ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS

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 :

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 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.

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 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.

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 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].

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 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

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), 77-90. 2. Banzaru, T. Multifunctions and M-product spaces (Romanian), Bull. Stiin. Teh. Inst. Politeh Timisoara Ser. Mat. Fiz Mec. Teor. Apl., 17 (31) (1972), 17-23. 3. Berge, C. Espaces Topologiques, Fonctions Multivoques, Dunod, Paris, 1959. 4. Dontchev, J. Contra-continuous functions and strongly S-closed spaces, Internat J. Math. Math. Sci., 19 (1996), 303-310. 5. Ekici, E.; Popa, V. Some properties of upper and lower clopen continuous multifunctions, Bul. Şt. Univ. Politehnica Timisoara, Seria Mat.-Fiz., 50 (64), (2005), 1-11. 6. Jafari, S.; Noiri, T. Contra-super-continuous functions, Anal. Univ. Sci. Budapest, 42 (1999), 27-34. 7. Kovačević, I. Subsets and paracompactness, Univ. u Novom Sadu, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 14 (1984), 79-87. 8. Levine, N. Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41. 9. Mashhour, A.S.; Abd El-Monsef, M.E.; El-Deeb, S.N. On precontinuous and weak precontinuous mappings, Proc. Phys. Soc. Egypt, 53 (1982), 47-53. 10. Mrsevic, M. On pairwise R 0 and pairwise R 1 bitopological spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie 30 (1986), 141-148. 11. Njåstad, O. On some classes of nearly open sets, Pacific J. Math., 15 (1965), 961-970. 12. Noiri, T.; Popa, V. Almost weakly continuous multifunctions, Demonstratio Math., 26 (1993), 363-380. 13. Noiri, T.; Popa, V. A unified theory of weak continuity for multifunctions, Stud. Cerc. St. Ser. Mat. Univ. Bacau, 16 (2006), 167-200.

11 ON UPPER AND LOWER CONTRA-CONTINUOUS MULTIFUNCTIONS 85 14. Ponomarev, V.I. Properties of topological spaces preserved under multivalued continuous mappings on compacta, Amer. Math. Soc. Translations, 38 (2) (1964), 119-140. 15. Popa, V. Weakly continuous multifunctions, Boll. Un. Mat. Ital. (5), 15-A (1978), 379-388. 16. Smithson, R.E. Almost and weak continuity for multifunctions, Bull. Calcutta Math. Soc., 70 (1978), 383-390. 17. Wine, D. Locally paracompact spaces, Glasnik Mat., 10 (30) (1975), 351-357. Received: 18.VI.2007 Revised: 14.IX.2007 Department of Mathematics, Canakkale Onsekiz Mart University, Terzioglu Campus, 17020 Canakkale, TURKEY eekici@comu.edu.tr College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, DENMARK jafari@stofanet.dk 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142, JAPAN t.noiri@nifty.com