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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 6= ∅} [3]. The graph multifunction GF : X → X ×Y of a multifunction F : X → Y is defined as follows GF (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 (1) (2) G+ F (A G− F (A 2 F + (B), × B) = 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 βcl(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 6= ∅ 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)) ⊂ for every subset A of Y , then F is upper contra-continuous. F − (ker(A)) 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 such that x ∈ F − (V ), there exists a clopen set U containing U ⊂ F − (V ). (2) upper clopen continuous if for each x ∈ X and each such that x ∈ F + (V ), there exists a clopen set U containing U ⊂ F + (V ). open set V x such that open set V x such that 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 Remark 11. The following diagram hold for a multifunction F : X → Y: 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 6= ∅, there exists an open set U containing x such that if y ∈ U , then F (y)∩A 6= ∅. 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 {Ak }k∈I be a closed cover of Y . Since F (x) is strongly S-closedSfor each x ∈ X, there exists a finite subset Ix of I such that F (x) ⊂ k∈Ix Ak (= Ax ). Since F is upper contra-continuous, there exists an open set Ux of X containing x such that F (Ux ) ⊂ Ax . The family {Ux }x∈X is an open cover of S X. Since X is compact, there exist x1 , x2 , x3 , ...,xn in X such that X = ni=1 Uxi . Thus, Y = F (X) = F ( n [ Uxi ) = i=1 and hence Y is strongly S-closed. n [ F (Uxi ) ⊂ i=1 n [ Axi = i=1 n [ [ Ak i=1k∈Ixi ¤ 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 {Ai }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 : Ai → 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) ∩ Ai is open in Ai . 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) ∩ Ai is open in Ai . Since + + A Si is open in+X, (F |Ai ) (B) is open in X for each i ∈ I and hence F (B) = ¤ i∈I (F |Ai ) (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 . Thus, F : X → F (X) is upper contra-continuous. ¤ 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 αregular 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 GF : 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 GF (x) ⊂ X × A, there exists an open set U containing x such + that GF (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 GF : X → X × Y is lower contra-continuous, then F is lower contra-continuous. Proof. Let GF 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 GF (x) ∩ (X × A) = ({x} × F (x)) ∩ (X × A) = {x} × (F (x) ∩ A) 6= ∅. Since GF 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, τQ) and (Xi , τi ) be topological spaces (i ∈ I). If a multifunction F : X → i∈I Xi is an upper (lower) contra-continuous multifunction, then Pi ◦ F is an upper Q (resp. lower) contra-continuous multifunction for each i ∈ I, where Pi : i∈I Xi → Xi is the projection for each i ∈ I. Proof. Let Ai0 be a closed set in (Xi0 , τi0 ). We have Q (Pi0 ◦ F )+ (Ai0 ) = F + (Pi+0 (Ai0 )) = F + (Ai0 × Xi ). i6=i0 Q Since F is an upper contra-continuous multifunction, then F + (Ai0 × i6=i0 Xi ) is open in (X, τ ). This implies that Pi0 ◦ F is an upper contra-continuous multifunction. Thus, Pi ◦ F is upper contra-continuous for each i ∈ I. The proof for lower contra-continuity is similar. ¤ Theorem 37. Let (Xi , τi ), (Yi , υi ) be topological spaces and F Q Qi : Xi → Yi be a multifunction Q for each i ∈ I. Suppose that F : i∈I Xi → i∈I Yi is defined by F ((xi )) = i∈I Fi (xi ). If F is upper (lower) contra-continuous, then Fi is upper (lower) contra-continuous for each i ∈ I. Proof. Let Ai ⊂ Yi be a closed set. Since F is upper contra-continuous, then Y Y F + (Ai × Yj ) = Fi+ (Ai ) × Xj i6=j i6=j 84 ERDAL EKICI, SAEID JAFARI and TAKASHI NOIRI 10 is an open set. Thus, Fi+ (Ai ) is an open set and hence Fi 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 βcontinuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77-90. 2. 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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 [email protected] College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, DENMARK [email protected] 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142, JAPAN [email protected]