (5): Let x~ be a fuzzy point in X and V be any fuzzy pre-semi-open set in Y such that f(x~) E V. Then x~ c f - J ( V ) . Since f is fuzzy pre-semi-irresolute, f I(V) is a fuzzy pre-semi-open set. Let f - l ( V ) = U. We have f ( U ) = f f l(V) ~
the following conditions are satisfied: (1) when x ~ y, either x~ has a fuzzy pre-semi-nbd U such that U qlY/~ or y/~ has a fuzzy pre-semi-nbd V such that V~ x~; (2) when x = y and e < f l (say), Y[3 has a fuzzy pre-semi-q-nbd V such that Vqi x~.
r = {0~, A, lx }, where
Consider the identity mapping f : (X, 6) ---+ (X,z). Then f is fuzzy continuous. Clearly, C is fuzzy presemi-open in (X,z), but f - 1 ( C ) = C is not fuzzy pre-semi-open in (X, 6). Thus f is not fuzzy presemi-irresolute.
4. Fuzzy pre-semi-separation axioms Definition 4.1. An fts X is called fuzzy pre-semi-To iff for every pair of distinct fuzzy points x~ and y[~,
Definition 4.2. An fts X is called fuzzy pre-semi-Ti iff for every pair of distinct fuzzy points x~ and Y/I, the following conditions are satisfied: (1) when x ¢; y, x~ has a fuzzy pre-semi-nbd U and y/~ has a fuzzy pre-semi-nbd V such that Uql y[~ and V~x~; (2) when x = y and ~ < f l (say), y/~ has a fuzzy pre-semi-q-nbd V such that V~ x~.
Definition 4.3. An fts X is called fuzzy pre-semi-T2 iff for every pair of distinct fuzzy points x~ and y/i, the following conditions are satisfied: (1) when x ~ y, x~ and y/~ have fuzzy pre-seminbds which are not q-coincident;
Bai Shi-Zhon9, Wan9 Wan-Liang/Fuzzy Sets and Systems 94 (1998) 261~68 (2) when x = y and 7 < f l (say), then x= has a fuzzy pre-semi-nbd U and y/~ has a fuzzy pre-semi-q-nbd V such that U~[ V. Remark 4.4. Obviously, fuzzy pre-semi-T2 ~ fuzzy pre-semi-T1 ~ fuzzy pre-semi-To. Also, fuzzy T~ axiom [6] ~ fuzzy semi-T/ axiom [7] ~ fuzzy presemi-T/axiom, for i = 0, 1,2. Theorem 4.5. X is fuzzy pre-semi-To iff for every pair of distinct fuzzy points x~ and Yl~, either x~ f[ (Yl~)~ or YI~ fg (x~)~. Proof. Let X be fuzzy pre-semi-To and x~ and y/~ be two distinct fuzzy points in X. When x # y, xl has a fuzzy pre-semi-nbd U such that Uq[ Yl*, or yl has a fuzzy pre-semi-nbd V such that V(lx~. Suppose xj has a fuzzy pre-semi-nbd U which is not q-coincident with y/~. Then U is a fuzzy pre-semi-q-nbd ofx~ and U~ YI~. Hence x~ ~ (Yl~)~. When x -- y and :~ < ~ (say), then y/~ has a fuzzy pre-semi-q-nbd which is not q-coincident with x~ and so in this case also Yl~ fg (x=)~. Conversely, let x~ and Y13 be two distinct fuzzy points in X. We suppose, without loss of generality, that x~ ~ (y/~)~. When x ¢ y, since x~ ~ (y/~)~, xl ~ (y/~)~ and hence ((y/~)~)t(x)= 1. Then ((y/~)~)' is a fuzzy presemi-nbd of x~ such that ((Yl~)~)'(1 Yl~" When x = y we have ~ >/~ and then x~ has a fuzzy pre-semi-q-nbd which is not q-coincident with YV Theorem 4.6. X is fuzzy pre-semi-T~ iff every fuzzy point x~ is fuzzy pre-semi-closed in X. Proof. Let X be fuzzy pre-semi-T1 and x= and y/~ be two distinct fuzzy points in X. When x # y, x~ and y/~ have fuzzy pre-semi-open pre-semi-nbd U and V, respectively such that U~ y/~ and V~x~. Then x~ ~ V'. Since V' is fuzzy presemi-closed, (x~)~ ~< V', equivalent Vql(x=)~. Thus (x~)~ ~< x~ i.e., x~ = (x~)~. Hence every fuzzy point x~ is fuzzy pre-semi-closed in X. When x = y, it is analogous to the proof of it above. Conversely, let x~ and y/~ be two distinct fuzzy points in X. When x # y, since x~ and y/~ are fuzzy pre-semiclosed in X, (x~) / and (y~)~ are fuzzy pre-semi-open.
265
Then (x~)' is a fuzzy pre-semi-nbd of Y13 and (y/~)~ is a fuzzy pre-semi-nbd of x~ such that (x~)'4 x~ and
(y~)'~ y~. When x = y and ~ < fl (say), obviously (x~)' is a fuzzy pre-semi-q-nbd of y/~ such that (x~)'qt x~. Theorem 4.7. X & fuzzy pre-semi-T2 iff for every
fuzzy point x~ in X,
v is a fuzzy pre-semi-nbd of
}
and for every x, y E X with x ~ y, there is a fuzzy pre-semi-nbd U of xl such that y ~ supp(U~). Proof. Let X be fuzzy pre-semi-T2. Let x~ and y~ be fuzzy points in X such that y~ ¢ (x~). To establish the required equality, it suffices to show the existence of a fuzzy pre-semi-nbd V o f x~ such that y~ ~ V~. If x # y, then there are fuzzy presemi-open sets U and V containing Yl and x~, respectively, such that Uq[ V. Then V is a fuzzy presemi-nbd ofx~ and U is a pre-semi-q-nbd o f y~ such that Ug[ V. Hence y/~ g' V~. If x = y, then fi > :~, and hence there are a fuzzy pre-semi-q-nbd U o f y/~ and a fuzzy pre-semi-nbd V of x~ such that Uq[ V. Then
y~¢ v_. Finally, for two distinct points x, y of X, since X is fuzzy pre-semi-T2, there exist fuzzy pre-semi-open sets U and V such that xl E U, Yl E V and U~[ V. Then V ' ( y ) = 0 and U~< V'. Since V ~ is fuzzy pre-semi-closed, U~ ~< V ~. Thus (U~)(y) = 0, i.e., y ~ supp(U~ ). Conversely, let x~ and y/~ be two distinct fuzzy points in X. When x ¢ y, we first suppose that at least one o f and/~ is less than 1, say 0 < e < 1. There exists a positive real number 2 with 0 < ~ + 2 < 1. By hypothesis, there exists a fuzzy pre-semi-nbd U of y/~ such that x;. ~ U_. Then x;~ has a fuzzy pre-semi-open presemi-q-nbd V such that V~ U. Now, 2 + V(x) > 1 so that V ( x ) > 1 - 2 > ~ and hence V is a fuzzy presemi-nbd ofx~ such that U~[ V, where U is a fuzzy presemi-nhd of YV In case ~ =/~ = 1, by hypothesis there is a fuzzy pre-semi-nbd U ofxt such that ( U ~ ) ( y ) = 0 . Then V = ( U ~ ) / is a fuzzy pre-semi-nbd of Yl such that U9[ V. When x = y and :~
266
Bai Shi-Zhong. Wang Wan-Liang/ Fuzzy Sets and Systems 94 (1998) 261-268
there exists a fuzzy pre-semi-q-nbd V of y/~ such that Uq[ V. Considering the above cases, we conclude that X is fuzzy pre-semi-T2. Theorem 4.8. Let f : X --, Y be a one-to-one fuzz), pre-semi-irresolute mapping. I f Y is Juzzy pre-semiTi then so is"X, for i = O, 1,2. Proof. We prove only i = 1. Let x~ and y/~ be two distinct fuzzy points in X. When x ~ Y, we have f ( x ) # f ( y ) . Since Y is fuzzy pre-semi-Tl,(f(x))~ and (J(Y))[3 have fuzzy pre-semi-open pre-semi-nbd U and V, respectively, such that V(l(f(x))~ and U(t(f(y))f~. Then f - I ( U ) and f - l ( v ) are fuzzy pre-semi-nbds of x~ and yf~, respectively, such that f - l ( v ) ( t x ~ and f - l ( U ) ( l y / ~ . When x = y and ~<[3 (say), f ( x ) = f ( y ) . Since Y is fuzzy pre-semi-Tl, there is a fuzzy presemi-open pre-semi-q-nbd V of (f(Y))t3 such that V(1 (f(x))~. Then f - ~ ( V ) is a fuzzy pre-semi-q-nbd of Y/~ in X such that f l(V)q4x~. Thus X is fuzzy pre-semi-T1.
5. Fuzzy pre-semi-connectedness Definition 5.1. Two non-null fuzzy sets A and B in X are said to be fuzzy pre-semi-separated iffA ~ B~ and
SqA_ Remark 5.2. Clearly, if fuzzy sets A and B in X are fuzzy semi-separated [7] then they are fuzzy pre-semiseparated. That the converse need not be true is shown by Example 5.5. If A and B are fuzzy separated then they are fuzzy semi-separated (cf. [7]) and so they are fuzzy pre-semi-separated. Definition 5.3. A fuzzy set which cannot be expressed as the union of two fuzzy pre-semi-separated sets is said to be fuzzy pre-semi-connected. Remark 5.4. Clearly every fuzzy pre-semi-connected set is fuzzy semi-connected [7]. That the converse need not be true is shown by Example 5.5. Every fuzzy semi-connected set is fuzzy connected [7] and so every fuzzy pre-semi-connected set is fuzzy connected.
Example 5.5. Let X = [0, 1] and A, B, C be fuzzy sets in X defined as follows: A(x) = { 00.5 i f x = 0 , if0
1;
B(x)=
0.6 1
if x = 0 , if0
1;
C(x)=
0.2 0
if x = 0 , if0
1.
Clearly, 6 = {Ox,A, lx} is a fuzzy topology on X. By easy computations it follows that B~ = B and C~ = C. Then B ~ C~ and C ~ B_. Hence B and C are fuzzy pre-semi-separated. But B_ = lx and C_ =A t, so that B q C _ and C q B _ . Thus B and C are not fuzzy semiseparated. Again, B - B U C and B,C are fuzzy presemi-separated. It implies that B is not fuzzy pre-semi-connected. We show that B is fuzzy semi-connected. In fact, let B = D U E, where D and E are non-null fuzzy sets in X. Then either D(0) = 0.6 or E(0) = 0.6. Suppose D(0) = 0.6, then D = lx. Clearly, E q D _ . Thus D and E cannot be fuzzy semi-separated. Hence B is fuzzy semiconnected. Theorem 5.6. Let A and B be non-null fuzzy sets in X. ( 1 ) I f A, B are fuzzy pre-semi-separated, and C, D are non-nullJuzzy sets such that C <~A, D <~B, then C and D are also Juzzy pre-semi-separated. (2) I r A (1B and either both are fuzzy pre-serniopen or both are fuzzy pre-semi-closed, then A and B are fuzzy pre-semi-separated. (3) I f A,B are either both Ji~zzy pre-semi-open or both Juzz)' pre-semi-closed, then A N B ~ and B N A ~ are Ji4zzy pre-semi-separated. Proof. (1) and (2) are obvious. (3) Let A and B be both fuzzy pre-semi-open. Since A N B ~ ~ B ~, (A N B~)~ ~
Bai Shi-Zhong, Wang Wan-Liang/Fuzzy Sets and Systems 94 (1998) 261~68
Theorem 5.7. The non-null fuzzy sets A and B are fuzzy pre-semi-separated iff there ex&t two fuzzy pre-semi-open sets U and V such that A <~ U, B <,%V, A ~ V a n d B ~ U . Proof. For two fuzzy pre-semi-separated sets A and B, B ~< (A~)' = V (say) and A ~< ( B ~ ) ' = U (say), where V and U are clearly fuzzy pre-semi-open, then V~A~ andU~B~.ThusA~VandB~U. Conversely, let U and V be fuzzy pre-semi-open sets such that A <~ U, B <~ V, A (t V and B ~ U. Then A ~< V ~ a n d B ~ < U ~. H e n c e A ~ ~< W a n d B ~ ~
Theorem 5.8. Let A be a non-null fuzzy pre-semiconnected set in X. I f A <~B <~A~, then B is also fuzzy pre-semi-connected. Proof. I f B is not fuzzy pre-semi-connected inX, then there exist fuzzy pre-semi-separated sets C and D in X such that B = C t 3 D . Let E = A N C andF= A f3 D. Then A = E U F. Since E ~< C and F ~< D, by Theorem 5.6(1 ), E and F are fuzzy pre-semi-separated contradicting the fuzzy pre-semi-connectedness o f A. Thus B is fuzzy pre-semi-connected.
267
we have Aj <~B, or Aj <~ C. We may assume that Aj <~B. Then for each i ~ j, Ai <~B. In fact, ifAj ~B, then Ai <~ C. By Theorem 5.6(1) Aj and Ai are fuzzy pre-semi-separated. This is a contradiction. Hence for each i C I , Ai <~B. It follows that A = [..J{Ai: i E I } ~< B, clearly C = 0x. Thus A is fuzzy pre-semiconnected. [] The following corollary is obvious.
Corollary 5.11. Let {Ai: i E I} be a collection of fuzzy pre-semi-connected sets in X. If N{ Ai: i E I} ¢ Ox, then U{Ai: i E I} is fuzzy pre-semi-connected. Theorem 5.12. Let f : X---~ Y be a one-to-one fuzzy pre-semi-irresolute mapping, l f A is fuzzy pre-semiconnected in X, so is f ( A ) in Y. ProoL I f possible, let f ( A ) be not fuzzy pre-semiconnected in Y. Then there exist fuzzy pre-semiseparated sets B and C in Y such that f ( A ) = B U C. Since B and C are fuzzy pre-semi-separated, by Theorem 5.7 there exist two fuzzy pre-semiopen sets U a n d V s u c h t h a t B ~ < U, C~< V , B ~ V a n d C ~ U . Now, f being fuzzy pre-semi-irresolute, f - l ( U ) and f - l ( V ) are fuzzy pre-semi-open sets in X, and
A = f-if(A)
= f - l ( B U C) = f - l ( B ) U f - l ( C ) .
Theorem 5.9. Let A be a non-null fuzzy pre-semiconnected set in X, and C and D be two fuzzy presemi-separated sets in X. IrA <~ C to D, then A <<,C orA <~D.
ForB(tVandCqiU, w e h a v e B ~ < V~ and C~< U ~, i.e., f - l ( B ) <~ ( f ( V ) ) ' and f - l ( c ) ~< ( f - l ( U ) ) ' . Hence f - l ( B ) (t f - l ( V ) and f - l ( C ) (t f - l ( U ) . By Theorem 5.7 f - l ( B ) and f - l ( C ) are fuzzy pre-semiseparated in X. Thus we arrive at a contradiction. []
Proof. Suppose that A N D ~ 0x and A A C ~ 0x. By Theorem 5.6(1 ), A A C and A N D become fuzzy presemi-separated sets such that A = (A N C) U (A A D), contradicting the fuzzy pre-semi-connectedness of A. HenceA ~
Corollary 5.13. Let f : X ~ Y be a fuzzy pre-semiirresolute mapping. I f X &fuzzy pre-semi-connected, so is f ( X ) . References
Theorem 5.10. Let {Ai: i E I } be a collection of fuzzy pre-semi-connected sets in X. Suppose there exists a j E I such that Ai and Aj are not fuzzy pre-semi-separated for each i E 1. Then A = U{Ai: i E I} is fuzzy pre-semi-connected. Proof. If A is not fuzzy pre-semi-connected, then A =Bt0 C, where B and C are fuzzy pre-semi-separated in X. Since Ai is fuzzy pre-semi-connected for each i E I, by Theorem 5.9, A i <~B or Ai <~ C. Specifically,
[ 1] K.K. Azad, On fuzzy semi-continuity,fuzzy almost continuity and fuzzy weak continuity, .L Math. Anal. Appl. 82 (1981) 14-32. [2] Bai Shi-Zhong, Fuzzy weak semicontinuity, Fuzz), Sets and Systems 47 (1992) 93-98. [3] Bai Shi-Zhong,Fuzzy strongly semiopen sets and fuzzy strong semicontinuity, Fuzz), Sets and Systems 52 (1992) 345 -351. [4] Bai Shi-Zhong, Fuzzy pre-semiopen sets and fuzzy presemicontinuity, Proc. ICIS'92 (1992) 918-920. [5] C.L. Chang, Fuzzy topological spaces, .L Math. Anal. Appl. 24 (1968) 182-190.
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[6] S. Ganguly and S. Saha, On separation axioms and Ti-fuzzy continuity, Fuzzy Sets and Systems 16 (1985) 265-275. [7] B. Ghosh, Semi-continuous and semi-closed mapping and semi-connectedness in fuzzy setting, Fuzzy Sets and Systems 35 (1990) 345 355. [8] M.N. Mukherjee and S.P. Sinha, Irresolute and almost open functions between fuzzy topological spaces, Fuzzy Sets and Systems 29 (1989) 381-388.
[9] Pu Bao-Ming and Liu Ying-Ming, Fuzzy topology I. Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571 599. [10] M.K. Singal and N. Prakash, Fuzzy preopen sets and fuzzy preseparation axioms, Fuzzy Sets and Systems 44 (1991) 273 281. [11] T.H. Yalvac, Semi-interior and semi-closure of a fuzzy set, J. Math. Anal AppL 132 (1988) 356 364.
sets and systems ELSEVIER
Fuzzy Sets and Systems 94 (1998) 261-268
Fuzzy non-continuous mappings and fuzzy pre-semi-separation axioms B a i S h i - Z h o n g a, W a n g W a n - L i a n g b a Department of Mathematics, Xiangtan University, Xiangtan Hunan, China b The Capital Normal University, Beijing, 100037 China Received February 1993; revised July 1996
Abstract The aim of the paper is mainly to introduce and study fuzzy pre-semi-open mapping, fuzzy pre-semi-irresolute mapping, fuzzy pro-semi-separation axioms and fuzzy pre-semi-connectedness in fuzzy topological spaces. © 1998 Elsevier Science B.V.
Keywords. Fuzzy topological space; Pre-semi-open set; Pre-semi-q-neighbourhood; Pre-semi-irresolute mapping; Pre-semi-Ti axioms; Pre-semi-connected
1. Introduction and preliminaries In 1980, Pu and Liu introduced a new concept of the so-called q-neighbourhood, which could reflect the features of neighbourhood structure in a fuzzy topological space (f.t.s., for short), and by this q-neighbourhood the adherent or cluster points of a fuzzy set were characterized [9]. This gives us a v e r y convenient and much sought for means for formulating closure of a fuzzy set, since such cluster points cannot be characterized in terms of fuzzy open neighbourhoods as is done in general topology. So it is natural to define pre-semi-closure of a fuzzy set in terms of pre-semi-q-neighbourhoods, and this we do in Section 2 and ultimately show, as one should expect, that pre-semi-closure of a fuzzy set A is the intersection of all fuzzy pre-semi-closed sets containing A. Fuzzy topological spaces and fuzzy continuity were introduced in Chang [5] early in 1968. The notion of fuzzy continuity has been proved to be of fundamental importance in the realm 0165-0114/98/$19.00 (~ 1998 Elsevier Science B.V. All rights reserved PH SO 165-01 14( 96 )00225-4
of fuzzy topology. Along with this, many workers [ 14 , 7, 8, 11 ] have studied fuzzy non-continuity. One of them [4] introduced and studied fuzzy pre-semiopen sets and fuzzy pre-semi-continuous mappings in fuzzy topological spaces. Here in Section 2 of this paper we establish some new characteristic properties of fuzzy pre-semi-continuous mapping by the concepts of fuzzy pre-semi-open q-neighbourhoods. And we introduce and study fuzzy pre-semi-open and fuzzy pre-semi-closed mappings. In Section 3 we introduce the fuzzy pre-semi-irresolute mapping and establish some of its characteristic properties and also discuss the relations between it and fuzzy continuous or fuzzy pre-semi-continuous mappings. In Section 4 we follow the definitions of [6], with the help of fuzzy pre-semi-q-neighbourhoods, to introduce fuzzy pre-semi-separation axioms and establish some of their properties. As a natural follow-up of the study of the fuzzy pre-semi-open sets, in Section 5 we introduce and study fuzzy pre-semi-connectedness of a fuzzy set.
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Throughout the paper by (X, 6) and (Y, r) or simply by X and Y we mean fuzzy topological spaces. A fuzzy point in X, whose value in e (0 < ~ ~< 1) at the support x E X, is denoted by x~. A fuzzy point x~ EA, where A is a fuzzy set in X, iff ~ ~< A(x). For two fuzzy sets A and B in X, we write A qB to mean that A is quasi-coincident (q-coincident, for short) with B, i.e., there exists at least one point x E X such that A(x) + B ( x ) > l [9]. Negation of such a statement is denoted as A ~B. B is said to be a quasi-neighbourhood (q-nbd, for short) of A iff there exists a fuzzy open set U such that A q U ~< B [9]. B is said to be a semi-q-nbd of a fuzzy point x~ in X iff there exists a fuzzy semi-open set V such that X~ q V ~< B [7]. The constant fuzzy sets taking on the values 0 and l on X are designated by 0x and lx, respectively. The notations A ° , A - , A o , A _ , A ~ and supp A will, respectively, stand for the interior, closure, semi-interior, semi-closure, complement and support of the fuzzy set A.
2. Fuzzy pre-semi-eontinuous and pre-semi-open mappings Definition 2.1 (Bai [4]). A fuzzy set A in X is said to be (1) fuzzy pre-semi-open iff A ~< (A-)o; (2) fuzzy pre-semi-closed iffA /> (A°)_. Obviously, A fuzzy set A is fuzzy pre-semi-open iff A ~is fuzzy pre-semi-closed. It is known that any union of fuzzy pre-semi-open sets is fuzzy pre-semiopen but the intersection of any two fuzzy presemi-open sets or even the intersection of a fuzzy pre-semi-open set with a fuzzy open set may not be fuzzy pre-semi-open [4]. Every fuzzy semi-open set [ l] is fuzzy pre-semi-open and every fuzzy pre-open set [3] is fuzzy pre-semi-open, and that none of the converses need be true [4].
Every fuzzy q-nbd of a fuzzy point is always a fuzzy semi-q-nbd of the fuzzy point [7]. Clearly every fuzzy semi-q-nbd of a fuzzy point is always a fuzzy pre-semi-q-nbd of the fuzzy point, but not conversely.
Definition 2.4. A fuzzy point x~ in X is called a fuzzy pre-semi-cluster point of a fuzzy set A in X iff every fuzzy pre-semi-q-nbd of x~ is q-coincident with A. The set of all fuzzy pre-semi-cluster points of A will be called the fuzzy pre-semi-closure of A and will be denoted by A~. Theorem 2.5. For a f u z z y set A in X, A~ is the intersection o f all f u z z y pre-semi-closed sets, each containing A. Proof. Let B denote the intersection of all fuzzy presemi-closed sets containing A. Suppose x~ E B and if possible, let there exist a fuzzy pre-semi-q-nbd N of x~ such that N 9[ A. Then there exists a fuzzy pre-semiopen set V in X such that x~ q V ~< N, which shows that V 9(A and hence A ~< Vt. As V~is fuzzy pre-semiclosed, B ~< VC Since x~ ¢ V~, we obtain x~ ~B which is a contradiction. Conversely, suppose x~ ~ B. Then there exists a fuzzy pre-semi-closed set D >~ A such that x~ ~D. We then have x~ q D ~ and Ag(D ~, where D ~ is fuzzy presemi-open. Thus x~ ~ A~. [] Corollary 2.6. A f u z z y set A in X is f u z z y pre-semiclosed iff A = A ~. It is clear that A~ ~< A_ ~< A - for each fuzzy set A.
Definition 2.7 (Bai [4]). A mapping f:X---~ Y is said to be fuzzy pre-semi-eontinuous if f - l ( B ) is fuzzy pre-semi-open in X, for each fuzzy open set B in Y; or, equivalently f - l ( D ) is fuzzy pre-semi-closed in X for each fuzzy closed set D in Y.
Definition 2.2. A fuzzy set A in X is called a fuzzy pre-semi-nbd of a fuzzy point x~ in X iff there exists a fuzzy pre-semi-open set B in X such that x~ E B ~< A.
Theorem 2.8 (Bai [4]). A mapping f : X---+ Y isf u z z y pre-semi-continuous iff for any f u z z y point x~ in X and any f u z z y open set V in Y with f ( x ~ ) E V, there exists a f u z z y pre-semi-open set U in X such that x~ E U and f ( U ) <~ V.
Definition 2.3. A fuzzy set A in X is called a fuzzy pre-semi-q-nbd of a fuzzy point x~ in X iffthere exists a fuzzy pre-semi-open set B in X such that x~ qB ~< A.
Theorem 2.9. A mapping f : X ~ Y is Juzzy presemi-continuous iff corresponding to any f u z z y open q-nbd V o f y~ in Y, there exists a f u z z y pre-semi-
Bai Shi-Zhong, Wan9 Wan-Liany/ Fuzzy Sets and Systems 94 (1998) 261-268 open q-nbd U of x~ in X such that f ( U ) <~ V, where y = f(x).
Proof. Let f be fuzzy pre-semi-continuous and let V be a fuzzy open q-nbd of y~ in Y. Then V(y) 4- ~ > 1 and hence there exists a positive real number 3 such that V(y) > 3 > 1 - c~, so that V is a fuzzy open nbd of Yl~" By Theorem 2.8 there exists a fuzzy pre-semi-nbd W ofx/~ such that f ( W ) <~ V. Now, m(x)
>>. ~ ~
W(x)>
l - ~ ~
W ( x ) + ~ > l.
Thus W is a fuzzy pre-semi-open q-nbd ofx~ in X. Conversely, let the given condition hold. Let V be a fuzzy open set in Y and W = f - l ( v ) . If W = 0x then it is fuzzy pre-semi-open in X. If supp W ¢ 0, then for any x Esupp W, put f ( x ) = y so that W(x) = V(y). There exists a positive integer m such that 1/m<~ W(x). We put ~ = 1 - W ( x ) + 1In for n~>m. Then 0<~n ~< 1, for all n~>m. This V(y) 4- ~ = 1 + 1/n > 1, for each n >~ m, and by given condition there exists a fuzzy pre-semi-open q-nbd U,, ofx~,, i n X such that f ( U , ) <~ V, for all n ~> m. Let us put u = U { U ~ : n ~> m}, then f ( U ) <~ V. Also, n ~> m implies U , ( X ) + ~ n > l so that U , ( x ) > W ( x ) - 1In and hence U ( x ) > W(x) - l/n. Thus U(x) >~ W(x) which in turn implies that W ~< U. Again from f ( U ) <~ V, U <~f - l ( V ) = W. Hence U = W and W becomes fuzzy pre-semi-open in X. Thus f is fuzzy pre-semi-continuous. Theorem 2.10. For a mapping f : X ~ Y the fol-
lowing are equivalent: (1) f is Juzzy pre-semi-continuous. (2) f ( A ~ ) <~( f ( A ) ) - for each fuzzy set A in X. (3) ( f - l ( B ) ) ~ <~f I(B ) f o r each Juzzy set B inY. Proof. (1) :::> (2): Let A be a fuzzy set in X. Then ( f ( A ) ) - is fuzzy closed in Y, and by Definition 2.7 f - J ( ( f ( A ) ) - ) is fuzzy pre-semi-closed in X. Furthermore,
A~<~ ( f - ~ f ( A ) ) ~ <~( f
J((f(A))-))~
=f-I((f(A))-). Thus f ( A ~ ) <~f f - l ( ( f ( A ) ) - ) <~( f ( A ) ) - . (2) :::> (3): Let B be a fuzzy set B in Y. By (2),
f((f-l(B))~)
<~( f f - l ( B ) ) -
<~B - .
Thus ( f - l ( B ) ) ~ <~. f - l f ( ( f - I ( B ) ) ~ )
<~f - l ( B - ) .
263
(3) ::~ (1): Let B be a fuzzy closed set in Y. By (3), f - l ( B ) <~( f - l ( B ) ) ~ <. f - l ( B - ) = f - l ( B ) i.e., f - ~ ( B ) = ( f - 1 ( B ) ) ~ . Hence f - l ( B ) is fuzzy presemi-closed in X and consequently, f is fuzzy presemi-continuous. Definition 2.11. A mapping f : X - ~ Y is said to be (1) fuzzy pre-semi-open if f ( A ) is fuzzy pre-semiopen in Y for each fuzzy open set A in X; (2) fuzzy pre-semi-elosed if f ( A ) is fuzzy presemi-closed in Y for each fuzzy closed set A in X. Theorem 2.12. For a mappinq f :X--+ Y the Jollow-
in9 are equivalent: (1) f is fuzzy pre-semi-open. (2) f ( A °) <~( ( f ( A ) ) - ) o for each fuzzy set A inX. (3) ( f I(B))° ~< f - l ( ( B - ) o ) for each fuzzy set BinY. Proof. This is analogous to the proof of Theorem 2.10. [] Theorem 2.13. For a mapping f : X --+ Y the follow-
in9 are equivalent: (1) f is fuzzy pre-semi-closed. (2) ( ( f ( A ) ) ° ) _ ~< f ( A - ) for each fuzzy set A inX. (3) ( f (A ) )~ <~f (A- ) for each fuzzy set A in X. Proof. This is analogous to the proof of Theorem 2.10. [] Theorem 2.14. Let f : X --+ Y be one-to-one and
onto. f is a fuzzy pre-semi-closed mapping iff f I ( B ~ ) < . ( f - I ( B ) ) - for each .fuzzy set B inY. Proof. Let f be fuzzy pre-semi-closed and B be any fuzzy set in Y. Then f ( ( f I ( B ) ) - ) is fuzzy pre-semi-closed in Y. By Corollary 2.6 and since f is onto we have f ( ( f - l ( B ) ) - ) = ( f ( ( f - J ( B ) ) - ) ) ~ >>, ( f f - l ( B ) ) ~ = B~. Again since f is one-to-one we have f - l ( B ~ ) <~f If ( ( , f - I ( B ) ) - ) = ( f - J ( B ) ) - . Conversely, let A be fuzzy closed set in X. By hypothesis f - l ( ( f ( A ) ) ~ ) <<,( f - i f ( A ) ) = A- = A and ( f ( A ) ) ~ = f f - l ( ( f ( A ))~ ) ~ f ( A ).
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Hence f ( A ) = ( f ( A ) ) ~ , i.e., f ( A ) is fuzzy presemi-closed in Y. Thus f is fuzzy pre-semi-closed. []
3. Fuzzy pre-semi-irresolute mappings Definition 3.1. A mapping f : X --~ Y is said to be
Remark 3.3. For the mapping f : X ---, Y, the following statements are valid: (1) f is fuzzy pre-semi-irresolute ~ f is fuzzy pre-semi-continuous. (2) f is fuzzy continuous 5+ f is fuzzy pre-semiirresolute.
fuzzy pre-semi-irresolute if f - l (B) is fuzzy pre-semiopen in X for each fuzzy pre-semi-open set B in Y.
Example 3.4. Let X = { a , b , c } , 6={Ox,A,B, Ix} and
Theorem 3.2. For a mapping f : X --+ Y the following are equivalent: (1) f is fuzzy pre-semi-irresolute. (2) f - ~(B) is fuzzy pre-semi-closed in X for each Juzzy pre-semi-closed set B in Y. (3) f ( A ~ ) <<.( f ( A ) ) ~ for each fuzzy set A in X. (4) ( f - l ( B ) ) ~ <. f - l ( B ~ ) Jor each fuzz), set B inY. (5) For each fuzzy point x~ in X and each Juzzy pre-semi-open set V in Y with f(x~) E V, there exists a fuzzy pre-semi-open set U in X such that x~ E U and f ( U ) <~ V. (6) For each fuzzy point x~ in X and each Juzzy pre-semi-open set V in Y satisfying f ( x ~ ) q V there exists a fuzzy pre-semi-open set U in X such that x~qU and f ( U ) <~ V.
A(a) = 0.2,
A(b) = 0.4,
A(c) = 0.5;
B(a) =0.8,
B ( b ) = 0.8,
B ( c ) = 0.6;
C(a) =0.2,
C(b) =0.1,
C(c) =0.3.
Proof. (1) <==>(2): Straightforward and hence omitted. ( 1) =~ (3) =~ (4) =~ ( 1 ) is analogous to the proof of Theorem 2.10. (1) =~ (5): Let x~ be a fuzzy point in X and V be any fuzzy pre-semi-open set in Y such that f(x~) E V. Then x~ c f - J ( V ) . Since f is fuzzy pre-semi-irresolute, f I(V) is a fuzzy pre-semi-open set. Let f - l ( V ) = U. We have f ( U ) = f f l(V) ~
the following conditions are satisfied: (1) when x ~ y, either x~ has a fuzzy pre-semi-nbd U such that U qlY/~ or y/~ has a fuzzy pre-semi-nbd V such that V~ x~; (2) when x = y and e < f l (say), Y[3 has a fuzzy pre-semi-q-nbd V such that Vqi x~.
r = {0~, A, lx }, where
Consider the identity mapping f : (X, 6) ---+ (X,z). Then f is fuzzy continuous. Clearly, C is fuzzy presemi-open in (X,z), but f - 1 ( C ) = C is not fuzzy pre-semi-open in (X, 6). Thus f is not fuzzy presemi-irresolute.
4. Fuzzy pre-semi-separation axioms Definition 4.1. An fts X is called fuzzy pre-semi-To iff for every pair of distinct fuzzy points x~ and y[~,
Definition 4.2. An fts X is called fuzzy pre-semi-Ti iff for every pair of distinct fuzzy points x~ and Y/I, the following conditions are satisfied: (1) when x ¢; y, x~ has a fuzzy pre-semi-nbd U and y/~ has a fuzzy pre-semi-nbd V such that Uql y[~ and V~x~; (2) when x = y and ~ < f l (say), y/~ has a fuzzy pre-semi-q-nbd V such that V~ x~.
Definition 4.3. An fts X is called fuzzy pre-semi-T2 iff for every pair of distinct fuzzy points x~ and y/i, the following conditions are satisfied: (1) when x ~ y, x~ and y/~ have fuzzy pre-seminbds which are not q-coincident;
Bai Shi-Zhon9, Wan9 Wan-Liang/Fuzzy Sets and Systems 94 (1998) 261~68 (2) when x = y and 7 < f l (say), then x= has a fuzzy pre-semi-nbd U and y/~ has a fuzzy pre-semi-q-nbd V such that U~[ V. Remark 4.4. Obviously, fuzzy pre-semi-T2 ~ fuzzy pre-semi-T1 ~ fuzzy pre-semi-To. Also, fuzzy T~ axiom [6] ~ fuzzy semi-T/ axiom [7] ~ fuzzy presemi-T/axiom, for i = 0, 1,2. Theorem 4.5. X is fuzzy pre-semi-To iff for every pair of distinct fuzzy points x~ and Yl~, either x~ f[ (Yl~)~ or YI~ fg (x~)~. Proof. Let X be fuzzy pre-semi-To and x~ and y/~ be two distinct fuzzy points in X. When x # y, xl has a fuzzy pre-semi-nbd U such that Uq[ Yl*, or yl has a fuzzy pre-semi-nbd V such that V(lx~. Suppose xj has a fuzzy pre-semi-nbd U which is not q-coincident with y/~. Then U is a fuzzy pre-semi-q-nbd ofx~ and U~ YI~. Hence x~ ~ (Yl~)~. When x -- y and :~ < ~ (say), then y/~ has a fuzzy pre-semi-q-nbd which is not q-coincident with x~ and so in this case also Yl~ fg (x=)~. Conversely, let x~ and Y13 be two distinct fuzzy points in X. We suppose, without loss of generality, that x~ ~ (y/~)~. When x ¢ y, since x~ ~ (y/~)~, xl ~ (y/~)~ and hence ((y/~)~)t(x)= 1. Then ((y/~)~)' is a fuzzy presemi-nbd of x~ such that ((Yl~)~)'(1 Yl~" When x = y we have ~ >/~ and then x~ has a fuzzy pre-semi-q-nbd which is not q-coincident with YV Theorem 4.6. X is fuzzy pre-semi-T~ iff every fuzzy point x~ is fuzzy pre-semi-closed in X. Proof. Let X be fuzzy pre-semi-T1 and x= and y/~ be two distinct fuzzy points in X. When x # y, x~ and y/~ have fuzzy pre-semi-open pre-semi-nbd U and V, respectively such that U~ y/~ and V~x~. Then x~ ~ V'. Since V' is fuzzy presemi-closed, (x~)~ ~< V', equivalent Vql(x=)~. Thus (x~)~ ~< x~ i.e., x~ = (x~)~. Hence every fuzzy point x~ is fuzzy pre-semi-closed in X. When x = y, it is analogous to the proof of it above. Conversely, let x~ and y/~ be two distinct fuzzy points in X. When x # y, since x~ and y/~ are fuzzy pre-semiclosed in X, (x~) / and (y~)~ are fuzzy pre-semi-open.
265
Then (x~)' is a fuzzy pre-semi-nbd of Y13 and (y/~)~ is a fuzzy pre-semi-nbd of x~ such that (x~)'4 x~ and
(y~)'~ y~. When x = y and ~ < fl (say), obviously (x~)' is a fuzzy pre-semi-q-nbd of y/~ such that (x~)'qt x~. Theorem 4.7. X & fuzzy pre-semi-T2 iff for every
fuzzy point x~ in X,
v is a fuzzy pre-semi-nbd of
}
and for every x, y E X with x ~ y, there is a fuzzy pre-semi-nbd U of xl such that y ~ supp(U~). Proof. Let X be fuzzy pre-semi-T2. Let x~ and y~ be fuzzy points in X such that y~ ¢ (x~). To establish the required equality, it suffices to show the existence of a fuzzy pre-semi-nbd V o f x~ such that y~ ~ V~. If x # y, then there are fuzzy presemi-open sets U and V containing Yl and x~, respectively, such that Uq[ V. Then V is a fuzzy presemi-nbd ofx~ and U is a pre-semi-q-nbd o f y~ such that Ug[ V. Hence y/~ g' V~. If x = y, then fi > :~, and hence there are a fuzzy pre-semi-q-nbd U o f y/~ and a fuzzy pre-semi-nbd V of x~ such that Uq[ V. Then
y~¢ v_. Finally, for two distinct points x, y of X, since X is fuzzy pre-semi-T2, there exist fuzzy pre-semi-open sets U and V such that xl E U, Yl E V and U~[ V. Then V ' ( y ) = 0 and U~< V'. Since V ~ is fuzzy pre-semi-closed, U~ ~< V ~. Thus (U~)(y) = 0, i.e., y ~ supp(U~ ). Conversely, let x~ and y/~ be two distinct fuzzy points in X. When x ¢ y, we first suppose that at least one o f and/~ is less than 1, say 0 < e < 1. There exists a positive real number 2 with 0 < ~ + 2 < 1. By hypothesis, there exists a fuzzy pre-semi-nbd U of y/~ such that x;. ~ U_. Then x;~ has a fuzzy pre-semi-open presemi-q-nbd V such that V~ U. Now, 2 + V(x) > 1 so that V ( x ) > 1 - 2 > ~ and hence V is a fuzzy presemi-nbd ofx~ such that U~[ V, where U is a fuzzy presemi-nhd of YV In case ~ =/~ = 1, by hypothesis there is a fuzzy pre-semi-nbd U ofxt such that ( U ~ ) ( y ) = 0 . Then V = ( U ~ ) / is a fuzzy pre-semi-nbd of Yl such that U9[ V. When x = y and :~
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Bai Shi-Zhong. Wang Wan-Liang/ Fuzzy Sets and Systems 94 (1998) 261-268
there exists a fuzzy pre-semi-q-nbd V of y/~ such that Uq[ V. Considering the above cases, we conclude that X is fuzzy pre-semi-T2. Theorem 4.8. Let f : X --, Y be a one-to-one fuzz), pre-semi-irresolute mapping. I f Y is Juzzy pre-semiTi then so is"X, for i = O, 1,2. Proof. We prove only i = 1. Let x~ and y/~ be two distinct fuzzy points in X. When x ~ Y, we have f ( x ) # f ( y ) . Since Y is fuzzy pre-semi-Tl,(f(x))~ and (J(Y))[3 have fuzzy pre-semi-open pre-semi-nbd U and V, respectively, such that V(l(f(x))~ and U(t(f(y))f~. Then f - I ( U ) and f - l ( v ) are fuzzy pre-semi-nbds of x~ and yf~, respectively, such that f - l ( v ) ( t x ~ and f - l ( U ) ( l y / ~ . When x = y and ~<[3 (say), f ( x ) = f ( y ) . Since Y is fuzzy pre-semi-Tl, there is a fuzzy presemi-open pre-semi-q-nbd V of (f(Y))t3 such that V(1 (f(x))~. Then f - ~ ( V ) is a fuzzy pre-semi-q-nbd of Y/~ in X such that f l(V)q4x~. Thus X is fuzzy pre-semi-T1.
5. Fuzzy pre-semi-connectedness Definition 5.1. Two non-null fuzzy sets A and B in X are said to be fuzzy pre-semi-separated iffA ~ B~ and
SqA_ Remark 5.2. Clearly, if fuzzy sets A and B in X are fuzzy semi-separated [7] then they are fuzzy pre-semiseparated. That the converse need not be true is shown by Example 5.5. If A and B are fuzzy separated then they are fuzzy semi-separated (cf. [7]) and so they are fuzzy pre-semi-separated. Definition 5.3. A fuzzy set which cannot be expressed as the union of two fuzzy pre-semi-separated sets is said to be fuzzy pre-semi-connected. Remark 5.4. Clearly every fuzzy pre-semi-connected set is fuzzy semi-connected [7]. That the converse need not be true is shown by Example 5.5. Every fuzzy semi-connected set is fuzzy connected [7] and so every fuzzy pre-semi-connected set is fuzzy connected.
Example 5.5. Let X = [0, 1] and A, B, C be fuzzy sets in X defined as follows: A(x) = { 00.5 i f x = 0 , if0
1;
B(x)=
0.6 1
if x = 0 , if0
1;
C(x)=
0.2 0
if x = 0 , if0
1.
Clearly, 6 = {Ox,A, lx} is a fuzzy topology on X. By easy computations it follows that B~ = B and C~ = C. Then B ~ C~ and C ~ B_. Hence B and C are fuzzy pre-semi-separated. But B_ = lx and C_ =A t, so that B q C _ and C q B _ . Thus B and C are not fuzzy semiseparated. Again, B - B U C and B,C are fuzzy presemi-separated. It implies that B is not fuzzy pre-semi-connected. We show that B is fuzzy semi-connected. In fact, let B = D U E, where D and E are non-null fuzzy sets in X. Then either D(0) = 0.6 or E(0) = 0.6. Suppose D(0) = 0.6, then D = lx. Clearly, E q D _ . Thus D and E cannot be fuzzy semi-separated. Hence B is fuzzy semiconnected. Theorem 5.6. Let A and B be non-null fuzzy sets in X. ( 1 ) I f A, B are fuzzy pre-semi-separated, and C, D are non-nullJuzzy sets such that C <~A, D <~B, then C and D are also Juzzy pre-semi-separated. (2) I r A (1B and either both are fuzzy pre-serniopen or both are fuzzy pre-semi-closed, then A and B are fuzzy pre-semi-separated. (3) I f A,B are either both Ji~zzy pre-semi-open or both Juzz)' pre-semi-closed, then A N B ~ and B N A ~ are Ji4zzy pre-semi-separated. Proof. (1) and (2) are obvious. (3) Let A and B be both fuzzy pre-semi-open. Since A N B ~ ~ B ~, (A N B~)~ ~
Bai Shi-Zhong, Wang Wan-Liang/Fuzzy Sets and Systems 94 (1998) 261~68
Theorem 5.7. The non-null fuzzy sets A and B are fuzzy pre-semi-separated iff there ex&t two fuzzy pre-semi-open sets U and V such that A <~ U, B <,%V, A ~ V a n d B ~ U . Proof. For two fuzzy pre-semi-separated sets A and B, B ~< (A~)' = V (say) and A ~< ( B ~ ) ' = U (say), where V and U are clearly fuzzy pre-semi-open, then V~A~ andU~B~.ThusA~VandB~U. Conversely, let U and V be fuzzy pre-semi-open sets such that A <~ U, B <~ V, A (t V and B ~ U. Then A ~< V ~ a n d B ~ < U ~. H e n c e A ~ ~< W a n d B ~ ~
Theorem 5.8. Let A be a non-null fuzzy pre-semiconnected set in X. I f A <~B <~A~, then B is also fuzzy pre-semi-connected. Proof. I f B is not fuzzy pre-semi-connected inX, then there exist fuzzy pre-semi-separated sets C and D in X such that B = C t 3 D . Let E = A N C andF= A f3 D. Then A = E U F. Since E ~< C and F ~< D, by Theorem 5.6(1 ), E and F are fuzzy pre-semi-separated contradicting the fuzzy pre-semi-connectedness o f A. Thus B is fuzzy pre-semi-connected.
267
we have Aj <~B, or Aj <~ C. We may assume that Aj <~B. Then for each i ~ j, Ai <~B. In fact, ifAj ~B, then Ai <~ C. By Theorem 5.6(1) Aj and Ai are fuzzy pre-semi-separated. This is a contradiction. Hence for each i C I , Ai <~B. It follows that A = [..J{Ai: i E I } ~< B, clearly C = 0x. Thus A is fuzzy pre-semiconnected. [] The following corollary is obvious.
Corollary 5.11. Let {Ai: i E I} be a collection of fuzzy pre-semi-connected sets in X. If N{ Ai: i E I} ¢ Ox, then U{Ai: i E I} is fuzzy pre-semi-connected. Theorem 5.12. Let f : X---~ Y be a one-to-one fuzzy pre-semi-irresolute mapping, l f A is fuzzy pre-semiconnected in X, so is f ( A ) in Y. ProoL I f possible, let f ( A ) be not fuzzy pre-semiconnected in Y. Then there exist fuzzy pre-semiseparated sets B and C in Y such that f ( A ) = B U C. Since B and C are fuzzy pre-semi-separated, by Theorem 5.7 there exist two fuzzy pre-semiopen sets U a n d V s u c h t h a t B ~ < U, C~< V , B ~ V a n d C ~ U . Now, f being fuzzy pre-semi-irresolute, f - l ( U ) and f - l ( V ) are fuzzy pre-semi-open sets in X, and
A = f-if(A)
= f - l ( B U C) = f - l ( B ) U f - l ( C ) .
Theorem 5.9. Let A be a non-null fuzzy pre-semiconnected set in X, and C and D be two fuzzy presemi-separated sets in X. IrA <~ C to D, then A <<,C orA <~D.
ForB(tVandCqiU, w e h a v e B ~ < V~ and C~< U ~, i.e., f - l ( B ) <~ ( f ( V ) ) ' and f - l ( c ) ~< ( f - l ( U ) ) ' . Hence f - l ( B ) (t f - l ( V ) and f - l ( C ) (t f - l ( U ) . By Theorem 5.7 f - l ( B ) and f - l ( C ) are fuzzy pre-semiseparated in X. Thus we arrive at a contradiction. []
Proof. Suppose that A N D ~ 0x and A A C ~ 0x. By Theorem 5.6(1 ), A A C and A N D become fuzzy presemi-separated sets such that A = (A N C) U (A A D), contradicting the fuzzy pre-semi-connectedness of A. HenceA ~
Corollary 5.13. Let f : X ~ Y be a fuzzy pre-semiirresolute mapping. I f X &fuzzy pre-semi-connected, so is f ( X ) . References
Theorem 5.10. Let {Ai: i E I } be a collection of fuzzy pre-semi-connected sets in X. Suppose there exists a j E I such that Ai and Aj are not fuzzy pre-semi-separated for each i E 1. Then A = U{Ai: i E I} is fuzzy pre-semi-connected. Proof. If A is not fuzzy pre-semi-connected, then A =Bt0 C, where B and C are fuzzy pre-semi-separated in X. Since Ai is fuzzy pre-semi-connected for each i E I, by Theorem 5.9, A i <~B or Ai <~ C. Specifically,
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