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Fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 67 (1994) 359 364
Fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings J i n H a n P a r k a'*, B u Y o u n g L e e b aDepartment of Natural Sciences, Pusan National University of Technology. Pusan. 608-739, South Korea bDepartment of Mathematics, Dong-A University', Pusan. 604-714. South Korea
Received January 1993; revised March 1994
Abstract The aim of this paper is to introduce and discuss the concepts of fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings. Keywords: Fuzzy semi-preopen sets and fuzzy semi-preclosed sets; Fuzzy semi-precontinuous; Fuzzy semi-pre-T o, fuzzy semi-pre-T 1 and fuzzy semi-pre-T2
1. Introduction and preliminaries Weaker forms of fuzzy continuity between fuzzy topological spaces have been considered by many authors [1,2,4,6,11,13] using the concepts of fuzzy semi-open sets [1], fuzzy regularly open sets [1], fuzzy preopen sets and fuzzy or-open sets [2]. Recently, Bin Shahna [2] introduced and investigated fuzzy strong semi-continuity and fuzzy precontinuity between fuzzy topological spaces, one of which was independent and the other strictly stronger than fuzzy semi-continuity [1]. Also Park et al. [12] studied some characterizing theorems for these two mappings and showed that fuzzy precontinuity and fuzzy almost continuity due to Mukherjee and Sinha [11] were equivalent concepts. In Section 2, we introduce the concept of fuzzy semi-preopen sets which is weaker than any of the concepts of fuzzy semi-open or fuzzy preopen.
*Corresponding author.
Using this concept, in Section 3 we define and study fuzzy semi-precontinuous mapping between fuzzy topological spaces. Let X be a nonempty set. Fuzzy sets of X will be denoted by capital letters such as A, B, C, etc. and fuzzy points will be denoted by x~ and yp. And x~ • A either means that fuzzy point x~ takes its nonzero value in (0, 1) at the support x and ct < A(x) (see [15]) or fuzzy point x~ takes its nonzero value in (0, 1] and ~ ~< A(x) (see [8]). In particular x, • 1A means c~< A(x) and 0 < ~ < 1. If for a fuzzy point x, and a fuzzy set A, we have • + A(x) > 1, then this case, which is denoted by Ming and Ming [8] as 'x, quasi-coincident with A', will be denoted by x, q A. For definitions and results not explained in this paper, we refer to the papers [1, 3, 5 8, 16], assuming them to be well known. The words 'neighborhood', 'fuzzy' and 'fuzzy topological space' will be abbreviated as 'nbd', 'f.' and 'fts' respectively. Simply by X, Y,Z we shall denote fts's (X, zx),(Y,~r),(Z, rz), a n d f : X ~ Y will mean that f i s a mapping from (X, zx) to (Y, zr). Also by Int A,
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CI A and A' we shall denote respectively the interior, closure and complement of the f.set A of fts. The following elementary results will be used in the sequel.
Theorem 1.1 [3, 16]. Let f: X ~ Y be a mapping, and A and B be f.sets of X and Y respectively. The following statements are true: (a) f(A)' <~f(A'), f - 1(B') = f - ~(B)'. (b) A <~f-I(f(A)), f ( f - ~ ( B ) ) <~ B. (c) l f f is injective, then f - 1(f(A)) = A. (d) If f is surjective, then f ( f - I(B)) = B. (e) l f f is bijective, then f(A)' = f(A'). Theorem 1.2 [16,1. If A is a f.set of X such that A(x) ~ Ofor x ~ X, then
A=
U
O
x~=
U
X~.
Theorem 2.1. If A is f.set of a fts X, then the following are equivalent: (a) A is f.semi-open, (b) A ~< Cl Int A, (c) C1A = C11nt A. Proof. (a) ¢~ (b) are proved in [1, Theorem 4.2]. (b) ¢~ (c) Since A ~< CI Int A, C1 A ~< C1 C11nt A = C1 Int A ~< C1 A. Hence C1 A = CI Int A. (c) ¢~ (b) Clear. [] Theorem 2.2. If A is a f.set of a fts X, then the following are equivalent: (a) B is f. semi-closed, (b) Int C1 B ~< B, (c) Int C1 B = Int B. Proof. Follows easily by taking complements from Theorem 2.1. []
0<~< A(x)
2. Fuzzy semi-preopen and fuzzy semi-preclosed sets Definition 2.1. A f. set A of fts X is called (a) f. semi-open set (f. semi-closed set) if there exists a f.open set if.closed set) U such that U ~< A ~< C1 U(Int U ~< A ~< U) [1], (b) f.preopen set (f.preclosed set) if A ~< I n t C l A (A ~> CI Int A) [2, 14], (c) f. semi-preopen set (f. semi-preclosed set) if there exists a f.preopen set (f.preclosed set) U such that U ~ < A ~ < C I U ( I n t U ~ < A ~ < U). It is obvious that every f.semi-open (f.preopen) set is a f. semi-preopen set. Example 2.1. shows that the converse is false. Example 2.1. Let U1,U2 and X = {a, b, c} defined as
U3
U~(a) = 0.8,
Ul(b) = 0.7,
U~(c) = 0.9,
U2(a) = 0.5,
U2(b) = 0.3,
U2(c) = 0.6,
U3(a) = 0.3,
Ua(b) = 0.4,
U3(c) = 0.3,
U4(a) = 0.2,
U,(b) = 0.6,
U4(c) = 0.2.
be f.sets on
Clearly zl = {1x,Ox, U1,U2,U3, U2 u U3, U 2 N U3} is f.topology on X. In (X, zl ), U4 is f.semi-preopen set but neither f.semi-open nor f.preopen.
Theorem 2.3. If A is f.semi-preopen (f.semi-preclosed) set of a fts X, then A <~ CIlntC1A (Int Cllnt A ~< A). Proof. Let A be f.semi-preopen set. Then there exists a f.preopen set U such that U ~< A ~< Ci U. This implies C1A = C1 U. Since U is f.preopen, we have A ~< CI U ~< CI Int Ci U ~< C1 Int C1A. Hence A ~< C11nt C1 A. For f.semi-preclosed set, the proof is similar. []
Theorem 2.4. (a) Any union off.semi-preopen sets is a f.semi-preopen set, and (b) any intersection of f.semi-preclosed sets is a f.preclosed set. Proof. (a) Let {A,} be a collection of f.semi-preopen sets of a fts X. Then there exists a f.preopen set U, such that U, ~< A, ~< CI U, for each ~. Thus, by Lemma 3.1 in [1] and Theorem 3.2 in [14"1, U, U, ~< U,A, ~< U,C1 U, ~< CI(U, U,), and U, U, is f.preopen set. Hence U,A, is f.semi-preopen set. (b) Follows easily by taking complements. [] Definition 2.2. A f. set A of a fts X is called f. semipre-nbd (f.semi-pre-q-nbd) of a f.point x~ if there exists a f.semi-preopen set U such that x~ E U ~< A (x~q U ~< A).
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Theorem 2.5. A f.set A is f.semi-preopen if and only iffor every f .point x, q A, A is f.semi-pre-q-nbd of x,. Proof. Let A be a f.semi-preopen set of a fts X. Then clearly if any f.point x, with x , q A, then A is f.semi-pre-q-nbd of x,. Conversely, let x, e 1A. Since 0 4: ~ < A(x), we have x~ , q A and so by hypothesis A is f.semipre-q-nbd of x~_,. Then there exists a f.semi-preopen set U such that x a _ , q U ~< A which implies x, 6 ~U ~< A. Hence, by Theorem 2.4, A is a f.semipreopen set. ~] Note that the closure of f.semi-preopen set is a f.regularly closed and the interior of a f.semipreclosed set is a f.regularly open set.
Theorem 2.6. Let X and Y be fts's such that X is product related to Y. Then the product U x V of a f .semi-preopen set U in X and a f.semi-preopen set V in Y is a f.semi-preopen set in the product fts XxY. Proof. Since U and V are f.semi-preopen sets, there exist f.preopen sets U1 and V1 in X and Y respectively such that U~ ~< U ~< CIU1 and I/1 ~< V~< Cll/1. By Theorem 1.5 in [2] and Theorem 3.10 in [1], U1 × V~ is f.preopen set and U 1 x V1 ~ U x V ~ C1 U1 × C1 V1 -- CI(Ux × Wl). Hence U x V is f.semi-preopen set.
[]
Theorem 2.7. If a f.set A is f.semi-preopen (f.semipreclosed) and f.semi-closed (f.semi-open), then A is f. semi-open (f. semi-closed).
361
Proof. We see that a f.semi-preopen set A ~< B is precisely the complement of a f.semi-preclosed set C/> A'. Thus Spint A = U {C'] C is f.semi-preclosed and C/> A'} = 1 - 0 { c a C is f.semi-preclosed and C ~> A'} = 1 - Spcl A'. Hence Spcl A' = (Spint A)'. The other proof is similar.
[]
Theorem 2.9. Let x, be any f.point in a fis X and A be anyf.set of X. Then x~ e Spcl A if and only iffor every f. semi-pre-q-nbd U of x,, U q A. Proof. Suppose that there exists a f.semi-pre-q-nbd U of x, such that U qt A. Then there exists a f. semipreopen set U1 such that x, q U1 ~< U and U1 qj A, Since U~ is f.semi-preclosed set containing A, we have Spcl A ~< U~. Also since x~¢ U~, we have x, ~ Spcl A which is a contradiction. Conversely, suppose x , ¢ SpclA. Then there exists a f.semi-preclosed set B such that x, ~ B and A ~< B. Hence B' is f.semi-preopen set such that x, q B' and A ~ B' which is a contradiction. []
Theorem 2.10. I r A is any f.set and B is a f.semipreopen set of a fis X such that A(IB, then Spcl A q[ B.
Proof. This follows from Theorem 2.1, 2.2 and 2.3 []
Proof, Suppose that Spcl A q B. Then there exists a x ~ X such that S p c l A ( x ) + B ( x ) > 1. Putting Spcl A(x) = ~, B is f. semi-pre-q-nbd of x, such that A ql B. Hence x, ~ Spcl A. Thus we arrive at a contradiction. []
Definition 2.3. Let A be any f.set of a fts X. Then f.semi-preclosure (Spcl) and f.semi-preinterior (Spint) of A are defined as follows:
3. Characterizations of fuzzy semi-precontinuous mappings
Spcl A = (~ {B{ B is f.semi-preclosed and A ~< B}, Spint A = U {BIB is f.semi-preopen and B ~< A}.
Theorem 2.8. Let A be any f.set of a fis X. Then Spcl A' = ( S p i n t A)' and Spint A' = (Spcl A)'.
Definition 3.1. A m a p p i n g f : X ~ Y is called (a) f.semi-continuous (f.s.c.) if the inverse of f.open set is f.semi-open set [1], (b) f.precontinuous (f.p.c.) if the inverse of f.open set is f.preopen set [23,
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(c) f.semi-precontinuous (f.s.p.c.) if the inverse of f.open set is f.semi-preopen set. The implications contained in following diagram are true.
Theorem 3.2. A mapping f : X ~ Y is f.s.p.c, if and only if for every f .point x~ in X and every f .nbd V of f ( x , ) = f ( x ) , , there exists f.semi-pre-nbd U of x, such that f ( U ) <~ V.
f.s.c. f.c. -o f.s.s.c,
f I ( C 1 B ' ) = f - l ( B ' ) . By Theorem 2.8, we have f - 1(B)' >/Spcl(f - ~(B)') =(Spint f - ~(n))'. Hence f - I ( B ) is f.semi-preopen set.
f.s.p.c. f.p.c.
where f.c. = f.continuity semi-continuity [2].
[3],
f.s.s.c. = f.strong
From [2, Remark 2.2] and the following Example 3.1 show that the reverse need not be true Example 3.1. Let U1, U2, U3 and U4 be f.sets of X = {a,b,c} as described in Example 2.1. Let zl = {1x,Ox, U1,U2,U3,U2f-~U3,U2k_)U3} and "C2= {lx,0x, U4} be f.topologies on X a n d f : ( X , zl)-o (X, z2) be a mapping defined as follows: f(a) = a,f(b) = b andf(c) = c. T h e n f i s f.s.p.c, but neither f.s.c, nor f.p.c. Theorem 3.1. Let f: X --* Y be a mapping. Then the following are equivalent: (a) f is f.s.p.c., (b) the inverse off.closed set of Y is f.semi-preclosed set, (c) f(Spcl A) <<,Cl f(A) for each f.set A of X, (d) Spclf - 1(B) ~
Proof. Let f : X ~ Y be f.s.p.c.. Suppose x, is any f.point in X and Vis any f.nbd off(x,). Then there exists f.open set B such thatf(x,) • B ~< V. By f.semiprecontinuity off, f - I(B) is f.semi-preopen in X and x~ • f - x(B) ~
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Proof. Let V = (j,.p(V, × Wt~),where V, and Wp are f. open sets in Y1 and Y2 respectively, be a f. open set in Y1 x Y2. From Lemma 2.1 and 2.3 in [1], we have
(A xA) '(v)= U(A
xA)-l(v
= Uif;'(V,)xA
,x
w#)
'(wp)).
363
Definition 4.3. A fts X is called f.semi-pre-T2 if for every distinct two f.points x, and y~, the following conditions are satisfied: (a) When x :~ y, x, and yo have f.semi-pre-nbds which are not q-coincident. (b) When x = y and ~ < f l (say), then x, has a f.semi-pre-nbd U and y~ has a f.semi-pre-qnbd V such that U q~ V.
ct,fl
Since fl a n d f z are f.s.p.c.,fz- 1(V,) and f2-1(W~) are f.semi-preopen sets, and by Theorem 2.4 and 2.6, it follows that (fl x f2) a(V) is f.semi-preopen set. Hence f~ ×f2 is f.s.p.c. []
Theorem 3.5. Let f: X ~ Y be a mapping from a fis X to a fis Y, and g : X ~ X × Y be the graph o f f If 9 is fs.p.c., then so is f Proof. Let V be a f.open set in Y. Then, b y Lemma 2.4 in [-1], we have f - l ( V ) = l ~ f - l ( V ) = 9-1(1 x V). Since 9 is f.s.p.c, and 1 x Vis a f.open set in X × Y, f - 1(V) is f.semi-preopen set of X. H e n c e f i s f.s.p.c. []
4. Separation axioms Definition 4.1. A fts X is called f. semi-pre-To if for every distinct two f.points x, and yp, the following conditions are satisfied: (a) When x :~ y, either x~ has a f.semi-pre-nbd which is not q-coincident with yp, or y~ has a f.semi-pre-nbd which is not q-coincident with Xa.
(b) When x = y and ~ < fl (say), there is a f.semipre-q-nbd of yp which is not q-coincident with X~t.
Definition 4.2. A fts X is called f.semi-pre-T1 if for every distinct two f.points x~ and ya, the following conditions are satisfied: (a) When x ¢ y, x~ has a f.semi-pre-nbd U and yp has a f.semi-pre-nbd V such that Yaql U and x~¢l V. (b) When x = y and ~ < fl (say), then there exists a f.semi-pre-q-nbd V of y~ such that x , ~ V.
It is obvious that f.semi-pre-T2 ~ f.semi-pre-T1 ~f.semi-pre-To, and f.Ti axiom [5] ~f.semi-Ti axiom [7] ~ f.semi-pre-T~ axiom, for i -- 0, l, 2.
Theorem 4.1. A fis X is f.semi-pre-To if and only if for every pair of distinct x~ and y~, either x~ q~Spcl(y~) or yp ~ Spcl(x~). Proof. Let X be f.semi-pre-To, and x, and yp be two distinct f.point in X. When x ¢ y, Xl has a f.semi-pre-nbd U such that ypflU, or Yl has a f.semi-pre-nbd V such that x , ~ V. Suppose xl has a f.semi-pre-nbd U which is not q-coincident with yp. Then U is f.semipre-q-nbd of x, and y ~ U. This implies that x~ ¢ Spcl (y~). When x = y and ~ < fl (say), then y~ has a semipre-q-nbd which is not q-coincident with x, and thus y~ ¢ Spcl(x,). Conversely, let x, and y~ be two distinct f.points in X. Without loss of generality we suppose that x, ¢ Spcl(yp). When x # y, since x, ¢ Spcl(y~), Xl¢Spcl(y~) and so [Spcl(y~)]'(x)= 1. Hence [Spcl(yp)]' is a f.semi-pre-nbd of x, such that y ~ [ S p c l ( y ~ ) ] ' . Also when x = y we must have > fl and then x, has a semi-pre-q-nbd which is not q-coincident with yp. []
Theorem 4.2. A fis X is f.semi-pre-Tl if and only if for every f.point x, is f.semi-preclosed in X. Proof. The proof is easy and hence omitted.
[]
Theorem 4.3. A fis X is f.semi-pre-T2 if and only if for every f.point x, in X, x~ = N{Spcl VI V is f.semi-pre-nbd of x,} and for every x, y e X with x v~y, there is af.semi-pre-nbd U of xl such that y ¢ (Spcl U)o, where (Spcl U)o is support of Spcl U.
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Proof. Let x~ and yg be f.points in X such that y~ ~ {x~}. If x :~ y, then there are f.semi-preopen sets U and V containing YI and x~ respectively such that U ¢ V. Then V is a f.semi-pre-nbd of x~ and U is a f.semi-pre-q-nbd of y~ such that U~l V. Hence y~ q~ Spcl V. If x = y, then ~ < fl, and hence there are a f.semi-pre-q-nbd U of y~ and a f.semipre-nbd V of x~ such that U ¢ V. Then yp ¢ Spcl V. Finally, for distinct two point x, y of X, since X is f.semi-pre-T2, there exist f.semi-preopen sets U and V such that x~ e U, y~ ~ V and U ¢ V. Since V' is f.semi-preclosed set containing U, Spcl U ~< V'. Hence y ¢ (Spcl U)o. Conversely, let x, and Yt~ be two distinct f.points in X. W h e n x # y. We first suppose that at least one of ct and fl is less than 1, say 0 < ~ < 1. Then there exists a positive real n u m b e r 2 with 0 < ~ + 2 < 1. By hypothesis, there exists a f.semi-pre-nbd U of y¢ such that xz ¢ Spcl U. Then there exists a f.semipreopen set V such that x a q V and V¢ U. Since x z q V, V(x) > 1 - 2 > ~ and hence V is a f.semipre-nbd of x, such that U ¢ V. Next if ~ = fl = 1, by hypothesis there exists a f.semi-pre-nbd U of Xl such that Spcl U ( y ) = O. Then V = (Spcl U)' is a f.semi-pre-nbd of Yl such that U ¢ V. W h e n x ¢ y and c~ < fl (say), then there exists a f.semi-pre-nbd U of x, such that y~ ~ Spcl U. Hence there exists a f.semi-pre-q-nbd V of yo such that U ¢ V. Therefore X is f.semi-pre-T2. []
Theorem 4.4. L e t f : X ~ Y be injective and fs.p.c. I f Y i s f . T i , then X i s f . s e m i - p r e - T i , f o r i = 0, 1, 2.
Proof. We give a p r o o f for i = 1 only; the other cases being similar, are omitted. Let x, and yp be distinct two f.points in X. W h e n x 4= y, we h a v e f ( x ) e l ( y ) , and by the f.T1 property of Y, there exist f.nbds U and V off(x)~ and f ( y ) ~ respectively such that f ( x ) , q t V and f(Y)t~ qi U. Since f is f.s.p.c., f - 1(U) and f - x(V) are f.semi-pre-nbds of x, and y~ respectively such that x , ~ l f - l ( V ) and y o q i f - l(U). When x = y and ~ < fl (say), thenf(x) = f(y). Since Y is f.T~, there exists a f.q-nbd V off(y)t3 such that
f ( x ) , ~ V. Then f - I(V) is f.semi-pre-q-nbd of yp such that x ~ f - l ( V ) . Hence X is f.semi-preT1. []
References [1] K.K. Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82 (1981) 14-32. [2] A.S. Bin Shahna, On fuzzy strong semicontinuity and fuzzy precontinuity, Fuzzy Sets and Systems 44 (1991) 303-308. [3] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182-190. [4] J.R. Choi, B.Y. Lee and J.H. Park, On fuzzy 0-continuous mappings, Fuzzy Sets and Systems 54 (1993) 107 113. [5] S. Ganguly and S. Saha, On separation axioms and Tifuzzy continuity, Fuzzy Sets and Systems 16 (1985) 265-275. [6] S. Ganguly and S. Saha, A note on semi-open sets in fuzzy topological spaces, Fuzzy Sets and Systems 18 (1986) 83-96. [7] B. Ghosh, Semi-continuous and semi-closed mappings and semi-connectedness in fuzzy setting, Fuzzy Sets and Systems 35 (1990) 345-355. [8] P.P. Ming and L.Y. Ming, Fuzzy topology I, Neighborhood structure of fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571 599. [9] M.N. Mukherjee and B. Ghosh, Some stronger forms of fuzzy continuous mappings on fuzzy topological spaces, Fuzzy Sets and Systems 38 (1990) 375 387. [10] M.N. Mukherjee and B. Ghosh, Fuzzy semiregularization topologies and fuzzy submaximal spaces, Fuzzy Sets and Systems 44 (1991) 283-294. [l 1] M.N. Mukherjee and S.P. Sinha, On some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topological spaces, Fuzzy Sets and Systems 32 (1989) 103-114. [12] J.H. Park, B.Y. Lee and J.R. Choi, On fuzzy strong semicontinuity, fuzzy precontinuity and fuzzy weakly semicontinuity, manuscript (submitted). [13] Z. Petricevic, On fuzzy semiregularization, separation properties and mappings, Indian J. Pure Appl. Math. 22 (1991) 971-982. [14] M.K. Singal and N. Prakash, Fuzzy preopen sets and fuzzy preseparation axioms, Fuzzy Sets and Systems 44 (1991) 273-281. [15] R. Srivastava, S.N. Lal and A.K. Srivastava, Fuzzy Hausdorff topological spaces, J. Math. Anal. Appl. gl (1981) 499 506. [16] T.H. Yalvac, Fuzzy sets and functions on fuzzy topological spaces, J. Math. Anal. Appl. 126 (1987) 409-423.
sets and systems ELSEVIER
Fuzzy Sets and Systems 67 (1994) 359 364
Fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings J i n H a n P a r k a'*, B u Y o u n g L e e b aDepartment of Natural Sciences, Pusan National University of Technology. Pusan. 608-739, South Korea bDepartment of Mathematics, Dong-A University', Pusan. 604-714. South Korea
Received January 1993; revised March 1994
Abstract The aim of this paper is to introduce and discuss the concepts of fuzzy semi-preopen sets and fuzzy semi-precontinuous mappings. Keywords: Fuzzy semi-preopen sets and fuzzy semi-preclosed sets; Fuzzy semi-precontinuous; Fuzzy semi-pre-T o, fuzzy semi-pre-T 1 and fuzzy semi-pre-T2
1. Introduction and preliminaries Weaker forms of fuzzy continuity between fuzzy topological spaces have been considered by many authors [1,2,4,6,11,13] using the concepts of fuzzy semi-open sets [1], fuzzy regularly open sets [1], fuzzy preopen sets and fuzzy or-open sets [2]. Recently, Bin Shahna [2] introduced and investigated fuzzy strong semi-continuity and fuzzy precontinuity between fuzzy topological spaces, one of which was independent and the other strictly stronger than fuzzy semi-continuity [1]. Also Park et al. [12] studied some characterizing theorems for these two mappings and showed that fuzzy precontinuity and fuzzy almost continuity due to Mukherjee and Sinha [11] were equivalent concepts. In Section 2, we introduce the concept of fuzzy semi-preopen sets which is weaker than any of the concepts of fuzzy semi-open or fuzzy preopen.
*Corresponding author.
Using this concept, in Section 3 we define and study fuzzy semi-precontinuous mapping between fuzzy topological spaces. Let X be a nonempty set. Fuzzy sets of X will be denoted by capital letters such as A, B, C, etc. and fuzzy points will be denoted by x~ and yp. And x~ • A either means that fuzzy point x~ takes its nonzero value in (0, 1) at the support x and ct < A(x) (see [15]) or fuzzy point x~ takes its nonzero value in (0, 1] and ~ ~< A(x) (see [8]). In particular x, • 1A means c~< A(x) and 0 < ~ < 1. If for a fuzzy point x, and a fuzzy set A, we have • + A(x) > 1, then this case, which is denoted by Ming and Ming [8] as 'x, quasi-coincident with A', will be denoted by x, q A. For definitions and results not explained in this paper, we refer to the papers [1, 3, 5 8, 16], assuming them to be well known. The words 'neighborhood', 'fuzzy' and 'fuzzy topological space' will be abbreviated as 'nbd', 'f.' and 'fts' respectively. Simply by X, Y,Z we shall denote fts's (X, zx),(Y,~r),(Z, rz), a n d f : X ~ Y will mean that f i s a mapping from (X, zx) to (Y, zr). Also by Int A,
0165-0114/94/$07.00 © 1994 ElsevierScience B.V. All rights reserved SSDI 0165-01 14(94)001 33-R
J.H. Park, B.Y. Lee / Fuzz)' Sets and Systems 67 (1994) 359-364
360
CI A and A' we shall denote respectively the interior, closure and complement of the f.set A of fts. The following elementary results will be used in the sequel.
Theorem 1.1 [3, 16]. Let f: X ~ Y be a mapping, and A and B be f.sets of X and Y respectively. The following statements are true: (a) f(A)' <~f(A'), f - 1(B') = f - ~(B)'. (b) A <~f-I(f(A)), f ( f - ~ ( B ) ) <~ B. (c) l f f is injective, then f - 1(f(A)) = A. (d) If f is surjective, then f ( f - I(B)) = B. (e) l f f is bijective, then f(A)' = f(A'). Theorem 1.2 [16,1. If A is a f.set of X such that A(x) ~ Ofor x ~ X, then
A=
U
O
x~=
U
X~.
Theorem 2.1. If A is f.set of a fts X, then the following are equivalent: (a) A is f.semi-open, (b) A ~< Cl Int A, (c) C1A = C11nt A. Proof. (a) ¢~ (b) are proved in [1, Theorem 4.2]. (b) ¢~ (c) Since A ~< CI Int A, C1 A ~< C1 C11nt A = C1 Int A ~< C1 A. Hence C1 A = CI Int A. (c) ¢~ (b) Clear. [] Theorem 2.2. If A is a f.set of a fts X, then the following are equivalent: (a) B is f. semi-closed, (b) Int C1 B ~< B, (c) Int C1 B = Int B. Proof. Follows easily by taking complements from Theorem 2.1. []
0<~< A(x)
2. Fuzzy semi-preopen and fuzzy semi-preclosed sets Definition 2.1. A f. set A of fts X is called (a) f. semi-open set (f. semi-closed set) if there exists a f.open set if.closed set) U such that U ~< A ~< C1 U(Int U ~< A ~< U) [1], (b) f.preopen set (f.preclosed set) if A ~< I n t C l A (A ~> CI Int A) [2, 14], (c) f. semi-preopen set (f. semi-preclosed set) if there exists a f.preopen set (f.preclosed set) U such that U ~ < A ~ < C I U ( I n t U ~ < A ~ < U). It is obvious that every f.semi-open (f.preopen) set is a f. semi-preopen set. Example 2.1. shows that the converse is false. Example 2.1. Let U1,U2 and X = {a, b, c} defined as
U3
U~(a) = 0.8,
Ul(b) = 0.7,
U~(c) = 0.9,
U2(a) = 0.5,
U2(b) = 0.3,
U2(c) = 0.6,
U3(a) = 0.3,
Ua(b) = 0.4,
U3(c) = 0.3,
U4(a) = 0.2,
U,(b) = 0.6,
U4(c) = 0.2.
be f.sets on
Clearly zl = {1x,Ox, U1,U2,U3, U2 u U3, U 2 N U3} is f.topology on X. In (X, zl ), U4 is f.semi-preopen set but neither f.semi-open nor f.preopen.
Theorem 2.3. If A is f.semi-preopen (f.semi-preclosed) set of a fts X, then A <~ CIlntC1A (Int Cllnt A ~< A). Proof. Let A be f.semi-preopen set. Then there exists a f.preopen set U such that U ~< A ~< Ci U. This implies C1A = C1 U. Since U is f.preopen, we have A ~< CI U ~< CI Int Ci U ~< C1 Int C1A. Hence A ~< C11nt C1 A. For f.semi-preclosed set, the proof is similar. []
Theorem 2.4. (a) Any union off.semi-preopen sets is a f.semi-preopen set, and (b) any intersection of f.semi-preclosed sets is a f.preclosed set. Proof. (a) Let {A,} be a collection of f.semi-preopen sets of a fts X. Then there exists a f.preopen set U, such that U, ~< A, ~< CI U, for each ~. Thus, by Lemma 3.1 in [1] and Theorem 3.2 in [14"1, U, U, ~< U,A, ~< U,C1 U, ~< CI(U, U,), and U, U, is f.preopen set. Hence U,A, is f.semi-preopen set. (b) Follows easily by taking complements. [] Definition 2.2. A f. set A of a fts X is called f. semipre-nbd (f.semi-pre-q-nbd) of a f.point x~ if there exists a f.semi-preopen set U such that x~ E U ~< A (x~q U ~< A).
J.H. Park, B.Y. Lee / Fuzz)' Sets and Systems 67 (1994) 35~364
Theorem 2.5. A f.set A is f.semi-preopen if and only iffor every f .point x, q A, A is f.semi-pre-q-nbd of x,. Proof. Let A be a f.semi-preopen set of a fts X. Then clearly if any f.point x, with x , q A, then A is f.semi-pre-q-nbd of x,. Conversely, let x, e 1A. Since 0 4: ~ < A(x), we have x~ , q A and so by hypothesis A is f.semipre-q-nbd of x~_,. Then there exists a f.semi-preopen set U such that x a _ , q U ~< A which implies x, 6 ~U ~< A. Hence, by Theorem 2.4, A is a f.semipreopen set. ~] Note that the closure of f.semi-preopen set is a f.regularly closed and the interior of a f.semipreclosed set is a f.regularly open set.
Theorem 2.6. Let X and Y be fts's such that X is product related to Y. Then the product U x V of a f .semi-preopen set U in X and a f.semi-preopen set V in Y is a f.semi-preopen set in the product fts XxY. Proof. Since U and V are f.semi-preopen sets, there exist f.preopen sets U1 and V1 in X and Y respectively such that U~ ~< U ~< CIU1 and I/1 ~< V~< Cll/1. By Theorem 1.5 in [2] and Theorem 3.10 in [1], U1 × V~ is f.preopen set and U 1 x V1 ~ U x V ~ C1 U1 × C1 V1 -- CI(Ux × Wl). Hence U x V is f.semi-preopen set.
[]
Theorem 2.7. If a f.set A is f.semi-preopen (f.semipreclosed) and f.semi-closed (f.semi-open), then A is f. semi-open (f. semi-closed).
361
Proof. We see that a f.semi-preopen set A ~< B is precisely the complement of a f.semi-preclosed set C/> A'. Thus Spint A = U {C'] C is f.semi-preclosed and C/> A'} = 1 - 0 { c a C is f.semi-preclosed and C ~> A'} = 1 - Spcl A'. Hence Spcl A' = (Spint A)'. The other proof is similar.
[]
Theorem 2.9. Let x, be any f.point in a fis X and A be anyf.set of X. Then x~ e Spcl A if and only iffor every f. semi-pre-q-nbd U of x,, U q A. Proof. Suppose that there exists a f.semi-pre-q-nbd U of x, such that U qt A. Then there exists a f. semipreopen set U1 such that x, q U1 ~< U and U1 qj A, Since U~ is f.semi-preclosed set containing A, we have Spcl A ~< U~. Also since x~¢ U~, we have x, ~ Spcl A which is a contradiction. Conversely, suppose x , ¢ SpclA. Then there exists a f.semi-preclosed set B such that x, ~ B and A ~< B. Hence B' is f.semi-preopen set such that x, q B' and A ~ B' which is a contradiction. []
Theorem 2.10. I r A is any f.set and B is a f.semipreopen set of a fis X such that A(IB, then Spcl A q[ B.
Proof. This follows from Theorem 2.1, 2.2 and 2.3 []
Proof, Suppose that Spcl A q B. Then there exists a x ~ X such that S p c l A ( x ) + B ( x ) > 1. Putting Spcl A(x) = ~, B is f. semi-pre-q-nbd of x, such that A ql B. Hence x, ~ Spcl A. Thus we arrive at a contradiction. []
Definition 2.3. Let A be any f.set of a fts X. Then f.semi-preclosure (Spcl) and f.semi-preinterior (Spint) of A are defined as follows:
3. Characterizations of fuzzy semi-precontinuous mappings
Spcl A = (~ {B{ B is f.semi-preclosed and A ~< B}, Spint A = U {BIB is f.semi-preopen and B ~< A}.
Theorem 2.8. Let A be any f.set of a fis X. Then Spcl A' = ( S p i n t A)' and Spint A' = (Spcl A)'.
Definition 3.1. A m a p p i n g f : X ~ Y is called (a) f.semi-continuous (f.s.c.) if the inverse of f.open set is f.semi-open set [1], (b) f.precontinuous (f.p.c.) if the inverse of f.open set is f.preopen set [23,
J.H. Park, B.Y. Lee / Fuzz)' Sets and Systems 67 (1994) 359-364
362
(c) f.semi-precontinuous (f.s.p.c.) if the inverse of f.open set is f.semi-preopen set. The implications contained in following diagram are true.
Theorem 3.2. A mapping f : X ~ Y is f.s.p.c, if and only if for every f .point x~ in X and every f .nbd V of f ( x , ) = f ( x ) , , there exists f.semi-pre-nbd U of x, such that f ( U ) <~ V.
f.s.c. f.c. -o f.s.s.c,
f I ( C 1 B ' ) = f - l ( B ' ) . By Theorem 2.8, we have f - 1(B)' >/Spcl(f - ~(B)') =(Spint f - ~(n))'. Hence f - I ( B ) is f.semi-preopen set.
f.s.p.c. f.p.c.
where f.c. = f.continuity semi-continuity [2].
[3],
f.s.s.c. = f.strong
From [2, Remark 2.2] and the following Example 3.1 show that the reverse need not be true Example 3.1. Let U1, U2, U3 and U4 be f.sets of X = {a,b,c} as described in Example 2.1. Let zl = {1x,Ox, U1,U2,U3,U2f-~U3,U2k_)U3} and "C2= {lx,0x, U4} be f.topologies on X a n d f : ( X , zl)-o (X, z2) be a mapping defined as follows: f(a) = a,f(b) = b andf(c) = c. T h e n f i s f.s.p.c, but neither f.s.c, nor f.p.c. Theorem 3.1. Let f: X --* Y be a mapping. Then the following are equivalent: (a) f is f.s.p.c., (b) the inverse off.closed set of Y is f.semi-preclosed set, (c) f(Spcl A) <<,Cl f(A) for each f.set A of X, (d) Spclf - 1(B) ~
Proof. Let f : X ~ Y be f.s.p.c.. Suppose x, is any f.point in X and Vis any f.nbd off(x,). Then there exists f.open set B such thatf(x,) • B ~< V. By f.semiprecontinuity off, f - I(B) is f.semi-preopen in X and x~ • f - x(B) ~
J.H. Park, B.Y. Lee / Fuzz), Sets and Systems 67 (1994) 359 364
Proof. Let V = (j,.p(V, × Wt~),where V, and Wp are f. open sets in Y1 and Y2 respectively, be a f. open set in Y1 x Y2. From Lemma 2.1 and 2.3 in [1], we have
(A xA) '(v)= U(A
xA)-l(v
= Uif;'(V,)xA
,x
w#)
'(wp)).
363
Definition 4.3. A fts X is called f.semi-pre-T2 if for every distinct two f.points x, and y~, the following conditions are satisfied: (a) When x :~ y, x, and yo have f.semi-pre-nbds which are not q-coincident. (b) When x = y and ~ < f l (say), then x, has a f.semi-pre-nbd U and y~ has a f.semi-pre-qnbd V such that U q~ V.
ct,fl
Since fl a n d f z are f.s.p.c.,fz- 1(V,) and f2-1(W~) are f.semi-preopen sets, and by Theorem 2.4 and 2.6, it follows that (fl x f2) a(V) is f.semi-preopen set. Hence f~ ×f2 is f.s.p.c. []
Theorem 3.5. Let f: X ~ Y be a mapping from a fis X to a fis Y, and g : X ~ X × Y be the graph o f f If 9 is fs.p.c., then so is f Proof. Let V be a f.open set in Y. Then, b y Lemma 2.4 in [-1], we have f - l ( V ) = l ~ f - l ( V ) = 9-1(1 x V). Since 9 is f.s.p.c, and 1 x Vis a f.open set in X × Y, f - 1(V) is f.semi-preopen set of X. H e n c e f i s f.s.p.c. []
4. Separation axioms Definition 4.1. A fts X is called f. semi-pre-To if for every distinct two f.points x, and yp, the following conditions are satisfied: (a) When x :~ y, either x~ has a f.semi-pre-nbd which is not q-coincident with yp, or y~ has a f.semi-pre-nbd which is not q-coincident with Xa.
(b) When x = y and ~ < fl (say), there is a f.semipre-q-nbd of yp which is not q-coincident with X~t.
Definition 4.2. A fts X is called f.semi-pre-T1 if for every distinct two f.points x~ and ya, the following conditions are satisfied: (a) When x ¢ y, x~ has a f.semi-pre-nbd U and yp has a f.semi-pre-nbd V such that Yaql U and x~¢l V. (b) When x = y and ~ < fl (say), then there exists a f.semi-pre-q-nbd V of y~ such that x , ~ V.
It is obvious that f.semi-pre-T2 ~ f.semi-pre-T1 ~f.semi-pre-To, and f.Ti axiom [5] ~f.semi-Ti axiom [7] ~ f.semi-pre-T~ axiom, for i -- 0, l, 2.
Theorem 4.1. A fis X is f.semi-pre-To if and only if for every pair of distinct x~ and y~, either x~ q~Spcl(y~) or yp ~ Spcl(x~). Proof. Let X be f.semi-pre-To, and x, and yp be two distinct f.point in X. When x ¢ y, Xl has a f.semi-pre-nbd U such that ypflU, or Yl has a f.semi-pre-nbd V such that x , ~ V. Suppose xl has a f.semi-pre-nbd U which is not q-coincident with yp. Then U is f.semipre-q-nbd of x, and y ~ U. This implies that x~ ¢ Spcl (y~). When x = y and ~ < fl (say), then y~ has a semipre-q-nbd which is not q-coincident with x, and thus y~ ¢ Spcl(x,). Conversely, let x, and y~ be two distinct f.points in X. Without loss of generality we suppose that x, ¢ Spcl(yp). When x # y, since x, ¢ Spcl(y~), Xl¢Spcl(y~) and so [Spcl(y~)]'(x)= 1. Hence [Spcl(yp)]' is a f.semi-pre-nbd of x, such that y ~ [ S p c l ( y ~ ) ] ' . Also when x = y we must have > fl and then x, has a semi-pre-q-nbd which is not q-coincident with yp. []
Theorem 4.2. A fis X is f.semi-pre-Tl if and only if for every f.point x, is f.semi-preclosed in X. Proof. The proof is easy and hence omitted.
[]
Theorem 4.3. A fis X is f.semi-pre-T2 if and only if for every f.point x, in X, x~ = N{Spcl VI V is f.semi-pre-nbd of x,} and for every x, y e X with x v~y, there is af.semi-pre-nbd U of xl such that y ¢ (Spcl U)o, where (Spcl U)o is support of Spcl U.
364
J.H. Park. B.Y. Lee / Fuzzy Sets and Systems 67 (1994) 359-364
Proof. Let x~ and yg be f.points in X such that y~ ~ {x~}. If x :~ y, then there are f.semi-preopen sets U and V containing YI and x~ respectively such that U ¢ V. Then V is a f.semi-pre-nbd of x~ and U is a f.semi-pre-q-nbd of y~ such that U~l V. Hence y~ q~ Spcl V. If x = y, then ~ < fl, and hence there are a f.semi-pre-q-nbd U of y~ and a f.semipre-nbd V of x~ such that U ¢ V. Then yp ¢ Spcl V. Finally, for distinct two point x, y of X, since X is f.semi-pre-T2, there exist f.semi-preopen sets U and V such that x~ e U, y~ ~ V and U ¢ V. Since V' is f.semi-preclosed set containing U, Spcl U ~< V'. Hence y ¢ (Spcl U)o. Conversely, let x, and Yt~ be two distinct f.points in X. W h e n x # y. We first suppose that at least one of ct and fl is less than 1, say 0 < ~ < 1. Then there exists a positive real n u m b e r 2 with 0 < ~ + 2 < 1. By hypothesis, there exists a f.semi-pre-nbd U of y¢ such that xz ¢ Spcl U. Then there exists a f.semipreopen set V such that x a q V and V¢ U. Since x z q V, V(x) > 1 - 2 > ~ and hence V is a f.semipre-nbd of x, such that U ¢ V. Next if ~ = fl = 1, by hypothesis there exists a f.semi-pre-nbd U of Xl such that Spcl U ( y ) = O. Then V = (Spcl U)' is a f.semi-pre-nbd of Yl such that U ¢ V. W h e n x ¢ y and c~ < fl (say), then there exists a f.semi-pre-nbd U of x, such that y~ ~ Spcl U. Hence there exists a f.semi-pre-q-nbd V of yo such that U ¢ V. Therefore X is f.semi-pre-T2. []
Theorem 4.4. L e t f : X ~ Y be injective and fs.p.c. I f Y i s f . T i , then X i s f . s e m i - p r e - T i , f o r i = 0, 1, 2.
Proof. We give a p r o o f for i = 1 only; the other cases being similar, are omitted. Let x, and yp be distinct two f.points in X. W h e n x 4= y, we h a v e f ( x ) e l ( y ) , and by the f.T1 property of Y, there exist f.nbds U and V off(x)~ and f ( y ) ~ respectively such that f ( x ) , q t V and f(Y)t~ qi U. Since f is f.s.p.c., f - 1(U) and f - x(V) are f.semi-pre-nbds of x, and y~ respectively such that x , ~ l f - l ( V ) and y o q i f - l(U). When x = y and ~ < fl (say), thenf(x) = f(y). Since Y is f.T~, there exists a f.q-nbd V off(y)t3 such that
f ( x ) , ~ V. Then f - I(V) is f.semi-pre-q-nbd of yp such that x ~ f - l ( V ) . Hence X is f.semi-preT1. []
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