ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS
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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIV, 2008, f.1 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS BY ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI Abstract. In 1996, Dontchev [4] introduced and investigated the notion of contracontinuity. In this paper we introduce and study the basic properties of upper (lower) contra-continuous multifunctions. Mathematics Subject Classification 2000: 54C60. Key words: strongly S-closed space, multifunction, contra-continuity. 1. Introduction. Throughout this paper, spaces X and Y mean topological spaces. For a subset A of X, cl(a) and int(a) represent the closure of A and the interior of A, respectively. In this paper, F : X Y presents a multifunction. For a multifunction F : X Y, we shall denote the upper and lower inverse of a set A of Y by F + (A) and F (A), respectively, that is, F + (A) = {x X : F (x) A} and F (A) = {x X : F (x) A } [3]. The graph multifunction G F : X X Y of a multifunction F : X Y is defined as follows G F (x) = {x} F (x) for every x X. Definition 1. ([10]) The set {A τ : B A} is called the kernel of a subset B of a space (X, τ) and is denoted by ker(b). A multifunction F : X Y is called upper semi-continuous (resp. lower semi-continuous) [14] if F + (V ) (resp. F (V )) is open in X for every open set V of Y. Lemma 2. ([12]) Let X and Y be topological spaces and let A X and B Y. The following properties hold for a multifunction F : X Y :
2 76 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 2 (1) G + F (A B) = A F + (B), (2) G F (A B) = A F (B). Definition 3. A subset A of a space X is said to be (1) α-open [11] if A int(cl(int(a))). (2) semi-open [8] if A cl(int(a)). (3) preopen [9] if A int(cl(a)). (4) β-open [1] if A cl(int(cl(a))). The intersection of all α-closed (resp. semi-closed, preclosed, β-closed) sets of X containing A is called the α-closure (resp. semi-closure, preclosure, β-closure) of A and is denoted by α-cl(a) (resp. s-cl(a), p-cl(a) and β- cl(a)). 2. Contra-continuous multifunctions Definition 4. A multifunction F : (X, τ) (Y, σ) is called (1) lower contra-continuous at x X if for each closed set A such that x F (A), there exists an open set U containing x such that U F (A), (2) upper contra-continuous at x X if for each closed set A such that x F + (A), there exists an open set U containing x such that U F + (A). (3) lower (upper) contra-continuous if F has this property at each point of X. Theorem 5. The following are equivalent for a multifunction F : (X, τ) (Y, σ): (1) F is upper contra-continuous, (2) F + (A) is an open set for any closed set A Y, (3) F (U) is a closed set for any open set U Y, (4) for each x X and each closed set A containing F (x), there exists an open set U containing x such that if y U, then F (y) A. Proof. (1) (2): Let A be a closed set in Y and x F + (A). Since F is upper contra-continuous, there exists an open set U containing x such that U F + (A). Thus, F + (A) is open. The converse of the proof is similar. (2) (3): This follows from the fact that F + (Y \A) = X\F (A) for every subset A of Y. (1) (4): Obvious.
3 3 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS 77 Lemma 6. ([6]) Let A, B be subsets of a space (X, τ). The following properties hold: (1) x ker(a) if and only if A B for any closed set B containing x. (2) If A τ, then A = ker(a). Theorem 7. Let F : (X, τ) (Y, σ) be a multifunction. If cl(f (A)) F (ker(a)) for every subset A of Y, then F is upper contra-continuous. Proof. Suppose that cl(f (A)) F (ker(a)) for every subset A of Y. Let A τ. By Lemma 6, cl(f (A) F (ker(a)) = F (A). Thus, cl((f (A)) = F (A) and hence F (A) is closed in X. Consequently, by Theorem 5, F is upper contra-continuous. Definition 8. ([5]) A multifunction F : X Y is called (1) lower clopen continuous if for each x X and each open set V such that x F (V ), there exists a clopen set U containing x such that U F (V ). (2) upper clopen continuous if for each x X and each open set V such that x F + (V ), there exists a clopen set U containing x such that U F + (V ). Definition 9. ([15, 16]) A multifunction F : X Y is said to be: (1) lower weakly continuous if for each x X and each open set V of Y such that x F (V ), there exists an open set U in X containing x such that U F (cl(v )). (2) upper weakly continuous if for each x X and each open set V of Y such that x F + (V ), there exists an open set U in X containing x such that U F + (cl(v )). Theorem 10. If F : X Y is upper/lower contra-continuous, then F is upper/lower weakly continuous. Proof. Let F be upper contra-continuous, x X and V any open set of Y contining F (x). Then cl(v ) is a closed set contining F (x). Since F is upper contra-continuous by Theorem 5 there exists an open set U containing x such that U F + (cl(v )). Hence F is upper weakly continuous. The proof for lower contra-coninuous is similar.
4 78 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 4 Y : Remark 11. The following diagram hold for a multifunction F : X upper/lower semi-continuous upper/lower weakly continuous upper/lower clopen continuous upper/lower contra-continuous None of these implications is reversible as shown in the following examples. Example 12. Let X = {a, b, c, d} and τ = {, X, {a}, {a, b}, {a, b, c}}. Define a multifunction F : X X by F (a) = {b, c}, F (b) = {a}, F (c) = {a, d}, F (d) = {a}. Then F is upper contra-continuous but it is not upper semi-continuous. Define a multifunction F : X X by F (a) = {a, b}, F (b) = {b}, F (c) = {a, b}, F (d) = {d}. Then F is upper semi-continuous but it is not upper contra-continuous. Example 13. Let X = {a, b, c} and τ = {, X, {a}, {c}, {a, c}, {b, c}}. Define a multifunction F : X X by F (a) = {b, c}, F (b) = {a, c}, F (c) = {a, b}. Then F is upper contra-continuous but it is not upper clopen continuous. Define a multifunction G : X X by G(a) = {b, c}, G(b) = {a, b}, G(c) = {a, c}. Then G is upper semi-continuous but it is not upper contra-continuous. Theorem 14. The following are equivalent for a multifunction F : X Y : (1) F is lower contra-continuous multifunction, (2) F (A) is an open set for any closed set A Y, (3) F + (U) is a closed set for any open set U Y, (4) for each x X and for each closed set A such that F (x) A, there exists an open set U containing x such that if y U, then F (y) A. Proof. The proof is similar to that of Theorem 5. Theorem 15. Suppose that one of the following properties holds for a multifunction F : (X, τ) (Y, σ): (1) F (cl(a)) ker(f (A)) for every subset A of X, (2) cl(f + (A)) F + (ker(a)) for every subset A of Y. Then F is lower contra-continuous.
5 5 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS 79 Proof. Suppose that F (cl(a)) ker(f (A)) for every subset A of X. Let A Y. Then, F (cl(f + (A))) ker(a) and thus cl(f + (A)) F + (ker(a)). Therefore, the implication (1) (2) holds. Suppose that cl(f + (A)) F + (ker(a)) for every subset A of Y. Let A τ. By Lemma 6, cl(f + (A) F + (ker(a)) = F + (A). Thus, cl((f + (A)) = F + (A) and hence F + (A) is closed in X. Consequently, by Theorem 14, F is lower contra-continuous. Corollary 16. ([4]) For a function f : (X, τ) (Y, σ), the following are equivalent: (1) f is contra-continuous, (2) f 1 (A) is closed for any open set A in Y, (3) for each x X and for each closed set A containing f(x), there exists an open set U containing x such that f(u) A. Corollary 17. Let f : (X, τ) (Y, σ) be a function. Suppose that one of the following properties hold: (1) f(cl(a)) ker(f(a)) for every subset A of X, (2) cl(f 1 (A)) f 1 (ker(a)) for every subset A of Y. Then f is contra-continuous. Definition 18. A topological space X is called strongly S-closed [4] if every closed cover of X has a finite subcover. Theorem 19. Let F : X Y be an upper contra-continuous surjective multifunction. Suppose that F (x) is strongly S-closed for each x X. If X is compact, then Y is strongly S-closed. Proof. Let {A k } k I be a closed cover of Y. Since F (x) is strongly S-closed for each x X, there exists a finite subset I x of I such that F (x) k I x A k (= A x ). Since F is upper contra-continuous, there exists an open set U x of X containing x such that F (U x ) A x. The family {U x } x X is an open cover of X. Since X is compact, there exist x 1, x 2, x 3,...,x n in X such that X = n i=1 U x i. Thus, n Y = F (X) = F ( U xi ) = i=1 n F (U xi ) i=1 n A xi = i=1 n A k i=1k I xi and hence Y is strongly S-closed.
6 80 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 6 Theorem 20. If F : X Y is an upper/lower contra-continuous punctually connected surjective multifunction and X is connected, then Y is connected. Proof. Since F is upper/lower contra-continuous, then by Theorem 10, F is upper/lower weakly continuous. Then, the conclusion follows from Theorem 11 of [16]. Corollary 21. If f : X Y is contra-continuous surjection and X is connected, then Y is connected. Theorem 22. Let F : X Y and G : Y Z be multifunctions. If F is upper (lower) semi-continuous and G is upper (lower) contra-continuous, then G F : X Z is upper (lower) contra-continuous. Proof. Let A Z be a closed set. We have (G F ) + (A) = F + (G + (A)) ((G F ) (A) = F (G (A))). Since G is upper (lower) contra-continuous, then G + (A) (G (A)) is an open set. Since F is upper (lower) semi-continuous, then F + (G + (A)) (F (G (A))) is an open set. Thus, G F is an upper (lower) contracontinuous multifunction. Theorem 23. Let F : X Y be a multifunction and let A X. If F is a lower (upper) contra-continuous multifunction, then the restriction multifunction F A : A Y is lower (upper) contra-continuous. Proof. Let B Y be a closed set and x A and let x (F A ) (B). Since F is lower contra-continuous multifunction, then there exists an open set U in X containing x such that U F (B). This implies that x U A is open in A and hence U A (F A ) (B). Thus, F A is lower contracontinuous. Theorem 24. The following are equivalent for an open cover {A i } i I of a space X: (1) A multifunction F : X Y is upper contra-continuous (resp. lower contra-continuous), (2) The restriction F Ai : A i Y is upper contra-continuous (resp. lower contra-continuous) for each i I.
7 7 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS 81 Proof. (1) (2): Let i I and B be any closed set of Y. Since F is upper contra-continuous, F + (B) is open in X. Then (F Ai ) + (B) = F + (B) A i is open in A i. Thus, F Ai is upper contra-continuous. (2) (1): Let B be a closed set in Y. Since F Ai is upper contracontinuous for each i I, (F Ai ) + (B) = F + (B) A i is open in A i. Since A i is open in X, (F Ai ) + (B) is open in X for each i I and hence F + (B) = i I (F A i ) + (B) is open in X. Thus, F is upper contra-continuous. Theorem 25. Let F : X Y be a multifunction. Suppose that F (X) is endowed with the subspace topology. If F is upper contra-continuous, then F : X F (X) is upper contra-continuous. Proof. Let F be an upper contra-continuous multifunction. Then F + (V F (X)) = F + (V ) F + (F (X)) = F + (V ) is open for each closed subset V of Y. contra-continuous. Thus, F : X F (X) is upper Definition 26. A subset A of a space X is called: (1) α-paracompact [17] if every open cover of A is refined by a cover of A which consists of open sets of X and locally finite in X, (2) α-regular [7] if for each x A and each open set U of X containing x, there exists an open set V of X such that x V cl(v ) U. Lemma 27. ([7]) If A is an α-regular α-paracompact set of a space X and U is an open neighbourhood of A, then there exists an open set V of X such that A V cl(v ) U. Definition 28. ([2]) For a multifunction F : X Y, a multifunction cl(f ) : X Y is defined by cl(f )(x) = cl(f (x)) for each point x X. Similarly, we denote s-cl(f ), p-cl(f ), α-cl(f ), β-cl(f ). Lemma 29. If F : X Y is a multifunction such that F (x) is α- regular α-paracompact for each x X, then (1) G + (U) = F + (U) for each open set U of Y, (2) G (K) = F (K) for each closed set K of Y, where G denotes cl(f ), s-cl(f ), p-cl(f ), α-cl(f ), β-cl(f ). Proof. The proof follows from Lemma 3.6 of [13].
8 82 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 8 Lemma 30. For a multifunction F : X Y the following properties hold: (1) G (U) = F (U) for each open set U of Y, (2) G + (K) = F + (K) for each closed set K of Y, where G denotes cl(f ), s-cl(f ), p-cl(f ), α-cl(f ), β-cl(f ). Proof. The proof follows from Lemma 3.7 of [13]. Theorem 31. Let F : X Y be a multifunction. The following are equivalent: (1) F is upper contra-continuous, (2) G is upper contra-continuous. Proof. (1) (2): Let K be a closed set of Y. Then by Theorem 5 and Lemma 30, G + (K) = F + (K) is an open set of X. Hence G is upper contra-continuous. (2) (1): Let K be a closed set of Y. Then by Theorem 5 and Lemma 30, F + (K) = G + (K) is an open set of X. Hence F is upper contracontinuous. Theorem 32. Let F : X Y be a multifunction such that F (x) is α-regular α-paracompact for each x X. The following are equivalent: (1) F is lower contra-continuous, (2) G is lower contra-continuous. Proof. (1) (2): Let K be a closed set of X. Then by Lemma 29 and Theorem 14, G (K) = F (K) is open in X. Hence G is lower contracontinuous. (2) (1): Let K be a closed set of Y. Then by Lemma 29 and Theorem l4, F (K) = G (K) is an open set of X. Hence F is lower contracontinuous. 3. The graph multifunction and the product spaces Theorem 33. Let F : X Y be a multifunction. If the graph multifunction of F is upper contra-continuous, then F is upper contra-continuous. Proof. Let G F : X X Y be upper contra-continuous and x X. Let A be any closed set of Y containing F (x). Since X A is closed in X Y and G F (x) X A, there exists an open set U containing x such that G F (U) X A. By Lemma 2, U G + F (X A) = F + (A) and F (U) A. Thus, F is upper contra-continuous.
9 9 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS 83 Theorem 34. Let F : X Y be a multifunction. If G F : X X Y is lower contra-continuous, then F is lower contra-continuous. Proof. Let G F be lower contra-continuous and x X. Let A be any closed set in Y such that x F (A). This implies that X A is closed in X Y and G F (x) (X A) = ({x} F (x)) (X A) = {x} (F (x) A). Since G F is lower contra-continuous, there exists an open set U containing x such that U G F (X A). By Lemma 2, U F (A). Thus, F is lower contra-continuous. Corollary 35. Let f : X Y be a function. If the graph function g : X X Y, defined by g(x) = (x, f(x)) for each x X, is contracontinuous, then f is contra-continuous Theorem 36. Let (X, τ) and (X i, τ i ) be topological spaces (i I). If a multifunction F : X i I X i is an upper (lower) contra-continuous multifunction, then P i F is an upper (resp. lower) contra-continuous multifunction for each i I, where P i : i I X i X i is the projection for each i I. Proof. Let A i0 be a closed set in (X i0, τ i0 ). We have (P i0 F ) + (A i0 ) = F + (P + i 0 (A i0 )) = F + (A i0 i i 0 X i ). Since F is an upper contra-continuous multifunction, then F + (A i0 i i 0 X i ) is open in (X, τ). This implies that P i0 F is an upper contra-continuous multifunction. Thus, P i F is upper contra-continuous for each i I. The proof for lower contra-continuity is similar. Theorem 37. Let (X i, τ i ), (Y i, υ i ) be topological spaces and F i : X i Y i be a multifunction for each i I. Suppose that F : i I X i i I Y i is defined by F ((x i )) = i I F i(x i ). If F is upper (lower) contra-continuous, then F i is upper (lower) contra-continuous for each i I. Proof. Let A i Y i be a closed set. Since F is upper contra-continuous, then F + (A i Y j ) = F i + (A i) j i j i jx
10 84 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 10 is an open set. Thus, F i + (A i) is an open set and hence F i is upper contracontinuous. The proof for lower contra-continuity is similar. Acknowledgment. We would like to express our sincere gratitude to the Referee for valuable suggestions and comments which improved the paper. REFERENCES 1. Abd El-Monsef, M.E.; El-Deeb, S.N.; Mahmoud, R.A. β-open sets and β- continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), Banzaru, T. Multifunctions and M-product spaces (Romanian), Bull. Stiin. Teh. Inst. Politeh Timisoara Ser. Mat. Fiz Mec. Teor. Apl., 17 (31) (1972), Berge, C. Espaces Topologiques, Fonctions Multivoques, Dunod, Paris, Dontchev, J. Contra-continuous functions and strongly S-closed spaces, Internat J. Math. Math. Sci., 19 (1996), Ekici, E.; Popa, V. Some properties of upper and lower clopen continuous multifunctions, Bul. Şt. Univ. Politehnica Timisoara, Seria Mat.-Fiz., 50 (64), (2005), Jafari, S.; Noiri, T. Contra-super-continuous functions, Anal. Univ. Sci. Budapest, 42 (1999), Kovačević, I. Subsets and paracompactness, Univ. u Novom Sadu, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 14 (1984), Levine, N. Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), Mashhour, A.S.; Abd El-Monsef, M.E.; El-Deeb, S.N. On precontinuous and weak precontinuous mappings, Proc. Phys. Soc. Egypt, 53 (1982), Mrsevic, M. On pairwise R 0 and pairwise R 1 bitopological spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie 30 (1986), Njåstad, O. On some classes of nearly open sets, Pacific J. Math., 15 (1965), Noiri, T.; Popa, V. Almost weakly continuous multifunctions, Demonstratio Math., 26 (1993), Noiri, T.; Popa, V. A unified theory of weak continuity for multifunctions, Stud. Cerc. St. Ser. Mat. Univ. Bacau, 16 (2006),
11 11 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS Ponomarev, V.I. Properties of topological spaces preserved under multivalued continuous mappings on compacta, Amer. Math. Soc. Translations, 38 (2) (1964), Popa, V. Weakly continuous multifunctions, Boll. Un. Mat. Ital. (5), 15-A (1978), Smithson, R.E. Almost and weak continuity for multifunctions, Bull. Calcutta Math. Soc., 70 (1978), Wine, D. Locally paracompact spaces, Glasnik Mat., 10 (30) (1975), Received: 18.VI.2007 Revised: 14.IX.2007 Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, Canakkale, TURKEY College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, DENMARK Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, , JAPAN
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