EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No. 1, 2008, (82-98) ISSN 1307-5543 – www.ejpam.com Honorary Invited Paper Some Forms of C-continuity for Multifunctions Takashi Noiri1,∗, Valeriu Popa2 1 Department of Mathematics, Yatsushiro College of Technology, Yatsushiro,Kumamoto,866-8501 JAPAN 2 Department of Mathematics, University of Bacǎu, 5500 Bacǎu, ROMANIA Abstract. Lipski [14] introduced the notion of c-quasi-continuous multifunctions as a generalization of c-continuous multifunctions [20] and quasi-continuous multifunctions [26]. In this paper we obtain the unified theory of multifunctions containing upper/lower c-quasi-continuous multifunctions and upper/lower c-continuous multifunctions. AMS subject classifications: 54C08, 54C60. Key words: c-continuous, c-quasi-continuous, upper/lower C-m -continuous, multifunction. 1. Introduction Semi-open sets, preopen sets, α-open sets, β-open sets and δ-open sets play an important role in researching of generalizations of continuity in topological spaces. By using these sets many au- thors introduced and investigated various types of noncontinuous functions and multifunctions. In 1970, Gentry and Hoyle III [9] defined a function f : X → Y to be c-continuous at a point x ∈ X if for each open set V of Y containing f(x) and having compact complement, there exists an open set U of X containing x such that f(U) ⊂ V . Some properties of c-continuous functions are studied in [15], [16], [24] and other papers. Neubrunn [20] and Holá et al. [11] extended this no- tion to the setting of multifunctions. In [14], Lipski introduced the notion of C -quasicontinuous multifunctions as a generalization of C-continuous multifunctions and quasi-continuous multi- functions [26]. Some properties of C-quasi-continuous multifunctions are studied in [36]. In this paper we introduce upper/lower C-m-continuous multifunctions as multifunctions de- fined on a set satisfying some minimal conditions. We obtain some characterizations and sev- eral properties of such multifunctions which turn out unify some results established in [11], [14] and [36]. In the last section, we recall some types of modifications of open sets and point out the possibility for new forms of C -continuous multifunctions. ∗Corresponding author. Email addresses: noiri@as.yatsushiro-nct.ac.jp (T.Noiri) vpopa@ub.ro (V.Popa) http://www.ejpam.com 82 c© 2007 EJPAM All rights reserved. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 83 2. Preliminaries Let (X, τ) be a topological space and A a subset of X . The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. Definition 2.1. Let (X, τ) be a topological space. A subset A of X is said to be α-open [22] (resp. semi-open [13], preopen [18], β-open [1] or semi-preopen [4], b-open [5]) if A ⊂ Int(Cl(Int(A))) (resp. A ⊂ Cl(Int(A)), A ⊂ Int(Cl(A)), A ⊂ Cl(Int(Cl(A))), A ⊂ Int(Cl(A)) ∪ Cl(Int(A))). The family of all semi-open (resp. preopen, α-open, β-open, semi-preopen, b-open) sets in X is denoted by SO(X) (resp. PO(X), α(X), β(X), SPO(X), BO(X)). Definition 2.2. The complement of a semi-open (resp. preopen, α-open, β-open , semi-preopen, b-open) set is said to be semi-closed [7] (resp. preclosed [8], α-closed [19], β-closed [1], semi-preclosed [4], b-closed [5]). Definition 2.3. The intersection of all semi-closed (resp. preclosed, α-closed, β-closed, semi- preclosed, b-closed) sets of X containing A is called the semi-closure [7] (resp. preclosure [8], α-closure [19], β-closure [2], semi-preclosure [4], b-closure [5]) of A and is denoted by sCl(A) (resp. pCl(A), αCl(A), βCl(A), spCl(A), bCl(A). Definition 2.4. The union of all semi-open (resp. preopen, α-open, β-open, semi-preopen, b-open) sets of X contained in A is called the semi-interior (resp. preinterior, α-interior, β-interior, semi- preinterior, b-interior) of A and is denoted by sInt(A) (resp. pInt(A), αInt(A), βInt(A), spInt(A), bInt(A)). Throughout the present paper, (X, τ) and (Y, σ) (briefly X and Y ) always denote topological spaces and F : X → Y (resp. f : X → Y ) presents a multivalued (resp. single valued) function. For a multifunction F : X → Y , we shall denote the upper and lower inverse of a subset B of a space Y by F+(B) and F−(B), respectively, that is F+(B) = {x ∈ X : F (x) ⊂ B} and F−(B) = {x ∈ X : F (x) ∩B 6= ∅}. Definition 2.5. A multifunction F : (X, τ) → (Y, σ) is said to be (1) upper semi-continuous (briefly u.s.c.) at a point x ∈ X if for each open set V containing F (x), there exists an open set U of X containing x such that F (U) ⊂ V , (2) lower semi-continuous (briefly l.s.c.) at a point x ∈ X if for each open set V meeting F (x), there exists an open set U of X containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower semi-continuous on X if it has this property at each point of X . T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 84 Definition 2.6. A multifunction F : (X, τ) → (Y, σ) is said to be (a) upper C-continuous (briefly u.c.c.) [20] (resp. upper C-quasi-continuous (briefly u.c.q.c.) [14], [36]) if for each open set V containing F (x) and having compact complement, there exists an open (resp. semi-open) set U of X containing x such that F (U) ⊂ V , (2) lower C-continuous (briefly l.c.c.) [20] (resp. lower C-quasi-continuous (briefly l.c.q.c.) [14], [36]) at a point x ∈ X if for each open set V meeting F (x) and having compact complement, there exists an open (resp. semi-open) set U of X containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower C-continuous (resp. upper/lower C-quasi-continuous) on X if it has this prop- erty at each point of X . Remark 2.1. For the multifunctions defined above, the following implications hold: u.s.c. ⇒ u.c.c. ⇒ u.c.q.c.; l.s.c. ⇒ l.c.c. ⇒ l.c.q.c. 3. C-m-continuous multifunctions Definition 3.1. A subfamily mX of the power set P(X) of a nonempty set X is called a minimal structure (briefly m-structure) [31], [33] on X if ∅ ∈ mX and X ∈ mX . By (X, mX) (briefly (X, m)), we denote a nonempty set X with a minimal structure mX on X and call it an m-space. Each member of mX is said to be mX -open (briefly m-open) and the complement of an mX -open set is said to be mX -closed (briefly m-closed). Remark 3.1. Let (X, τ) be a topological space. Then the families τ , SO(X), PO(X), α(X), BO(X) and SPO(X) are all m-structures on X . Definition 3.2. Let (X, mX) be an m-space. For a subset A of X , the mX -closure of A and the mX -interior of A are defined in [17] as follows: (1) mX -Cl(A) = ∩{F : A ⊂ F, X − F ∈ mX}, (2) mX -Int(A) = ∪{U : U ⊂ A,U ∈ mX}. Remark 3.2. Let (X, τ) be a topological space and A be a subset of X . If mX = τ (resp. SO(X), PO(X), α(X), BO(X), SPO(X)), then we have (a) mX -Cl(A) = Cl(A) (resp. sCl(A), pCl(A), αCl(A), bCl(A), spCl(A)), (b) mX -Int(A) = Int(A) (resp. sInt(A), pInt(A), αInt(A), bInt(A), spInt(A)). Lemma 3.1. (Maki et al. [17]). Let (X, mX) be an m-space. For subsets A and B of X, the following properties hold: T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 85 (1) mX -Cl(X −A) = X −mX -Int(A) and mX -Int(X −A) = X −mX -Cl(A), (2) If (X −A) ∈ mX , then mX -Cl(A) = A and if A ∈ mX , then mX -Int(A) = A, (3) mX -Cl(∅) = ∅,mX -Cl(X) = X , mX -Int(∅) = ∅ and mX -Int(X) = X , (4) If A ⊂ B, then mX -Cl(A) ⊂ mX -Cl(B) and mX -Int(A) ⊂ mX -Int(B), (5) A ⊂ mX -Cl(A) and mX -Int(A) ⊂ A, (6) mX -Cl(mX -Cl(A)) = mX -Cl(A) and mX -Int(mX -Int(A)) = mX -Int(A). Lemma 3.2. (Popa and Noiri [31]). Let (X, mX) be an m-space and A a subset of X. Then x ∈ mX -Cl(A) if and only if U∩A 6= ∅ for every U ∈ mX containing x. Definition 3.3. A minimal structure mX on a nonempty set X is said to have property B [17] if the union of any family of subsets belonging to mX belongs to mX . Lemma 3.3. (Popa and Noiri [33]). For an m-structure mX on a nonempty set X , the following properties are equivalent: (1) mX has property B; (2) If mX -Int(A) = A, then A ∈ mX ; (3) If mX -Cl(A) = A, then A is mX -closed. Definition 3.4. Let (X, mX) be an m-space and (Y, σ) a topological space. A multifunction F : (X, mX) → (Y, σ) is said to be (1) upper C-m-continuous (briefly u.C.m.c.) at a point x ∈ X if for each open set V containing F (x) and having compact complement, there exists an mX -open set U containing x such that F (U) ⊂ V , (2) lower C-m-continuous (briefly l.C.m.c.) at a point x ∈ X if for each open set V meeting F (x) and having compact complement, there exists an mX -open set U containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower C-m-continuous on X if it has this property at every point of X . T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 86 Remark 3.3. Let (X, τ) and (Y, σ) be topological spaces. (1) If mX = τ (resp. SO(X)) and is upper/lower C-m-continuous, then F is upper/lower C-continuous (resp. upper/lower C-quasi-continuous). (2) For mX = α(X), PO(X), SPO(X) or BO(X), we can define new types of modifications of upper/lower C-continuous multifunctions. The definitions will be given in the last section. Theorem 3.1. For a multifunction F : (X, mX) → (Y, σ), the following properties are equiva- lent: (1) F is u.C.m.c. at x ∈ X; (2) x ∈ mX -Int(F+(V )) for each open set V containing F(x) and having compact comple- ment; (3) x ∈ F−(Cl(B)) for each subset B of Y having the compact closure such that x ∈ mX - Cl(F−(B)); (4) x ∈ mX -Int(F+(B)) for each subset B of Y such that Y − Int(B) is compact and x ∈ F+(Int(B)). Proof. (1) ⇒ (2): Let V be any open set of Y containing F (x) and having compact comple- ment. There exists an mX -open set U containing x such that F (U) ⊂ V . Thus x ∈ U ⊂ F+(V ). Since U ∈ mX , we have x ∈ mX -Int(F+(V )). (2) ⇒ (3): Suppose that B is any subset of Y having the compact closure. Then Cl(B) is closed and Y -Cl(B) is an open set having compact complement. Let x /∈ F−(Cl(B)). Then x ∈ X −F−(Cl(B)) = F+(Y −Cl(B)). This implies F (x) ⊂ Y −Cl(B). Since Y −Cl(B) is an open set having compact complement, by (2) we have x ∈ mX -Int(F+(Y − Cl(B))) = mX -Int(X − F−(Cl(B)) = X −mX -Cl(F−(Cl(B))) ⊂ X −mX -Cl(F−(B)). Hence x /∈ mX -Cl(F−(B)). (3) ⇒ (4): Let B be any subset of Y such that Y -Int(B) is compact and let x /∈ mX - Int(F+(B)). Then we have x ∈ X − mX -Int(F+(B)) = mX -Cl(X − F+(B)) = mX - Cl(F−(Y −B)). By (3), we have x ∈ F−(Cl(Y −B)) = F−(Y − Int(B)) = X−F+(Int(B)). Hence x /∈ F+(Int(B)). (4) ⇒ (1): Let V be any open set of Y containing F (x) and having compact complement. We have F+(V ) = F+(Int(V )) . Then Y − Int(V ) = Y − V which is compact and by (4) x ∈ mX - Int( F+(V )). Therefore, there exists an mX -open set U containing x such that x ∈ U ⊂ F+(V ). Thus F (U) ⊂ V . This shows that F is u.C.m.c. at x. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 87 Theorem 3.2. For a multifunction F : (X, mX) → (Y, σ), the following properties are equiva- lent: (1) F is l.C.m.c. at x ∈ X; (2) x ∈ mX -Int(F−(V )) for each open set V containing F(x) and having compact comple- ment; (3) x ∈ F+(Cl(B)) for each subset B of Y having the compact closure such that x ∈ mX - Cl(F+(B)); (4) x ∈ mX -Int(F−(B)) for each B of Y such that Y−Int(B) is compact and x ∈ F−(Int(B)). Proof. The proof is similar to that of Theorem 3.1 Theorem 3.3. For a multifunction F : (X, mX) → (Y, σ), the following properties are equiva- lent: (1) F is u.C.m.c.; (2) F+(V ) = mX -Int(F+(V )) for each open set V of Y having compact complement; (3) F−(K) = mX -Cl(F−(K)) for every compact closed set K of Y; (4) mX -Cl(F−(B)) ⊂ F−(Cl(B)) for every subset B of Y having the compact closure; (5) F+(Int(B)) ⊂ mX -Int(F+(B)) for every subset B of Y such that Y − Int(B) is compact. Proof. (1) ⇒ (2): Let V be any open set of Y having compact complement and x ∈ F+(V ). Then F (x) ⊂ V and by Theorem 3.1, x ∈ mX -Int(F+(V )). By Lemma 3.1, we have mX - Int(F+(V )) ⊂ F+(V ). Therefore, we obtain F+(V ) = mX -Int(F+(V )). (2)⇒ (3): Let K be any compact closed set of Y . Then, by Lemma 3.1 we have X−F−(K) = F+(Y −K) = mX -Int(F+(Y −K)) = mX -Int(X −F−(K)) = X −mX -Cl(F−(K)). There- fore, we obtain F−(K) = mX -Cl(F−(K)). (3) ⇒ (4): Let B be any subset of Y having the compact closure. By Lemma 3.1, we have F−(B) ⊂ F−(Cl(B)) = mX -Cl(F−(Cl(B))). Hence mX -Cl(F−(B)) ⊂ mX -Cl(F−(Cl(B))) = F−(Cl(B)). (4) ⇒ (5): Let B be a subset of Y such that Y − Int(B) is compact. Then by Lemma 3.1 we have X −mX -Int(F+(B)) = mX -Cl(X − F+(B)) = mX -Cl(F−(Y −B)) ⊂ ⊂ mX -Cl(F−(Y − Int(B))) ⊂ F−(Y − Int(B)) = X − F+(Int(B)). Therefore, we obtain F+(Int(B)) ⊂ mX -Int(F+(B)). T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 88 (5) ⇒ (1): Let x ∈ X and V be any open set of Y containing F (x) and having compact complement. Then x ∈ F+(V ) = F+(Int(V )) ⊂ mX -Int(F+(V )). By Theorem 3.1, F is u.C.m.c. at x. Theorem 3.4. For a multifunction , the following properties are equivalent: (1) F is l.C.m.c.; (2) F−(V ) = mX -Int(F−V )) for each open set V of Y having compact complement; (3) F+(K) = mX -Cl(F+(K)) is for every compact closed set K of Y; (4) mX -Cl(F+(B)) ⊂ F+(Cl(B)) for every subset B of Y having the compact closure; (5) F−(Int(B)) ⊂ mX -Int(F−(B)) for every subset B of Y such that Y − Int(B) is compact. Proof. The proof is similar to that of Theorem 3.3. Corollary 3.1. Let (X, mX) be an m-space and mX have property B. For a multifunction F : (X, mX) → (Y, σ), the following properties are equivalent: (1) F is u.C.m.c. (resp. l.C.m.c.); (2) F+(V ) (resp. F−(V )) is mX -open for each open set V of Y having compact complement; (3) F−(K) (resp. F+(K)) is mX -closed for every compact closed set K of Y. Proof. This is an immediate consequence of Theorems 3.3 and 3.4 and Lemma 3.3. Remark 3.4. Let (X, τ) and (Y, σ) be topological spaces. If mX = τ (resp. SO(X)) and is upper/lower C-m-continuous, then by Theorems 3.3 and 3.4 and Corollary 3.1 we obtain the results established in Proposition 1 of [11] (resp. Theorem 1 of [14], Theorems 3.3 and 3.4 of [36]). Definition 3.5. A function f : (X,mX) → (Y, σ) is said to be c-m-continuous if for each point x ∈ X and each open set V containing f(x) and having compact complement, there exists an mX -open set U containing x such that f(U) ⊂ V . Corollary 3.2. For a function f : (X,mX) → (Y, σ), the following properties are equivalent: (1) f is c-m-continuous; (2) f−1(V ) = mX -Int(f−1(V )) for each open set V of Y having compact complement; (3) f−1(K) = mX -Cl(f−1(K)) for every compact closed set K of Y; T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 89 (4) mX -Cl(f−1(B)) ⊂ f−1(Cl(B)) for every subset B of Y having the compact closure; (5) f−1(Int(B)) ⊂ mX -Int(f−1(B)) for every subset B of Y such that Y −Int(B) is compact. Remark 3.5. Let (X, τ) and (Y, σ) be topological spaces. If mX = τ and f : (X,mX) → (Y, σ) is c-m-continuous, then by Corollary 3.2 we obtain the results established in Theorem 1 of [9] and Theorems 2 of [15]. Corollary 3.3. A multifunction is u.C.m.c. (resp. l.C.m.c.) if F−(K) = mX -Cl(F−(K)) (resp. F+(K) = mX -Cl(F+(K))) for every compact set K of Y. Proof. Let G be any open set of Y having compact complement. Then Y − G is a compact closed set. By the hypothesis, X − F+(G) = F−(Y − G) = mX -Cl(F−(Y − G)) = mX - Cl(X − F+(G)) = X − mX -Int(F+(G)) and hence, F+(G) = mX -Int(F+(G)). It follows from Theorem 3.3 that F is u.C.m.c. The proof of lower C-m-continuity is entirely similar. Remark 3.6. (1) Let mX = τ (resp. SO(X)), then by Corollary 3.3 we obtain the results estab- lished in Proposition 2 of [20] (resp. Corollary 3.3 of [36]). (2) It is shown in Remark 4 of [11] that the converse of Corollary 3.3 is not true. Definition 3.6. A subset A of a topological space (X, τ) is said to be (1) α-paracompact [40] if every cover of A by open sets of X is refined by a cover of A which consists of open sets of X and is locally finite in X , (2) α-regular [12] if for each a ∈ A and each open set U of X containing a, there exists an open set G of X such that a ∈ G ⊂ Cl(G) ⊂ U . Lemma 3.4. (Kovačević [12]) If A is an α-regular α-paracompact set of a topological space X and U is an open neighbor- hood of A, then there exists an open set G of X such that A ⊂ G ⊂ Cl(G) ⊂ U . For a multifunction F : (X, mX) → (Y, σ), by ClF : (X, mX) → (Y, σ) we denote a multifunction defined as follows: (ClF )(x) = Cl(F (x)) for each point x ∈ X . Similarly, we can define αClF , sClF , pClF , spClF , bClF . Lemma 3.5. If is a multifunction such that F (x) is α-paracompact and α-regular for each x ∈ X , then for each open set V of Y F+(V ) = G+(V ), where G denotes ClF, αClF, sClF, pClF, bClF or spClF. Proof. The proof is similar to that of Lemma 3.3 of [30]. Theorem 3.5. Let be a multifunction such that F (x) is α-regular and α-paracompact for each x ∈ X . Then the following properties are equivalent: (1) F is u.C.m.c.; (2) ClF is u.C.m.c.; (3) αClF is u.C.m.c.; (4) sClF is u.C.m.c.; (5) pClF is u.C.m.c.; (6) bClF is u.C.m.c.; (7) spClF is u.C.m.c. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 90 Proof. We set G = ClF , αClF , sClF , pClF , bClF or spClF . Suppose that F is u.C.m.c. Let V be any open set of Y containing G(x) and having compact complement. By Lemma 3.5, we have x ∈ G+(V ) = F+(V ) and by Theorem 3.1 there exists U ∈ mX containing x such that F (U) ⊂ V . Since F (u) is α-paracompact and α-regular for each u ∈ U , by Lemma 3.4 there exists an open set H such that F (u) ⊂ H ⊂ Cl(H) ⊂ V ; hence G(u) ⊂ Cl(H) ⊂ V for every u ∈ U . Therefore, we obtain G(U) ⊂ V . This shows that G is u.C.m.c. Conversely, suppose that G is u.C.m.c. Let x ∈ X and V be any open set of Y containing F (x) and having compact complement. By Lemma 3.5, we have x ∈ F+(V ) = G+(V ) and hence G(x) ⊂ V . By Theorem 3.1, there exists U ∈ mX containing x such that G(U) ⊂ V . Therefore, we obtain U ⊂ G+(V ) = F+(V ) and hence F (U) ⊂ V . This shows that F is u.C.m.c. Lemma 3.6. If is a multifunction, then for each open set V of Y G−(V ) = F−(V ), where G = ClF, αClF, sClF, pClF, bClF or spClF. Proof. The proof is similar to that of Lemma 3.4 of [30]. Theorem 3.6. For a multifunction , the following properties are equivalent: (1) F is l.C.m.c.; (2) ClF is l.C.m.c.; (3) αClF is l.C.m.c.; (4) sClF is l.C.m.c.; (5) pClF is l.C.m.c.; (6) bClF is l.C.m.c.; (7) spClF is l.C.m.c. Proof. By using Lemma 3.6 this is shown similarly as in Theorem 3.5. Remark 3.7. Let (X, τ) and (Y, σ) be topological spaces and mX = SO(X). By Theorems 3.5 and 3.6, we obtain the results established in Theorems 3.5 and 3.6 of [36]. 4. The set of points of m-c-discontinuity For a multifunction F : (X, mX) → (Y, σ), the sets D+ mc(F ) and D− mc(F ) are defined as follows: D+ mc(F ) = {x ∈ X : F is not upper C-m-continuous at x }, D− mc(F ) = {x ∈ X : F is not lower C-m-continuous at x }. Theorem 4.1. For a multifunction , the following properties hold: D+ mc(F ) = ⋃ G∈cσ{F+(G)− [mX -Int(F+(G))]} = ⋃ B∈ iP (Y ) {F+(Int(B))− [mX -Int(F+(B))]} = ⋃ B∈ cP (Y ) {mX -Cl(F−(B))− F−(Cl(B))} = ⋃ H∈ cF {mX -Cl(F−(H))− F−(H)}, where cσ is the family of open set G having compact complement, iP(Y ) is the family of subset B of Y such that Y − Int(B) is compact, cP(Y ) is the family of subset B of Y with the compact closure and cF is the family of closed compact subsets of Y. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 91 Proof. We shall show only the first equality and the last since the proof of any other equality is similar to the first. Let x ∈ D+ mc(F ). By Theorem 3.1, there exists an open set V of Y having compact comple- ment such that x ∈ F+(V ) and x /∈ mX - Int(F+(V )). Therefore, we obtain x ∈ F+(V )− [mX - Int(F+(V ))] ⊂ ⋃ G∈cσ{F+(G) − [mX -Int(F+(G))]}. Conversely, let x ∈ ⋃ G∈cσ{F+(G) − [mX -Int(F+(G))]}. There exists V ∈ cσ such that x ∈ F+(V )− [mX -Int(F+(V ))]. By Theo- rem 3.1, we obtain x ∈ D+ mc(F ). We prove the last equality.⋃ H∈ cF {mX -Cl(F−(H))−F−(H)}⊂ ⋃ B∈ cP (Y ) {mX -Cl(F−(B))−F−(Cl(B))} = D+ mc(F ). Conversely, by Lemma 3.1 we have D+ mc(F ) = ⋃ B∈ cP (Y ) {mX - Cl(F−(B))− F−(Cl(B))} ⊂ ⋃ H∈ cF {mX -Cl(F−(H))− F−(H)}. Theorem 4.2. For a multifunction , the following properties hold: D− mc(F ) = ⋃ G∈cσ{F−(G)− [mX -Int(F−(G))]} = ⋃ B∈ iP (Y ) {F−(Int(B))− [mX -Int(F−(B))]} = ⋃ B∈ cP (Y ) {mX -Cl(F+(B))− F+(Cl(B))} = ⋃ H∈ cF {mX -Cl(F+(H))− F+(H)}. Proof. The proof is similar to that of Theorem 4.1 Remark 4.1. If is a multifunction and mX = τ (resp. SO(X)), then the set of points of upper/lower C-discontinuity (resp. c-quasi-discontinuity) is obtained. Definition 4.1. Let (X, mX) be an m-space and A a subset of X . The mX -frontier of A [35], denoted by mX -Fr(A), is defined as follows: mX -Fr(A) = mX -Cl(A) ∩mX -Cl(X −A) = mX -Cl(A)−mX -Int(A). Theorem 4.3. The set of all points x ∈ X at which a function is not u.C.m.c. (resp. l.C.m.c.) is identical with the union of the mX -frontiers of the u.C.m.c. (resp. l.C.m.c.) inverse images of open sets containing (resp. meeting) F(x) and having compact complement. Proof. Suppose that F is not u.C.m.c. at x ∈ X . Then, there exists an open set V of Y containing F (x) and having compact complement such that U ∩ (X − F+(V )) 6= ∅ for every mX -open set U containing x. Hence, by Lemma 3.2 we have x ∈ mX -Cl(X − F+(V )). On the other hand, we have x ∈ F+(V ) ⊂ mX -Cl(F+(V )) and hence x ∈ mX -Fr(F+(V )). Conversely, suppose that V is an open set of Y containing F (x) and having compact com- plement such that x ∈ mX -Fr(F+(V )). If F is u.C.m.c. at x ∈ X , then there exists U ∈ mX containing x such that U ⊂ F+(V ) and hence, x ∈ mX -Int(F+(V )). This is a contradiction and hence, F is not u.C.m.c. The proof for l.C.m.c. is similar. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 92 5. m-continuity and C-m-continuity Definition 5.1. A multifunction is said to be (1) upper m-continuous (briefly u.m.c.) at x ∈ X [34] if for each open set V containing F (x), there exists U ∈ mX containing x such that F (U) ⊂ V , (2) lower m-continuous (briefly l.m.c.) at x ∈ X [34] if for each open set V such that F (x) ∩ V 6= ∅, there exists U ∈ mX containing x such that F (u) ∩ V 6= ∅ for every u ∈ U , (3) upper/lower m-continuous on X if it has the properties at each point of X . Remark 5.1. Let (X, τ) be a topological space and mX = τ (resp. SO(X), PO(X), α(X), SPO(X), BO(X)). If a multifunction is upper/lower m-continuous, then F is upper/lower contin- uous (resp. upper/lower semi-continuous [27] or upper/lower quasi-continuous [28], upper/lower precontinuous [29], upper/lower α-continuous [21], upper/lower β-continuous [30], upper/lower b-continuous or upper/lowerγ-continuous [3]). A topological space (Y, σ) is called a KC-space [39] if every compact set of Y is closed. Definition 5.2. A multifunction is said to be m-bounded at the point p ∈ X if there exists U ∈ mX containing p and a compact set C of Y such that F (x) ⊂ C for each x ∈ U . Theorem 5.1. Let (Y, σ) be a KC space and X a nonempty set with two minimal structures m1 X and m2 X such that U ∩ V ∈ m2 X for every U ∈ m1 X and V ∈ m2 X . Then F : (X,m2 X) → (Y, σ) is u.m.c. (resp. l.m.c.) at p ∈ X if the following conditions satisfy: (1) F : (X,m1 X) → (Y, σ) is m-bounded at p ∈ X , (2) F : (X,m2 X) → (Y, σ) is u.C.m.c. (resp. l.C.m.c.) at p ∈ X . Proof. We prove only the first case, the proof of the second being entirely analogous. Let U ∈ m1 X containing p and C be a compact set of Y such that F (x) ⊂ C for each x ∈ U . Let V be any open set of Y such that F (p) ⊂ V . Put G = V ∪ (Y − C). Then G is open and Y −G is compact. By the condition (2), there exists W ∈ m2 X containing p such that F (x) ⊂ G for every x ∈ W . Put H = W ∩ U , then H ∈ m2 X containing p and F (x) ⊂ G ∩ C for any x ∈ H . Then F (x) ⊂ V for any x ∈ H . Therefore, F : (X, m2 X) → (Y, σ) is u.m.c. at p ∈ X . Remark 5.2. If m1 X = m2 X = τ , then by Theorem 5.1 we obtain the result established in Propo- sition 5 of [11]. Definition 5.3. An m-space (X, mX) is said to be m-saturated if for any x ∈ X the intersection of all mX -open sets containing x is mX -open. Theorem 5.2. Let (X, mX) be an m-saturated m-space and (Y, σ) a T1-space. If is u.C.m.c., then F is u.m.c. T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 93 Proof. Suppose that F is not u.m.c. at some point x0 ∈ X . There exists an open set V of Y such that F (x0) ⊂ V and F (U) ∩ (Y − V ) 6= ∅ for every U ∈ mX containing x0. Let U0 be the intersection of all mX -open sets containing x0. Then U0 ∈ mX and there exists z1 ∈ U0 such that F (z1) ∩ (Y − V ) 6= ∅. Hence there exists y ∈ F (z1) ∩ (Y − V ). The set Y − {y} is an open set with compact complement. Since F (x0) ⊂ Y −{y} and F is u.C.m.c. at x0, there exists G ∈ mX containing x0 such that for any x ∈ G we have F (x) ⊂ Y − {y}. This is a contradiction. Since U0 ⊂ G, z1 ∈ G and F (z1) ⊂ Y − {y}. This contradicts that y ∈ F (z1). Remark 5.3. If mX = τ , then by Theorem 5.2 we obtain the result established in Proposition 8 of [11]. Theorem 5.3. Let (X,mX) be an m-saturated m-space and (Y, σ) a locally compact Hausdorff space. If is an u.C.m.c. and closed valued multifunction, then F is u.m.c. Proof. Suppose that F is not u.m.c. at x0 ∈ X . Then, there exists an open set V of Y such that F (x0) ⊂ V and F (U) ∩ (Y − V ) 6= ∅ for every U ∈ mX containing x0. Let U0 be the intersection of all mX -open sets containing x0. Then U0 ∈ mX and there exists z1 ∈ U0 such that F (z1) ∩ (Y − V ) 6= ∅. Hence there exists y ∈ F (z1) ∩ (Y − V ). Since (Y, σ) is locally compact Hausdorff, (Y, σ) is regular. Since F (x0) is a closed set and y /∈ F (x0), there exists an open set W containing y such that Cl(W ) is a compact set and Cl(W ) ⊂ Y − F (x0). Since F (x0) ⊂ Y − Cl(W ) and F is u.C.m.c. at x0, there exists an mX -open set G containing x0 and F (x) ⊂ Y − Cl(W ) for each x ∈ G. This is a contradiction. Since z1 ∈ U0 ⊂ G, F (z1) ⊂ Y − Cl(W ). This contradicts that F (z1) ∩ Cl(W ) 6= ∅. Remark 5.4. If mX = τ , then by Theorem 5.3 we obtain the result established in Proposition 10 of [11]. Theorem 5.4. Let (X, mX) be an m-saturated m-space and (Y, σ) a KC space. If is l.C.m.c. and for each x ∈ X there exists a compact set Cx such that F (x) ⊂ Cx, then F is l.m.c. Proof. Suppose that F is not l.m.c. at x0 ∈ X . Then, there exists an open set V of Y such that F (x0) ∩ V 6= ∅ and for each U ∈ mX containing x0 there exists u ∈ U such that F (u) ∩ V = ∅. Let U0 be the intersection of all mX -open sets containing x0. Then U0 ∈ mX and there exists x ∈ U0 such that F (x)∩ V = ∅. By the hypothesis, there exists a compact set Cx such that F (x) ⊂ Cx. Therefore, we have F (x) ⊂ Cx − V and Cx − V is a compact set.The set Y − (Cx − V ) is open and F (x0) ∩ (Y − (Cx − V )) 6= ∅. Since F is l.C.m.c. at x0, there exists an mX -open set G containing x0 such that for any z ∈ G we have F (z) ∩ (Y − (Cx − V )) 6= ∅. This is a contradiction because x ∈ U0 ⊂ G and F (x) ⊂ Cx − V . Remark 5.5. If mX = τ , then by Theorem 5.4 we obtain the result established in Proposition 11 of [11]. 6. Some properties Definition 6.1. A multifunction is said to be upper C-m-rarely continuous at a point x ∈ X if for each open set G of Y containing F (x) and having compact complement, there exists a rare T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 94 set RG with Cl(RG) ∩ G = ∅ and an mX -open set U containing x such that F (U) ⊂ G ∪ RG. A multifunction is said to be upper C-m-rarely continuous if it has this property at each point x ∈ X . Theorem 6.1. Let X be a nonempty set with two minimal structures m1 X and m2 X such that U ∩ V ∈ m2 X for every U ∈ m1 X and V ∈ m2 X . Then F : (X,m2 X) → (Y, σ) is u.C.m.c. if the following conditions satisfy: (1) F : (X,m1 X) → (Y, σ) is upper C-m-rarely continuous and (2) for each open set G containing F(x) and having compact complement, F−(Cl(RG)) is an m2 X -closed set of X, where RG is the rare set of Definition 6.1. Proof. Let x ∈ X and G be any open set of Y containing F (x) and having compact comple- ment. By the condition (1), there exists V ∈ m1 X containing x and a rare set RG with Cl(RG) ∩ G = ∅ such that F (V ) ⊂ G ∪ RG. If we suppose that x ∈ F−(Cl(RG)), then Cl(RG) ∩G 6= ∅. This is a contradiction. Thus x /∈ F−(Cl(RG)). Put U = V ∩ (X − F−(Cl(RG))) . Then U ∈ m2 X and x ∈ U since x ∈ V and x ∈ X − F−(Cl(RG)). Let u ∈ U , then F (u) ⊂ G ∪ RG and F (u) ∩ Cl(RG) = ∅. Therefore, we have F (u) ∩ RG = ∅ and hence, F (u) ⊂ G for each u ∈ U . Since U ∈ m2 X containing x, it follows that F : (X, m2 X) → (Y, σ) is u.C.m.c. Definition 6.2. For a multifunction , the graph G(F ) = {(x, F (x)) : x ∈ X} is said to be strongly m-closed [32] if for each (x, y) ∈ (X × Y ) −G(F ), there exist an mX -open set U containing x and an open set V of Y containing y such that [U × Cl(V )] ∩G(F ) = ∅. Lemma 6.1. A multifunction has a strongly m-closed graph if and only if for each (x, y) ∈ (X × Y )−G(F ), there exist an mX -open set U containing x and an open set V of Y containing y such that F (U) ∩ Cl(V ) = ∅. Theorem 6.2. Let (Y, σ) be a locally compact Hausdorff space. If a multifunction is u.C.m.c. and F (x) is closed for each x ∈ X , then G(F ) is strongly m-closed. Proof. Let (x, y) ∈ (X×Y )−G(F ). Then y /∈ F (x). Since Y is locally compact Hausdorff, Y is regular. Since F (x) is a closed set and y /∈ F (x), there exists an open set V in Y containing y such that Cl(V ) is a compact set and Cl(V ) ⊂ X−F (x) and hence, F (x) ⊂ Y −Cl(V ). Since F is u.C.m.c. at x and Y − Cl(V ) is an open set having compact complement, there exists U ∈ mX containing x such that F (U) ⊂ Y −Cl(V ). This implies that F (U) ∩Cl(V ) = ∅ and by Lemma 6.1 G(F ) is strongly m-closed. 7. New modifications of C-continuous multifunctions For modifications of open sets defined in Definition 2.1, the following relationships are known: open ⇒ α-open ⇒ preopen ⇓ ⇓ semi-open ⇒ b-open ⇒ semi-preopen T.Noiri,V.Popa / Eur. J. Pure Appl. Math, 1 (2008), (82-98) 95 First, we can define the following modifications of upper/lower C -continuous multifunctions. Definition 7.1. A multifunction F : (X, τ) → (Y, σ) is said to be (1) upper C-α-continuous (resp. upper C-precontinuous, upper C-b-continuous, upper C- sp-continuous) at a point x ∈ X if for each open set V containing F (x) and having compact complement, there exists an α-open (resp. preopen, b-open, semi-preopen) set U containing x such that F (U) ⊂ V , (2) lower C-α-continuous (resp. lower C-precontinuous, lower C-b-continuous, lower C-sp- continuous) at a point x ∈ X if for each open set V meeting F (x) and having compact comple- ment, there exists an α-open (resp. preopen, b-open, semi-preopen) set U containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower C-α-continuous (resp. upper/lower C-precontinuous, upper/lower C-b-continuous, upper/lower C-sp-continuous) on X if it has this property at each x ∈ X . For multifunctions defined in Definition 7.1, the following relationships hold: upper C-con. ⇒ upper C-α-con. ⇒ upper C-precon. ⇓ ⇓ upper C-quasi-con. ⇒ upper C-b-con. ⇒ upper C-sp-con. Remark 7.1. In the diagram above, ”con.” means continuity and the analogous diagram holds for the case ”lower”. Let define the further modifications of upper/lower C-continuous multifunctions. For the pur- pose, we recall the definitions of the θ -closure and the δ-closure due to Veličko [38]. Let (X, τ) be a topological space and A a subset of X . A point x ∈ X is called a θ-cluster (resp. δ-cluster) point of A if Cl(V ) ∩A 6= ∅ (resp. Int(Cl(V )) ∩A 6= ∅) for every open set V containing x. The set of all θ-cluster (resp. δ-cluster) points of A is called the θ-closure (resp. δ-closure) of A and is denoted by Clθ(A) (resp. Clδ(A)) [38]. A subset A is said to be θ-closed (resp. δ-closed) if Clθ(A) = A (resp. Clδ(A) = A). The complement of a θ-closed (resp. δ-closed) set is said to be θ-open (resp. δ-open). The union of all θ-open (resp. δ-open) sets contained in the subset A is called the θ-interior (resp. δ-interior) of A and is denoted by Intθ(A) (resp. Intδ(A)). Definition 7.2. A subset A of a topological space (X, τ) is said to be (1) δ-semiopen [25] (resp. θ-semiopen [6]) if A ⊂ Cl(Intδ(A)) (resp. A ⊂ Cl(Intθ(A))), (2) δ-preopen [37] (resp. θ-preopen [23]) if A ⊂ Int(Clδ(A)) (resp. A ⊂ Int(Clθ(A))), (3) δ-sp-open [10] (resp. θ-sp-open [23]) if A ⊂ Cl(Int(Clδ(A))) (resp. A ⊂ Cl(Int(Clθ(A)))). By δSO(X) (resp. δPO(X), δSPO(X), θ SO(X), θPO(X), θSPO(X)), we denote the collec- tion of all δ-semiopen (resp. δ-preopen, δ-sp-open, θ -semiopen, θ-preopen, θ-sp-open) sets of a topological space (X, τ). These six collections are all m-structures with property B. It is known that the families of all θ-open sets and δ-open sets of (X, τ) are topologies for X , respectively. In [23] and [6], the following relationships are known: REFERENCES 96 θ-open ⇒ δ -open ⇒ open ⇒ preopen ⇒ δ -preopen ⇒ θ-preopen ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ θ-semiopen ⇒ δ-semiopen ⇒ semi-open ⇒ sp-open ⇒ δ-sp-open ⇒ θ-sp-open Definition 7.3. A multifunction F : (X, τ) → (Y, σ) is said to be (1) upper C-θ-continuous (resp. upper C-θ-precontinuous, upper C-θ-semi-continuous, upper C-θ-sp-continuous) at a point x ∈ X if for each open set V containing F (x) and having compact complement, there exists a θ-open (resp. θ-preopen, θ-semiopen, θ-sp-open) set U containing x such that F (U) ⊂ V , (2) lower C-θ-continuous (resp. lower C-θ-precontinuous, lower C-θ-semi-continuous, lower C-θ-sp-continuous) at a point x ∈ X if for each open set V meeting F (x) and having compact complement, there exists a θ-open (resp. θ-preopen, θ-semiopen, θ-sp-open) set U containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower C-θ-continuous (resp. upper/lower C-θ-precontinuous, upper/lower C-θ- semi-continuous, upper/lower C-θ-sp-continuous) on X if it has this property at each x ∈ X . Definition 7.4. A multifunction F : (X, τ) → (Y, σ) is said to be (1) upper C-δ-continuous (resp. upper C-δ-precontinuous, upper C-δ-semi-continuous, upper C-δ-sp-continuous) at a point x ∈ X if for each open set V containing F (x) and having compact complement, there exists a δ-open (resp. δ-preopen, δ-semiopen, δ-sp-open) set U containing x such that F (U) ⊂ V , (2) lower C-δ-continuous (resp. lower C-δ-precontinuous, lower C-δ-semi-continuous, lower C-δ-sp-continuous) at a point x ∈ X if for each open set V meeting F (x) and having compact complement, there exists a δ-open (resp. δ-preopen, δ-semiopen, δ-sp-open) set U containing x such that F (u) ∩ V 6= ∅ for each u ∈ U , (3) upper/lower C-δ-continuous (resp. upper/lower C-δ-precontinuous, upper/lower C-δ-semi- continuous, upper/lower C-δ-sp-continuous) on X if it has this property at each x ∈ X . 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