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Fuzzy θ-connectedness
Fuzzy Sets and Systems 59 (1993) 237-244 North-Holland
237
Fuzzy O-connectedness Jin Han Park, Bu Young Lee and Jae Rong Choi Department of Mathematics, Dong-A University, Hadan-dong, Saha-ku, Pusan, 604-714, South Korea Received February 1992 Revised February 1993
Abstract: In this paper the concept of fuzzy 0-connectedness in a fuzzy topological space is introduced and discussed to some extent.
Keywords: Fuzzy 6-closure and fuzzy 0-closure; fuzzy 0-continuous; fuzzy strong 0-continuous and fuzzy weakly 6-continuous mappings; fuzzy 0-connected sets.
1. Introduction and preliminaries The concepts of fuzzy 6-closure and fuzzy 0-closure of fuzzy sets in fuzzy topological space (fts, for short) were introduced by Ganguly and Saha [5] and Mukherjee and Sinha [12], respectively. Subsequently several authors [3, 7, 9] investigated some properties for near-fuzzy continuous mappings on fts's using these concepts. Ganguly and Saha [4, 5] also introduced the concepts of fuzzy connected sets and fuzzy 6-connected sets in fts. In this paper we investigate, in Section 2, some properties of fuzzy strong 0-continuous, fuzzy 0-continuous and fuzzy weakly ~-continuous mappings on fts's. In Section 3, we introduce the concept of fuzzy 0-connected sets, by the help of the concept of fuzzy 0-closure, and study the relationships between fuzzy 0-connected sets and such fuzzy subsets in fts. Throughout the paper, by (X, v) (or simply X) we mean afts in Chang's [2] sense. A fuzzy point in X with support x ~ X and value a (0 < a ~< 1) is denoted by x,~. For a fuzzy set A in X, C1A and (1 - A) will respectively denote the closure and complement of A, whereas the constant fuzzy sets taking on the values 0 and 1 on X are denoted by 0x and Ix respectively. A fuzzy set A in fts X is said to be q-coincident with a fuzzy set B, denoted by A q B, if there exists x e X such that A(x) + B(x) > 1 [8]. It is known [8] that A ~
Theorem 1.1 [2, 15]. Let f :X---~ Y be a mapping and A and B be a fuzzy set of X and Y, respectively. Then the following statements are true: (a) 1 - f ( A ) ~
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Theorem 1.2 [15]. Let f :X---> Y be a mapping and x~ be a fuzzy point in X. (a) I f B is fuzzy set in Y and f ( x ~ ) q B, then x~ q f - l ( B ) . (b) I r A is fuzzy set in X and x~ qA, then f ( x , ) qf(A). Definition 1.3 [6, 12]. A fuzzy point x~ in a fts X is said to be a fuzzy 0-cluster point (6-cluster point) of a fuzzy set A if for each fuzzy open q-nbd U of x~, Cl U qA (resp. Int C1 U qA). The set of all fuzzy 0-cluster points (6-cluster points) of A is called the fuzzy 0-closure (resp. 6-closure) of A and is denoted by [A]o (resp. [A]8). A fuzzy set A is fuzzy 0-closed (6-closed) if A = [A]o (resp. A = [A],) and the complement of a fuzzy 0-closed set (6-closed set) is fuzzy 0-open (resp. 6-open). Definition 1.4 [12]. A fuzzy set A is said to be a fuzzy 0-nbd (6-nbd) of x~ if there exists a fuzzy open q-nbd U of x~ such that C1 U~I (1 - A) (resp. Int CI Uq~ (1 - A)). Theorem 1.5. A fuzzy set A in X is fuzzy O-open if and only if for each fuzzy point x~ in X with x~ q A, A is fuzzy 0-nbd of x~. Proof. Let A be a fuzzy 0-open and x~ be a fuzzy point in X with x, q A. Since 1 - A is fuzzy 0-closed, x,~ ~t (1 - A) = [1 - A]o. Then there exists fuzzy open q-nbd U of x~ such that CI U ~1(1 - A). Hence A is a fuzzy 0-nbd of x~. Conversely, if x, ~ ( 1 - A ) , then x~ qA. Since A is a fuzzy 0-nbd of x~, there exists a fuzzy open q-nbd U ofx~ such that CI Uqi(1 - A ) and so x~ ~ [1 -A]o. Hence 1 - A is fuzzy 0-closed and then A is fuzzy 0-open.
Theorem 1.6. A fuzzy set A in X is fuzzy 6-open if and only if for each fuzzy point x~ in X with x,~ q A, A is a fuzzy 6-nbd ofx~. Proof. It is similar to the proof of Theorem 1.5.
2. Characterizations of fuzzy strong O-continuity, O-continuity and weakly G-continuity Definition 2.1 [9, 12]. A mapping f:X---> Y is said to be (a) fuzzy strong 0-continuous (f.s.0.c., for short) if for each fuzzy point x~ in X and each fuzzy open q-nbd V of f(x~), there exists fuzzy open q-nbd U of x~ such that f(Cl U) ~< V, (b) fuzzy 0-continuous (f.0.c., for short) if for each fuzzy point x,, in X and each fuzzy open q-nbd V of f(x,), there exists a fuzzy open q-nbd U of x, such that f(C1 U)~< C1V. (c) fuzzy weakly 6-continuous (f.w.&c., for short) if for each fuzzy point x,, in X and each fuzzy open q-nbd V of f(x~), there exists a fuzzy open q-nbd U of x~ such that f(Int Cl U) ~
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Proof. (a) ~ (b). Let x,, • [A]o and V be fuzzy open q-nbd of f(x~). By (a), there exists a fuzzy open q-nbd U of x,, such that f(Cl U) ~< V. Now, we have x~e[A]o ~ CIUqA
~ f(C1U)qf(A) ~ Vqf(A)
f(x,~) • e l f ( A ) ~ x,~ • f - l ( C l f ( A ) ) .
Hence [A]o <~f-l(Clf(A)) and so f([A]o) <~Clf(A). (b) ~ (c). Clear. ( c ) ~ ( d ) . Let B be a fuzzy closed in Y. By (c), we have [f-l(B)]o<~f-~(C1B)=f-~(B ) which implies that f - l ( B ) = [f-l(B)]o. Hence f - l ( B ) is fuzzy 0-closed. (d) ~ (e). Let B be a fuzzy open in Y. Then 1 - B is fuzzy closed and by (d), f-1(1 - B) = 1 - f - l ( B ) is fuzzy 0-closed. Hence f-a(B) is fuzzy 0-open. (e) ~ (a). Let x~ be any fuzzy point in X and V be a fuzzy open q-nbd of f(x~). By (e), f - l ( v ) is fuzzy 0-open in X. Now, f(x~) q V ~ x,, q f - l ( V ) ~ x,~ ~ 1 - f - a ( V ) .
Hence 1 - f - ~ ( V ) is fuzzy 0-closed set such that x~ ~ 1 - f - ~ ( V ) . Then there exists fuzzy open q-nbd U of x, such that C1 Uq~ (1 - f - a ( v ) ) which implies f(C1 U) ~< V. This shows that f i s f.s.0.c. Theorem 2.3 [12]. For a mapping f :X---~ Y, each of the following statements implies the next one:
(a) (b) (c) (d) (e) (f)
f i s f.0.c. For each fuzzy For each fuzzy For each fuzzy For each fuzzy For each fuzzy
set A in X, f([A]o) <~[f(A)]o. set B in Y, [f-l(B)] o <~f-l([B]o ). O-closed set B in Y, f - l ( B ) is fuzzy O-closed in X. O-open set B in Y, f - l ( B ) is fuzzy O-open in X. open set B in Y, [f-l(B)] o ~
The above theorem fails to characterize fuzzy 0-continuous mappings. For this it is imperative that equivalence of (a) and (f) is established and this we do as follows. Theorem 2.4. (a), (b), (c), (d), (e) and (f) in Theorem 2.3 are equivalent.
Proof. It is sufficient to show that (f) ~ (a). Let x, be a fuzzy point in X and V be a fuzzy open q-nbd of f(x~). Then 1 - CI V is fuzzy open in Y. By (f), we have [f-l(1 - C1V)]o ~
x,~ q(1 - [1 - f-l(C1V)]o) ~ x,~ ~ [1 - f - l ( C l V)]o. Then there exists a fuzzy open q-nbd U of x,~ such that Cl Uq~ ( 1 - f - l ( C 1 V ) ) so that f(Cl U)<~C1V. Hence f is f. 0.c. Theorem 2.5. For a mapping f : X ~ Y, the following are equivalent:
(a) f / s f.w.&c. (b) For each fuzzy point x~ in X and each fuzzy open q-nbd V of f(x,~), there exists a fuzzy regularly open q-nbd U of x,, such that f ( U) <<-C1V. (c) For each fuzzy point x,~ in X and each fuzzy open q-nbd V of f(x~), there exists a fuzzy g-open q-nbd U of x,, such that f ( U) <~C1 V. (d) f([A]8) ~< [f(A)]ofor each fuzzy set A of X. (e) [f-l(B)]8 <~f-l([B]o) for each fuzzy set B of Y. (f) [f-l(B)]~ <~f-l(C1 B) for each fuzzy open set B of Y.
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Proof. (a) ~ (b). Clear. (b) ~ (c). Follows from the fact that fuzzy regularly open sets are fuzzy ~-open. (c) ~ (a). Let x~ be a fuzzy point in X and V be a fuzzy open q-nbd of f(x~). By hypothesis, there exists a fuzzy ~-open q-nbd U of x~ such that f(U) <~CI V. Then there exists fuzzy open q-nbd W of x, such that Int C1 W~ (1 - U) which implies f(Int C1 W) <~f(U) <~C1 V. Hence f i s f.w.&c. (a)<:> (d)¢:> (e) is proved in Theorem 2.15 of [12]. (e) ~ (f). Straightforward. (f) ~ (a). Similar to the proof of Theorem 2.4. Theorem 2.6. If f :X---> Y is f.w.8.c., then the following are true: (a) For each fuzzy O-closed set B of Y, f-X(B) is fuzzy 8-closed in X. (b) For each fuzzy O-open set B of Y, f - l ( B ) is fuzzy 6-open in X. Proof. (a). Let B be a fuzzy 0-closed set of Y. By Theorem 2.5(e), we have [f-l(B)]~ <~f-a(B) and hence f - l ( B ) is fuzzy 6-closed. (b) Straightforward. It is shown that the converse of Theorem 2.6 need not be true in the following example.
Example 2.7. Let X = [0, 1], zl = {lx, Ox, A} and r2 = {lx, Ox, B}, where
x--O, tO,
x~0,
x--O, tO,
x~0.
Consider the identity mapping f : ( X , zl)--, (X, z2). Since lx and 0x are only fuzzy 0-closed sets in (X, Zz), f l(lx) = lx and f l(0x) = 0x are fuzzy 6-closed sets in (X, zl). Hence f satisfies (a) of Theorem 2.6. Now consider the fuzzy point x,~, where x = 0 and oz = 4. Then B is a fuzzy open q-nbd of f(x,). Let U be any fuzzy open q-nbd of x~. Then U = lx or A, and f(Int C1 U) = lx ~ C1B = B. Hence f is not f.w.&c.
Definition 2.8 [4]. A prefilter ~ on X is a nonempty collection of fuzzy sets in X with the properties: (a) IfA, B E ~ , t h e n A n B • ~ . (b) I f A • Y a n d A ~ < B , thenBE~. (c) 0x ~ ~. Definition 2.9 [4]. A collection N of fuzzy sets in X is said to be a prefilter-base on X if N ¢ ~b and (a) If B1, B2 • ~3, then B3 ~ Y be a mapping and ~ be a prefilter on X. Then {f(F) [ F • ~} is prefilter-based on Y [4]. The prefilter generated by {f(F) I F E ~} denoted by f ( ~ ) .
Definition 2.10. A prefilter ~ is said to converge (0-converge, 6-converge) to the fuzzy point x~ if for each fuzzy open q-nbd U of x~, U E ~ (resp. C1 U ~ ~, Int C1 U ~ ~). Theorem 2.11. L e t f : X ~ Y be a mapping. (a) f is f.0.c, if and only if for each fuzzy point x~ in X and each prefilter ~ which O-converges to x~, f ( ~) O-converges to f(x~). (b) f is f.s.0.c, if and only if for each fuzzy point x, in X and each prefilter ~ which O-converges to x~, f ( ~) converges to f(x~).
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(c) f is f.w.&c, if and only if for each fuzzy point x~ in X and each prefilter ~ which &converges to x,,, f(o~) O-converges to f(x~).
Proof. (a). Let V be fuzzy open q-nbd of f(x~). Then there exists a fuzzy open q-nbd U of x~ such that f(Cl U) ~
3. Fuzzy 0-connected sets Definition 3.1. (a) Two non-null fuzzy sets A and B in a f t s X are said to be fuzzy 0-separated if A ~ [B]o and B qi [A]o. (b) Two-null fuzzy sets A and B in a f t s X are fuzzy separated (fuzzy &separated) if A qi C1B and B ~ CI A (resp. A ~ [B]8 and B ~1[A]a) [5, 6]. For any two non-null fuzzy sets A and B, the following implications hold: fuzzy 0-separated ~ fuzzy &separated ~ fuzzy separated. The following example shows that the concept of fuzzy &separated is weaker than that of fuzzy 0-separated. Example 3.2. Let X = [0, 1] and r = {lx, Ox, At, A2, A3}, where
x=o, Al(x) = t0,
x~O,
x--o, A2(x)=tO,
x¢O,
x--o, A3(x)=tO,
x¢:O.
Then (X, ~) is afts. We consider two fuzzy sets B and C in X given by
B ( x ) = ~ 2, x=O, [0 ' x ¢ O ,
C(x)=f~, LO,
x=O, x~O.
Since [B]a = 1 - A 3 and [C]~ = 1 - A 2 , we have [B]~ ~1C and [C]~ ~IB. Hence B and C are fuzzy &separated. But [B]o = 1 - A: and [C]o = 1 - A1, so that [B]o q C and [Clo q B. Hence B and C are not fuzzy 0-separated.
Theorem 3.3. Let A and B be non-null fuzzy sets in a fts X. (a) If A and B are fuzzy O-separated, and A~ and B1 are non-null fuzzy sets such that A1 <- A and B 1 ~ B, then A1 and Ba are also fuzzy O-separated. (b) If A ~ B and either both are fuzzy O-open or both are fuzzy O-closed, then A and B are fuzzy O-separated. (c) I r A and B are either both fuzzy O-open or both fuzzy O-closed and if CA(B) = A f3 (1 - B) and CB(A) = B N (1 - A), then CA(B) and Ca(A) are fuzzy O-separated. Proof. (a). Since n (t [Z]o ~
A 1 ~
nl
we have [A1]o <<-[A]o. Then
[A]o ~ B, ~t [Ado.
Similarly, A1 qi [B1]o. Hence A1 and B1 are fuzzy O-separated. (b). When A and B are fuzzy 0-closed, then A = [A]o and B = [B]o. Since A qi B, we have [A]o ~t B and [B]o ~ A.
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When A and B are fuzzy 0-open, 1 - A and 1 - B are fuzzy 0-closed. Then
A~tB ~ A~<(1-B)
~
[A]o<~[1-B]o=I-B
~ [A]o~l B.
Similarly, [B]o OA. Hence A and B are fuzzy 0-separated. (c). When A and B are fuzzy 0-open, 1 - A and 1 - B are fuzzy 0-closed. Since CA(B)<~ 1 - B, [CA(B)]o <<-[1 - B]e = 1 - B and so [CA(B)]o ~1B. Thus CB(A) q~[CA(B)]o. Similarly, CA(B) ~t [CB(A)]o. Hence CA(B) and CB(A) are fuzzy 0-separated. When A and B are fuzzy 0-closed, A = [A]o and B = [B]o. Since CA(B) <~(1 - B), [B]o ~1CA(B) and hence [CB(A)]o ~1CA(B). Similarly, [CA(B)]o ~1CB(A). Hence CA(B) and CB(A) are fuzzy 0-separated. Theorem 3.4. Two non-null fuzzy sets A and B are fuzzy O-separated if and only if there exist two fuzzy
O-open sets U and V such that A <<-U, B <<-V, A q~V and B ~I U. Proof. Let A and B be fuzzy 0-separated sets. Putting V = 1 - [A]o and U = 1 - [B]o, then U and V are fuzzy 0-open such that A <~ U, b ~< V, A q~V and B ~ U. Conversly, let U and V be fuzzy 0-open sets such that A ~< U, B ~< V, A q~V and B qi U. Since 1 - V and 1 - U are fuzzy 0-closed, we have [A]o <<-(1 - V) ~< (1 - B) and [B]o <~(1 - U) ~< (1 - A). Thus [A]o ~ B and [B]o q~A. Hence A and B are fuzzy 0-separated. Definition 3.5. (a) A fuzzy set which cannot be expressed as the union of two fuzzy 0-separated sets is said to be a fuzzy 0-connected set. (b) A fuzzy set A in a f t s X is said to be fuzzy connected (fuzzy 8-connected) if A cannot be expressed as the union of two fuzzy separated (resp. fuzzy 6-separated) sets [5, 6]. For a fuzzy set A in a fts X, the following implications hold: fuzzy connected ~ fuzzy 6connected ~ fuzzy 0-connected. If A is a fuzzy open set, then these three properties are equivalent. Theorem 3.6. Let A be a non-null fuzzy O-connected set in a fts X. I r A is contained in the union of two fuzzy O-separated sets B and C, then exactly one o f the following conditions (a) and (b) holds: (a) A < ~ B a n d A n C = O x . (b) A < ~ C a n d A N B = O x . Proof. We first note that when A n C = 0x, then A <~B, since A ~ B U C. Similarly, when A n B = 0x, we have A ~< C. Since A ~< B U C, both A n B = 0x and A N C = 0x cannot hold simultaneously. Again, if A n B ~ 0x and A O C ~ 0 x , then, by T h e o r e m 3.3(a), A N B and A n C are fuzzy 0-separated sets such that A = ( A n B ) U ( A n C), contradicting the fuzzy 0-connectedness of A. Hence exactly one of the conditions (a) and (b) must hold. Theorem 3.7. Let {A,~ I o~ ~ A} be a collection of fuzzy O-connected sets in X. If there exists [3 ~ A such
that A~ n At3 ~ Ox for each a e A, then A = U {A~ I a c A} is fuzzy O-connected. Proof. Suppose that A is not fuzzy 0-connected set. Then there exist fuzzy 0-separated sets B and C such that A = B U C. By T h e o r e m 3.6, we have either (a) A~ ~< B with As n C = 0x or (b) A~ ~< C with A~AB=0x for each a e A . Similarly, either (c) A s~
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Definition 3.8 [10, 11]. A f t s X is said to be (a) fuzzy regular if for each fuzzy open set V and each fuzzy point p q V, there exists a fuzzy open U such that p q U <~C1 U ~< V, (b) fuzzy almost regular if for each fuzzy regularly open set V and each fuzzy point p q V, there exists a fuzzy regularly open U such that p q U ~
A = f - l ( f ( A ) ) = f - l ( B f'3 C) = f - l ( B ) t..Jf-l(C). Hence this is contrary to the fact that A is fuzzy O-connected. The other cases are similar to (a) and omitted.
References [1] K.K. Azad, On fuzzy semicontinuity,fuzzy almost continuity and fuzzy weakly continuity,J. Math. Anal. Appl. 82 (1981) 14-32. [2] C.L. Chang, Fuzzy topologicalspaces, J. Math. Anal. Appl. 24 (1968) 182-190. [3] J.R. Choi, B.Y. Lee and J.H. Park, On fuzzy 0-continuousmappings, Fuzzy Sets and Systems54 (1993) 107-114. [4] M.A. De Prada and M. Saralegui, Fuzzy filters,J. Math. Anal, Appl. 129 (1988) 560-568. [5] S. Ganguly and S. Saha, On separation axioms and separation of connected sets in fuzzy topological spaces, Bull Calcutta Math. Soc. 79 (1987) 215-225. [6] S. Ganguly and S. Saha, A note on 8-continuityand ~5-connectedsets in fuzzy set theory, Simon Stevin 62 (1988) 127-141.
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[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 of 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. [11] 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] M.N. Mukherjee and S.P. Sinha, On some near-fuzzy continuous functions between fuzzy topological spaces, Fuzzy Sets and Systems 34 (1990) 245-254. [13] T. Noiri, A weak form of connected sets, Math. Nachr. 116 (1984) 7-11. [14] R. Srivastava, S.N. Lal and A.K. Srivastava, Fuzzy Hausdorff topological spaces, J. Math. Anal. Appl. 81 (1981) 497-506. [15] T.H. Yalvac, Fuzzy sets and functions on fuzzy topological spaces, J. Math. Anal, Appl. 126 (1987) 409-423. [16] T.H. Yalvac, Semi-interior and semi-closure of a fuzzy set, J. Math. Anal Appl. 132 (1988) 356-364.
237
Fuzzy O-connectedness Jin Han Park, Bu Young Lee and Jae Rong Choi Department of Mathematics, Dong-A University, Hadan-dong, Saha-ku, Pusan, 604-714, South Korea Received February 1992 Revised February 1993
Abstract: In this paper the concept of fuzzy 0-connectedness in a fuzzy topological space is introduced and discussed to some extent.
Keywords: Fuzzy 6-closure and fuzzy 0-closure; fuzzy 0-continuous; fuzzy strong 0-continuous and fuzzy weakly 6-continuous mappings; fuzzy 0-connected sets.
1. Introduction and preliminaries The concepts of fuzzy 6-closure and fuzzy 0-closure of fuzzy sets in fuzzy topological space (fts, for short) were introduced by Ganguly and Saha [5] and Mukherjee and Sinha [12], respectively. Subsequently several authors [3, 7, 9] investigated some properties for near-fuzzy continuous mappings on fts's using these concepts. Ganguly and Saha [4, 5] also introduced the concepts of fuzzy connected sets and fuzzy 6-connected sets in fts. In this paper we investigate, in Section 2, some properties of fuzzy strong 0-continuous, fuzzy 0-continuous and fuzzy weakly ~-continuous mappings on fts's. In Section 3, we introduce the concept of fuzzy 0-connected sets, by the help of the concept of fuzzy 0-closure, and study the relationships between fuzzy 0-connected sets and such fuzzy subsets in fts. Throughout the paper, by (X, v) (or simply X) we mean afts in Chang's [2] sense. A fuzzy point in X with support x ~ X and value a (0 < a ~< 1) is denoted by x,~. For a fuzzy set A in X, C1A and (1 - A) will respectively denote the closure and complement of A, whereas the constant fuzzy sets taking on the values 0 and 1 on X are denoted by 0x and Ix respectively. A fuzzy set A in fts X is said to be q-coincident with a fuzzy set B, denoted by A q B, if there exists x e X such that A(x) + B(x) > 1 [8]. It is known [8] that A ~
Theorem 1.1 [2, 15]. Let f :X---~ Y be a mapping and A and B be a fuzzy set of X and Y, respectively. Then the following statements are true: (a) 1 - f ( A ) ~
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Theorem 1.2 [15]. Let f :X---> Y be a mapping and x~ be a fuzzy point in X. (a) I f B is fuzzy set in Y and f ( x ~ ) q B, then x~ q f - l ( B ) . (b) I r A is fuzzy set in X and x~ qA, then f ( x , ) qf(A). Definition 1.3 [6, 12]. A fuzzy point x~ in a fts X is said to be a fuzzy 0-cluster point (6-cluster point) of a fuzzy set A if for each fuzzy open q-nbd U of x~, Cl U qA (resp. Int C1 U qA). The set of all fuzzy 0-cluster points (6-cluster points) of A is called the fuzzy 0-closure (resp. 6-closure) of A and is denoted by [A]o (resp. [A]8). A fuzzy set A is fuzzy 0-closed (6-closed) if A = [A]o (resp. A = [A],) and the complement of a fuzzy 0-closed set (6-closed set) is fuzzy 0-open (resp. 6-open). Definition 1.4 [12]. A fuzzy set A is said to be a fuzzy 0-nbd (6-nbd) of x~ if there exists a fuzzy open q-nbd U of x~ such that C1 U~I (1 - A) (resp. Int CI Uq~ (1 - A)). Theorem 1.5. A fuzzy set A in X is fuzzy O-open if and only if for each fuzzy point x~ in X with x~ q A, A is fuzzy 0-nbd of x~. Proof. Let A be a fuzzy 0-open and x~ be a fuzzy point in X with x, q A. Since 1 - A is fuzzy 0-closed, x,~ ~t (1 - A) = [1 - A]o. Then there exists fuzzy open q-nbd U of x~ such that CI U ~1(1 - A). Hence A is a fuzzy 0-nbd of x~. Conversely, if x, ~ ( 1 - A ) , then x~ qA. Since A is a fuzzy 0-nbd of x~, there exists a fuzzy open q-nbd U ofx~ such that CI Uqi(1 - A ) and so x~ ~ [1 -A]o. Hence 1 - A is fuzzy 0-closed and then A is fuzzy 0-open.
Theorem 1.6. A fuzzy set A in X is fuzzy 6-open if and only if for each fuzzy point x~ in X with x,~ q A, A is a fuzzy 6-nbd ofx~. Proof. It is similar to the proof of Theorem 1.5.
2. Characterizations of fuzzy strong O-continuity, O-continuity and weakly G-continuity Definition 2.1 [9, 12]. A mapping f:X---> Y is said to be (a) fuzzy strong 0-continuous (f.s.0.c., for short) if for each fuzzy point x~ in X and each fuzzy open q-nbd V of f(x~), there exists fuzzy open q-nbd U of x~ such that f(Cl U) ~< V, (b) fuzzy 0-continuous (f.0.c., for short) if for each fuzzy point x,, in X and each fuzzy open q-nbd V of f(x,), there exists a fuzzy open q-nbd U of x, such that f(C1 U)~< C1V. (c) fuzzy weakly 6-continuous (f.w.&c., for short) if for each fuzzy point x,, in X and each fuzzy open q-nbd V of f(x~), there exists a fuzzy open q-nbd U of x~ such that f(Int Cl U) ~
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Proof. (a) ~ (b). Let x,, • [A]o and V be fuzzy open q-nbd of f(x~). By (a), there exists a fuzzy open q-nbd U of x,, such that f(Cl U) ~< V. Now, we have x~e[A]o ~ CIUqA
~ f(C1U)qf(A) ~ Vqf(A)
f(x,~) • e l f ( A ) ~ x,~ • f - l ( C l f ( A ) ) .
Hence [A]o <~f-l(Clf(A)) and so f([A]o) <~Clf(A). (b) ~ (c). Clear. ( c ) ~ ( d ) . Let B be a fuzzy closed in Y. By (c), we have [f-l(B)]o<~f-~(C1B)=f-~(B ) which implies that f - l ( B ) = [f-l(B)]o. Hence f - l ( B ) is fuzzy 0-closed. (d) ~ (e). Let B be a fuzzy open in Y. Then 1 - B is fuzzy closed and by (d), f-1(1 - B) = 1 - f - l ( B ) is fuzzy 0-closed. Hence f-a(B) is fuzzy 0-open. (e) ~ (a). Let x~ be any fuzzy point in X and V be a fuzzy open q-nbd of f(x~). By (e), f - l ( v ) is fuzzy 0-open in X. Now, f(x~) q V ~ x,, q f - l ( V ) ~ x,~ ~ 1 - f - a ( V ) .
Hence 1 - f - ~ ( V ) is fuzzy 0-closed set such that x~ ~ 1 - f - ~ ( V ) . Then there exists fuzzy open q-nbd U of x, such that C1 Uq~ (1 - f - a ( v ) ) which implies f(C1 U) ~< V. This shows that f i s f.s.0.c. Theorem 2.3 [12]. For a mapping f :X---~ Y, each of the following statements implies the next one:
(a) (b) (c) (d) (e) (f)
f i s f.0.c. For each fuzzy For each fuzzy For each fuzzy For each fuzzy For each fuzzy
set A in X, f([A]o) <~[f(A)]o. set B in Y, [f-l(B)] o <~f-l([B]o ). O-closed set B in Y, f - l ( B ) is fuzzy O-closed in X. O-open set B in Y, f - l ( B ) is fuzzy O-open in X. open set B in Y, [f-l(B)] o ~
The above theorem fails to characterize fuzzy 0-continuous mappings. For this it is imperative that equivalence of (a) and (f) is established and this we do as follows. Theorem 2.4. (a), (b), (c), (d), (e) and (f) in Theorem 2.3 are equivalent.
Proof. It is sufficient to show that (f) ~ (a). Let x, be a fuzzy point in X and V be a fuzzy open q-nbd of f(x~). Then 1 - CI V is fuzzy open in Y. By (f), we have [f-l(1 - C1V)]o ~
x,~ q(1 - [1 - f-l(C1V)]o) ~ x,~ ~ [1 - f - l ( C l V)]o. Then there exists a fuzzy open q-nbd U of x,~ such that Cl Uq~ ( 1 - f - l ( C 1 V ) ) so that f(Cl U)<~C1V. Hence f is f. 0.c. Theorem 2.5. For a mapping f : X ~ Y, the following are equivalent:
(a) f / s f.w.&c. (b) For each fuzzy point x~ in X and each fuzzy open q-nbd V of f(x,~), there exists a fuzzy regularly open q-nbd U of x,, such that f ( U) <<-C1V. (c) For each fuzzy point x,~ in X and each fuzzy open q-nbd V of f(x~), there exists a fuzzy g-open q-nbd U of x,, such that f ( U) <~C1 V. (d) f([A]8) ~< [f(A)]ofor each fuzzy set A of X. (e) [f-l(B)]8 <~f-l([B]o) for each fuzzy set B of Y. (f) [f-l(B)]~ <~f-l(C1 B) for each fuzzy open set B of Y.
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Proof. (a) ~ (b). Clear. (b) ~ (c). Follows from the fact that fuzzy regularly open sets are fuzzy ~-open. (c) ~ (a). Let x~ be a fuzzy point in X and V be a fuzzy open q-nbd of f(x~). By hypothesis, there exists a fuzzy ~-open q-nbd U of x~ such that f(U) <~CI V. Then there exists fuzzy open q-nbd W of x, such that Int C1 W~ (1 - U) which implies f(Int C1 W) <~f(U) <~C1 V. Hence f i s f.w.&c. (a)<:> (d)¢:> (e) is proved in Theorem 2.15 of [12]. (e) ~ (f). Straightforward. (f) ~ (a). Similar to the proof of Theorem 2.4. Theorem 2.6. If f :X---> Y is f.w.8.c., then the following are true: (a) For each fuzzy O-closed set B of Y, f-X(B) is fuzzy 8-closed in X. (b) For each fuzzy O-open set B of Y, f - l ( B ) is fuzzy 6-open in X. Proof. (a). Let B be a fuzzy 0-closed set of Y. By Theorem 2.5(e), we have [f-l(B)]~ <~f-a(B) and hence f - l ( B ) is fuzzy 6-closed. (b) Straightforward. It is shown that the converse of Theorem 2.6 need not be true in the following example.
Example 2.7. Let X = [0, 1], zl = {lx, Ox, A} and r2 = {lx, Ox, B}, where
x--O, tO,
x~0,
x--O, tO,
x~0.
Consider the identity mapping f : ( X , zl)--, (X, z2). Since lx and 0x are only fuzzy 0-closed sets in (X, Zz), f l(lx) = lx and f l(0x) = 0x are fuzzy 6-closed sets in (X, zl). Hence f satisfies (a) of Theorem 2.6. Now consider the fuzzy point x,~, where x = 0 and oz = 4. Then B is a fuzzy open q-nbd of f(x,). Let U be any fuzzy open q-nbd of x~. Then U = lx or A, and f(Int C1 U) = lx ~ C1B = B. Hence f is not f.w.&c.
Definition 2.8 [4]. A prefilter ~ on X is a nonempty collection of fuzzy sets in X with the properties: (a) IfA, B E ~ , t h e n A n B • ~ . (b) I f A • Y a n d A ~ < B , thenBE~. (c) 0x ~ ~. Definition 2.9 [4]. A collection N of fuzzy sets in X is said to be a prefilter-base on X if N ¢ ~b and (a) If B1, B2 • ~3, then B3 ~ Y be a mapping and ~ be a prefilter on X. Then {f(F) [ F • ~} is prefilter-based on Y [4]. The prefilter generated by {f(F) I F E ~} denoted by f ( ~ ) .
Definition 2.10. A prefilter ~ is said to converge (0-converge, 6-converge) to the fuzzy point x~ if for each fuzzy open q-nbd U of x~, U E ~ (resp. C1 U ~ ~, Int C1 U ~ ~). Theorem 2.11. L e t f : X ~ Y be a mapping. (a) f is f.0.c, if and only if for each fuzzy point x~ in X and each prefilter ~ which O-converges to x~, f ( ~) O-converges to f(x~). (b) f is f.s.0.c, if and only if for each fuzzy point x, in X and each prefilter ~ which O-converges to x~, f ( ~) converges to f(x~).
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(c) f is f.w.&c, if and only if for each fuzzy point x~ in X and each prefilter ~ which &converges to x,,, f(o~) O-converges to f(x~).
Proof. (a). Let V be fuzzy open q-nbd of f(x~). Then there exists a fuzzy open q-nbd U of x~ such that f(Cl U) ~
3. Fuzzy 0-connected sets Definition 3.1. (a) Two non-null fuzzy sets A and B in a f t s X are said to be fuzzy 0-separated if A ~ [B]o and B qi [A]o. (b) Two-null fuzzy sets A and B in a f t s X are fuzzy separated (fuzzy &separated) if A qi C1B and B ~ CI A (resp. A ~ [B]8 and B ~1[A]a) [5, 6]. For any two non-null fuzzy sets A and B, the following implications hold: fuzzy 0-separated ~ fuzzy &separated ~ fuzzy separated. The following example shows that the concept of fuzzy &separated is weaker than that of fuzzy 0-separated. Example 3.2. Let X = [0, 1] and r = {lx, Ox, At, A2, A3}, where
x=o, Al(x) = t0,
x~O,
x--o, A2(x)=tO,
x¢O,
x--o, A3(x)=tO,
x¢:O.
Then (X, ~) is afts. We consider two fuzzy sets B and C in X given by
B ( x ) = ~ 2, x=O, [0 ' x ¢ O ,
C(x)=f~, LO,
x=O, x~O.
Since [B]a = 1 - A 3 and [C]~ = 1 - A 2 , we have [B]~ ~1C and [C]~ ~IB. Hence B and C are fuzzy &separated. But [B]o = 1 - A: and [C]o = 1 - A1, so that [B]o q C and [Clo q B. Hence B and C are not fuzzy 0-separated.
Theorem 3.3. Let A and B be non-null fuzzy sets in a fts X. (a) If A and B are fuzzy O-separated, and A~ and B1 are non-null fuzzy sets such that A1 <- A and B 1 ~ B, then A1 and Ba are also fuzzy O-separated. (b) If A ~ B and either both are fuzzy O-open or both are fuzzy O-closed, then A and B are fuzzy O-separated. (c) I r A and B are either both fuzzy O-open or both fuzzy O-closed and if CA(B) = A f3 (1 - B) and CB(A) = B N (1 - A), then CA(B) and Ca(A) are fuzzy O-separated. Proof. (a). Since n (t [Z]o ~
A 1 ~
nl
we have [A1]o <<-[A]o. Then
[A]o ~ B, ~t [Ado.
Similarly, A1 qi [B1]o. Hence A1 and B1 are fuzzy O-separated. (b). When A and B are fuzzy 0-closed, then A = [A]o and B = [B]o. Since A qi B, we have [A]o ~t B and [B]o ~ A.
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When A and B are fuzzy 0-open, 1 - A and 1 - B are fuzzy 0-closed. Then
A~tB ~ A~<(1-B)
~
[A]o<~[1-B]o=I-B
~ [A]o~l B.
Similarly, [B]o OA. Hence A and B are fuzzy 0-separated. (c). When A and B are fuzzy 0-open, 1 - A and 1 - B are fuzzy 0-closed. Since CA(B)<~ 1 - B, [CA(B)]o <<-[1 - B]e = 1 - B and so [CA(B)]o ~1B. Thus CB(A) q~[CA(B)]o. Similarly, CA(B) ~t [CB(A)]o. Hence CA(B) and CB(A) are fuzzy 0-separated. When A and B are fuzzy 0-closed, A = [A]o and B = [B]o. Since CA(B) <~(1 - B), [B]o ~1CA(B) and hence [CB(A)]o ~1CA(B). Similarly, [CA(B)]o ~1CB(A). Hence CA(B) and CB(A) are fuzzy 0-separated. Theorem 3.4. Two non-null fuzzy sets A and B are fuzzy O-separated if and only if there exist two fuzzy
O-open sets U and V such that A <<-U, B <<-V, A q~V and B ~I U. Proof. Let A and B be fuzzy 0-separated sets. Putting V = 1 - [A]o and U = 1 - [B]o, then U and V are fuzzy 0-open such that A <~ U, b ~< V, A q~V and B ~ U. Conversly, let U and V be fuzzy 0-open sets such that A ~< U, B ~< V, A q~V and B qi U. Since 1 - V and 1 - U are fuzzy 0-closed, we have [A]o <<-(1 - V) ~< (1 - B) and [B]o <~(1 - U) ~< (1 - A). Thus [A]o ~ B and [B]o q~A. Hence A and B are fuzzy 0-separated. Definition 3.5. (a) A fuzzy set which cannot be expressed as the union of two fuzzy 0-separated sets is said to be a fuzzy 0-connected set. (b) A fuzzy set A in a f t s X is said to be fuzzy connected (fuzzy 8-connected) if A cannot be expressed as the union of two fuzzy separated (resp. fuzzy 6-separated) sets [5, 6]. For a fuzzy set A in a fts X, the following implications hold: fuzzy connected ~ fuzzy 6connected ~ fuzzy 0-connected. If A is a fuzzy open set, then these three properties are equivalent. Theorem 3.6. Let A be a non-null fuzzy O-connected set in a fts X. I r A is contained in the union of two fuzzy O-separated sets B and C, then exactly one o f the following conditions (a) and (b) holds: (a) A < ~ B a n d A n C = O x . (b) A < ~ C a n d A N B = O x . Proof. We first note that when A n C = 0x, then A <~B, since A ~ B U C. Similarly, when A n B = 0x, we have A ~< C. Since A ~< B U C, both A n B = 0x and A N C = 0x cannot hold simultaneously. Again, if A n B ~ 0x and A O C ~ 0 x , then, by T h e o r e m 3.3(a), A N B and A n C are fuzzy 0-separated sets such that A = ( A n B ) U ( A n C), contradicting the fuzzy 0-connectedness of A. Hence exactly one of the conditions (a) and (b) must hold. Theorem 3.7. Let {A,~ I o~ ~ A} be a collection of fuzzy O-connected sets in X. If there exists [3 ~ A such
that A~ n At3 ~ Ox for each a e A, then A = U {A~ I a c A} is fuzzy O-connected. Proof. Suppose that A is not fuzzy 0-connected set. Then there exist fuzzy 0-separated sets B and C such that A = B U C. By T h e o r e m 3.6, we have either (a) A~ ~< B with As n C = 0x or (b) A~ ~< C with A~AB=0x for each a e A . Similarly, either (c) A s~
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Definition 3.8 [10, 11]. A f t s X is said to be (a) fuzzy regular if for each fuzzy open set V and each fuzzy point p q V, there exists a fuzzy open U such that p q U <~C1 U ~< V, (b) fuzzy almost regular if for each fuzzy regularly open set V and each fuzzy point p q V, there exists a fuzzy regularly open U such that p q U ~
A = f - l ( f ( A ) ) = f - l ( B f'3 C) = f - l ( B ) t..Jf-l(C). Hence this is contrary to the fact that A is fuzzy O-connected. The other cases are similar to (a) and omitted.
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