Upper and Lower α I Continuous Multifunctions

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1 International Mathematical Forum, Vol. 9, 2014, no. 5, HIKARI Ltd, Upper and Lower α I Continuous Multifunctions Metin Akdağ and Fethullah Erol Cumhuriyet University, Science Faculty Department of Mathematics, Sivas, Turkey Copyright c 2014 Metin Akdağ and Fethullah Erol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we study a new form of continuity of multifunctions by using α I open sets and obtain several characterizations and some properties of these multifunctions. Also we investigate their relationship with other types of I continuity of multifunctions. Matghematics Subject Classification: Primary 54C10; secondary 54C60 Keywords: Multifunctions, α I continuous multifunctions, α I open set, I continuous functions 1. Introduction and Preliminaries The theory of multifunctions was first codified by [3]. A multifunction is a set-valued function. The concept of multifunctions has advanced in a variety of ways and applications of this theory can be found, specially in functional analysis and fixed point theory. In 1990, D. Janković and T. R. Hamlett [4] introduced the notion of I open sets in topological spaces. In 1992, Abd El-Monsef et al.[15] further investigated I open sets and I continuous functions. Dontchev [13] introduced the concept of pre-i open sets and obtained a decomposition of I continuity. Hatır and Noiri [10] introduced the notion of semi-i open sets and α I open sets to obtain decomposition of continuity. In addition these concepts have been extended to the setting of multifunctions [16]. The initiation of I- continuous multifunctions is due to Akdag [16]. He

2 226 Metin Akdağ and Fethullah Erol studied upper (lower) I continuous [16] (semi-i continuous[17]) multifunctions and obtained several characterizations and properties of this continuity. In the present paper, we study the concepts of upper (lower) α I continuous multifunctions on ideal topological spaces and obtain some characterizations and properties. Also, we investigate its the relationships with other types of I continuities of multifunctions. Throughout this paper, int (A) and Cl(A) denote the interior and closure of A respectively. An ideal is defined as a nonempty collection I of subsets of X satisfying the following two conditions: (1) If A I and B A, then B I; (2) If A I and B I, then A B I. An ideal topological space is a topological space (X, τ) with an ideal I on X and is denoted by (X, τ, I). For a subset A X, A (I) ={x X : U A/ I for each neighborhood U of x} is called the local function of A with respect to I and τ [21]. We simply write A instead of A (I) to be brief. X is often a proper subset of X. The hypothesis X = X [11] is equivalent to the hypothesis τ I = [19]. For every ideal topological space (X, τ, I), there exists a topology τ (I), finer than τ, generated by β (I,τ)={U I:U τ and I I}, but in general β (I,τ) is not always a topology [4]. Additionally, Cl (A) =A (A) defines a Kuratowski closure operator for τ (I). For a subset A X, A is called dence-in-itself [6] (resp. τ closed [4], perfect [11]) iff A A (resp. A A, A = A ). Given a space (X, τ, I) and A X, A is called I open if A int (A ) and a subset K is called I closed if its complement is I open [15]. A multifunction F of a set X into Y is a correspondence such that F (x) is a nonempty subset of Y for each x X. We will denote such a multifuntion by F : X Y. For a multifunction F, the upper and lower inverse set of a set B of Y will be denoted by F + (B) and F (B) respectively that is F + (B) ={x X : F (x) B} and F (B) ={x X : F (x) B }. The graph G(F ) of the multifunction F : X Y is strongly closed if for each (x, y) / G(F ), there exist open sets U and V containing x and containing y respectively such that (U Cl(V )) G(F )=. A multifunction F : X Y is said to be upper semi continuous (briefly u.s.c.) at a point x X if for each open set V in Y with F (x) V, there exists an open set U containing x such that F (U) V ; lower semi continuous (briefly l.s.c.) at a point x X if for each open set V in Y with F (x) V, there exists an open set U containing x such that F (z) V for every z U [22]. Throughout this paper, the spaces (X, τ) and (Y,σ) (or simply X and Y ) always mean topological spaces and F : X Y presents a multivalued function. 2. α I Continuity of Multifunctions Definition 1. A subset A of an ideal topological space (X, τ, I) is said to be (a) I open [15] if A int (A ), (b) α I open [10] if A Int(Cl (int (A)))

3 Upper and lower α I continuous multifunctions 227 (c) pre I open [14] if A Int(Cl (A)) (d) semi I open [10] if A Cl (int (A)) (e) pre open [2] if A Int(Cl(A)) (f) semi open [18] if A Cl(int (A)) In [10], Hatır and Noiri obtained the following diagram: open α I open semi I open semi open I open pre I open pre open Definition 2. A multifunction F :(X, τ, I) (Y,σ) is said to be upper (resp. lower) α I continuous iff for each x X and each open set V in Y with F (x) V (F (x) V ), there exists a α I open set U containing x such that F (U) = {F (u) :u U} V (if u U, then F (u) V ). Theorem 1. For a multifunction F :(X, τ, I) (Y,σ), the following statements are equivalent: (a) F is upper (resp. lower) α I continuous. (b) For each x X and each open set V in Y with x F + (V )(x F (V )), there exists an α I open set U containing x such that U F + (V )(U F (V )). (c) For every open set V in Y, F + (V )(F (V )) is an α I open set in X. (d) For every closed set V in Y, F (V ) (F + (V )) is an α I closed set in X. (e) Cl(int (Cl(F (V )))) F (Cl(V )) (Cl(int (Cl(F + (V )))) F + (Cl(V )), for subset V Y. (f) F (Cl(int (Cl(U)))) Cl(F (U)), for each U of X. Proof. (a) = (b) : Let x X and V be any open set in Y with x F + (V )(x F (V )). Then F (x) V (F (x) V ). Since F is upper (lower) α I continuous, there exists a α I open set U containing x such that F (U) V (if u U, then F (u) V ). Thus U F + (V ) (U F (V )). (b) = (c) : Let V be any open set in Y and let x F + (V )(x F (V )). Then by (b), there exists a α I open set U x containing x such that F (U x ) V (U x F (V )). Since the union of α I open sets is a α I open, (F + (V )) = U x (F (V )) = U x )isaα I open set in X. (c) = (d) : Let V be a closed set in Y. Set Y V is an open set in Y. Then by (c), F + (Y V )=X F (V )(F (Y V )=X F + (V )) is a α I open set in X. Thus F + (V )(F (V )) is an α I closed set in X.

4 228 Metin Akdağ and Fethullah Erol (d) = (e) : Let V Y any subset of Y. Since Cl(V ) is closed set in Y, by (d), F (Cl(V )) (F + (Cl(V ))) is α I closed set in X. Thus X F (Cl(V )) int (Cl (int (X F (Cl(V ))))) = X Cl(int (Cl(F (V )))) (X F + (Cl(V )) int (Cl (int (X F + (Cl(V ))))) = X Cl(int (Cl(F + (V ))))). Hence, we obtain Cl(int (Cl(F (V )))) F (Cl(V )) (Cl(int (Cl(F + (V )))) F + (Cl(V )). (e) = (f) Let U be any subset of X. By (e), we have Cl(int (Cl(U))) Cl(int (Cl(F (F (U))))) F (Cl(F (U))) (Cl(int (Cl(U))) Cl(int (Cl(F + (F (U))))) F + (Cl(F (U)))) and hence F (Cl(int (Cl(U)))) Cl(F (U)). (f) = (a) Let V be any open subset of Y. Then by (f), F (Cl(int (Cl(F (Y V ))))) Cl(F (F (Y V ))) Cl(Y V ) (F (Cl(int (Cl(F + (Y V ))))) Cl(F (F + (Y V ))) Cl(Y V )). Therefore, we have Cl(int (Cl(F (Y V )))) Cl(F (Y V )) X F + (V ) (Cl(int (Cl(F + (Y V )))) Cl(F + (Y V )) X F (V )). Consequently, we obtain that F + (V ) int (Cl (int (F + (V ))) (F (V ) int (Cl (int (F (V )))). This shows that F + (V )(F (V )) is α I open. Thus, F is upper (lower) α I continuous. Definition 3. A multifunction F : X Y is called (a) [23] upper(resp. lower) pre-continuous if for every open set V in Y, F + (V ) (F (V )) is a pre-open set in X. (b) [16] upper(resp. lower) I-continuous if for every open set V in Y, F + (V ) (F (V )) is an I open set in X. (c) [17] upper(resp. lower) pre-i continuous if for every open set V in Y, F + (V ) (F (V )) is a pre- I open set in X. (d) [17] upper(resp. lower) semi-i continuous if for every open set V in Y, F + (V ) (F (V )) is an semi-i open set in X. (e) [22] upper(resp. lower) semi continuous if for every open set V in Y, F + (V ) (F (V )) is a semi open set in X. Corollary 1. [17] continuity of multifunctions implies α I continuity of multifunctions. Corollary 2. [17]α I continuity of multifunctions implies semi I continuity of multifunctions. Corollary 3. [17]α I continuity of multifunctions implies pre I continuity of multifunctions.

5 Upper and lower α I continuous multifunctions 229 Proposition 1. [1] Let A be a subset of an ideal topological space (X, τ, I). Then A is α I open if and only if A is semi-i open and pre I open in (X, τ, I). Corollary 4. For a multifunction F :(X, τ, I) (Y,σ,J), the following properties are equivalent: Theorem 2. (a) F is upper (resp. lower) α-i continuous (b) F is upper (resp. lower) semi-i continuous and upper (resp. lower) pre I continuous Proof. It follows from Proposition1. continuity α I continuity semi I continuity semi continuity I continuity pre I continuity pre continuity The converses of the corollaries are not true as shown by the following examples. Example 1. [17]Let X = {a, b, c},τ= {,X,{a}, {c}, {a, c}},i = {, {c}} and σ = {,X,{a, b}}. Then the multifunction F :(X, τ, I) (X, σ) defined by F (x) ={x} for each x X is upper semi-i continuous but it is not upper α I continuous. Example 2. [17]Let X = Y = {a, b, c, d, e},τ={,x,{a, c, d}, {d}, {a, c}}, I = {, {c}, {d}, {c, d}} and σ = {X,, {d}}. Then the multifunction F : (X, τ, I) (Y,σ), F (a) =F (c) =F (d) =F (e) ={d} and F (b) =Y. Then F is upper α I continuous but it is not upper continuous. Example 3. Let X = {a, b, c},τ= {,X,{a}, {b, c}, },I = {, {c}}} and σ = {,X,{a, b}}. Then the multifunction F :(X, τ, I) (X, σ) defined by F (x) ={x} for each x X is upper pre-i continuous but it is not upper α I continuous. Definition 4. [9] An ideal topological spaces (X, τ, I) is called I submaximal if every dense subset of X is open. Theorem 3. [9] For an ideal space (X, τ, I), the following properties are equivalent: (1) X is I submaximal, (2) Every pre-i open set is open,

6 230 Metin Akdağ and Fethullah Erol (3) Every pre-i open set is semi-i open and every α-i open set is open. Corollary 5. [17]Let (X, τ, I) is I submaximal space and F :(X, τ, I) (Y,σ) be a multifunction. Then we have the followings: (1) F is pre-i continuous if and only if F is continuous. (2) F is pre-i continuous if and only if F is semi-i continuous. (3) F is continuous if and only if F is α I continuous Theorem 4. If F :(X, τ, I) (Y,σ) is upper α I continuous (lower α I continuous) and F (X) is endowed with subspace topology, then F : X F (X) is upper α I continuous (lower α I continuous). Proof. Since F :(X, τ, I) (Y,σ) is upper α I continuous (lower α I continuous), for every open subset V of Y, F + (V F (X)) = F + (V ) F + (F (X)) = F + (V )(F (V F (X)) = F (V ) F (F (X)) = F (V )) is α I open. Hence F : X F (X) is upper α I continuous (lower α I continuous). Theorem 5. Let F :(X, τ, I) (Y,σ) and G :(Y,σ) (Z,μ) be two multifunctions. Then G F is upper (or lower) α I continuous, if G is semi continuous and F is α I continuous. Proof. Let V be an open set in Z. Since G is semi continuous then (G + (V )) (G (V )) is an open set in Y and since F is α I continuous then F + (G + (V )) = (G F ) + (V ) ( F (G (V )) = (G F ) (V ) ) is an α I open set in X. Thus G F is upper (lower) α I continuous. Theorem 6. Let {F α :(X, τ, I) ( (X α,τ α )),α Δ} be a family of multifunctions and let F :(X, τ, I) X α,τ ς be a multifunction defined by α Δ F (x) =(F α (x)). Then F is upper(lower) α I continuous if and only if F α :(X, τ, I) (X α,τ α ) is upper(lower) α I continuous for every α Δ. Proof. (Necessity) Let G α be an open set of X α for some α Δ and let P α denotes the projection of X such that P α : X α X α. Then α Δ (P α F ) + (G α )=F + (Pα 1 (G α )) = F + (G α X β ) β α,α Δ ( (P α F ) (G α )=F (Pα 1 (G α )) = F (G α ) X β ). Since G α β α,α Δ is open set and F is upper (lower) α I continuous, then F + (G α X β ) β α,α Δ ( F (G α ) X β ) is α I open in X. Thus P α F = F α is upper(or β α,α Δ lower) α I continuous for every α Δ. X β β α,α Δ

7 Upper and lower α I continuous multifunctions 231 (Sufficiency)Let F α : X X α is upper (or lower) α I continuous for every α Δ. Let U β X α be a subbasic open set in X α. Since F β α β,α Δ. is upper(or lower) α I continuous then F + (U β X α )=F + (P 1 β (U β)) = F + β (U β) α β,α Δ ( F (U β ) X α )=F (P 1 β (U β)) = F β (U β) α β,α Δ is α I open set in X. Hence F is upper(or lower) α I continuous. Theorem 7. Let F :(X, τ, I) (Y,σ) be a multifuntion and let G F : X X Y defined by G F (x) =(x, F (x)) for each x X be the graph function. Then F is upper α I continuous if and only if G F is upper α I continuous. Proof. (Necessity) Let x X and let U V be any open set in X Y such that G F (x) U V. Then x U and F (x) V. Since F is upper α I continuous then there exists a α I open set U 1 X such that x U 1 and F (U 1 ) V. Since U is open set and U 1 is α I open set then U 1 U is α I open in X and x U 1 U. Also G F (U 1 U) U V. This shows that G F is upper α I continuous. (Sufficiency) Suppose that G F is upper α I continuous and x X. Let V be open set in Y such that F (x) V. Then X V is open set in X Y and G F (x) X V Since G F is upper α I continuous, then there exists a α I open set U X such that x U and G F (U) X V. Thus F (U) V. Therefore, F is upper α I continuous. Theorem 8. Let F :(X, τ, I) (Y,σ) be a multifuntion and A be an open subset of X. If F is upper (lower) α I continuous, then F A : A Y is upper (lower) α I continuous multifunction. Proof. Let V be any open subset of Y. Since F is upper (lower) α I continuous, then F + (V )(F (V )) is α I open in X. Since A F + (V )=F + A (V ) (A F (V )=F A (V )), then F + A (V )(F A (V )) is α I open. Hence F A is upper (lower) α I continuous multifunction. Theorem 9. Let F :(X, τ, I) (Y,σ,J) be a multifuntion and {U α : α Δ} be an open cover of X. If the restriction function F Uα is upper α I continuous for each α Δ, then F is upper α I continuous. Proof. Let V be any open subset of Y. Since F Uα is upper α I continuous for each α Δ, then F + Uα (V )=U α F + (V )isα I open set. Then (U α ) F + (V )=X F + (V )=F + (V )isα I open (U α F + (V )) = α Δ α Δ set. Hence F is upper α I continuous. Definition 5. [19] An ideal topological space (X, τ, I) is said to be I compact if for every I open cover {W α : α Δ}, there exists a finite subset Δ 0 of Δ such that (X {W α : α Δ 0 }) I. α Δ

8 232 Metin Akdağ and Fethullah Erol Definition 6. An ideal topological space (X, τ, I) is said to be α I compact if for every α I open cover {W i : i Δ} of X, there exists a finite subset Δ 0 of Δ such that (X {W i : i Δ 0 }) I. Lemma 1. [16] For any surjective multifunction F :(X, τ, I) (Y,σ),F(I) is an ideal on Y. Theorem 10. Let (X, τ, I) is α I compact space and F :(X, τ, I) (Y,σ) is upper α I continuous surjection. Then (Y,σ) is F (I) compact. Proof. Let F : X Y be a upper α I continuous surjection and {W α : α Δ} be an open cover of Y. Then {F + (W α ):α Δ} is a α I open cover of X due to our assumption on F. Since X is α I compact, then there exists a finite subset Δ 0 of Δ such that (X {F + (W α ):α Δ 0 }) I. Therefore by lemma 1 F ( X { F + (W α ):α Δ 0 }) =(Y {Wα : α Δ 0 }) F (I) which shows that (Y,σ,F (I)) is F (I) compact. Definition 7. A ideal topological space (X, τ, I) is called α I Hausdorff if for each two distinct points x y there exists α I open sets U and V containing x and y respectively, such that U V =. Then we say that the points x and y are α I seperated. Theorem 11. F :(X, τ, I) (Y,σ,J) be a image closed multifunction. If Y is normal space then X is α I Hausdorff where F (x) F (y) = for each distinct x, y X. Proof. Let x, y X and x y. Then F (x) F (y) =. Since Y is normal space then there exists distinct open sets U and V containing F (x) and F (y) respectively. Thus F + (U ) and F + (V ) are disjoint α I open sets containing x and y respectively. Then X is α I Hausdorff Definition 8. [16]An ideal topological space (X, τ, I) is said to be I connected if there are no nonempty disjoint I-open sets U, V such that U V = X. Definition 9. An ideal topological space (X, τ, I) is said to be α I connected if there are no nonempty disjoint α I-open sets U, V such that U V = X. Theorem 12. Let F : X Y be a upper α I continuous. If (X, τ, I) is α I connected, then (Y,σ) is connected. Proof. Suppose there are two nonempty disjoint open sets U, V of Y, such that U V = Y. Since F is upper α I continuous, so F + (U) and F + (V ) are α I-open sets of X. Also F + (U V )=F + (U) F + (V )=F + ( ) = and F + (U V )=F + (U) F + (V )=F + (Y )=X. So (X, τ, I) is α I disconnected. Therefore (Y,σ) is connected

9 Upper and lower α I continuous multifunctions 233 Definition 10. A multifunction F :(X, τ, I) (Y,σ,J) is said to be (a) upper α I irresolute if F + (V ) is α I open in X for each α I open set V in Y (b) lower α I irresolute if (F (V )) is α I open in X for each α I open set V in Y (c) α I irresolute if F is upper α I irresolute and lower α I irresolute. Theorem 13. Definition 11. Remark 1. If F is upper α I irresolute multifunction then F is upper α I continuous multifunction. If F is lower α I irresolute multifunction then F is lower α I continuous multifunction. Proof. The proof is obvious since for any open set is α I open set. Theorem 14. Let F :(X, τ, I) (Y,σ,J) is upper α I irresolute multifunction and G :(Y,σ,J) (Z,μ) is upper α I continuous multifunction then GoF is upper α I continuous multifunction Proof. Let V be any open set in Z. Since G is upper α I continuous. Then G + (V )isα J open in Y. Since F is upper α I irresolute, then F + (G + (V )) = (GoF ) + (V )isα I open in X. Thus G F is upper α I continuous. Definition 12. A multifunction F :(X, τ, I) (Y,σ,J) is said to be (a) α I open if F (U) isα J open in Y for each open set U in X. (b) α I closed if F (K) isα J closed in Y for each closed set K in X. (c) point α I open (closed) if F (x) isα J open (closed) in Y for each x X. Remark 2. If F is point α I open (closed) multifunction then F is point α open (closed). Theorem 15. A multifunction F :(X, τ, I) (Y,σ,J) is semi-i open if and only if for each x X and for each U neighborhood of x there exists a semi- I open set V containing F (x) such that V F (U). Proof. (Necessity) Suppose that F be α I open multifunction. Let x X and U be any neighborhood of x Then there exists U 0 τ such that x U 0 U. Since F is α I open multifunction, then F (x) F (U 0 ) F (U) and F (U 0 )=V Y is α I open. Hence V F (U). (Sufficiency) Let U τ. Since U is neighborhood of x for each x U, then there exists a α I open set V x containing F (x) such that V x F (U). Thus F (U) = x U V x and since the union of α I open sets is α I open set, then F (U) isα I open set. Therefore F is α I open multifunction.

10 234 Metin Akdağ and Fethullah Erol Theorem 16. Let F :(X, τ, I) (Y,σ,J) is a α I open (α I closed) multifunction. For any subset W of Y and for any closed (resp. open) subset of X with F + (W ) K (F (W ) K) then there exist a α I closed (α I open) subset H of Y such that F + (H) K (F (H) K). Proof. Suppose that F is a α I open. Let W be any subset of Y and let K X is closed set with F + (W ) K. Since X K is open and F is a α I open, then F (X K) isα I open. Thus H = Y F (X K) is α I closed. Since F + (W ) K, then W F + (H) and F + (H) K. For α I closed multifunction, the proof is similar. References [1] A. Açıkgöz, T. Noiri and Ş. Yüksel, On α I Continuous and α I open Functions, Acta Math. Hungar., 105 (1-2) (2004), [2] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, On pre-continuous and weak pre-continuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), [3] C. Berge, Topological Spaces, Macmillian, New York, (1963). [4] D. Jankovic and T. R. Hamlett; New Toplogies From Old Via Ideals, Amer. Math. Monthly, 97, No:4 (1990), [5] D. Jankovic and T. R. Hamlett.; Compactness With Respect To an Ideal, Boll. Univ. Mat. Italy., 4-B (1990), [6] D. Jankovic and T. R. Hamlett, Compatible extensions of ideals, Bull. Un. Mat. Ital., (7), 6-B (1992), [7] E. Ekici, On Pre-I open Sets, Semi-I open Sets and b I open Sets In Ideal Topological Spaces, Acta Uni. Apulensis, 30 (2012), [8] E. Ekici and T. Noiri, extremally Disconnected Ideal Topological Spaces, Acta Math. Hungar., 122 (1-2) (2009), [9] E. Ekici and T. Noiri, Properties of I submaximal Ideal Topological Spaces, Filomat, 24:4 (2010), [10] E. Hatır and T. Noiri, On Decompositions of Continuity Via Idealization, Acta Math. Hungar., 96 (2002) [11] E. Hayashi; Topologies Defined by Local Properties, Math. Ann., 156 (1964), [12] G. Aslım, A. C. Guler and T. Noiri, On Decompositions Of Continuity and Some Weaker Forms Of Via Idealization, Acta Math. Hungar., 109 (3) (2005), [13] J. Dontchev, On pre-i open sets and a decomposition of I continuity, Banyan Math. J., 2 (1996). [14] J. Dontchev, Idealization of Ganster-Reilly Decomposition Theorems, Arxiv:math, GN/ vl, (1999). [15] M. E. Abd El-Monsef., E. F. Lashien. and A. A. Nasef; On I open Sets and I continuous Functions, Kyungpook Math. J., Vol:32 No:1 (1992), [16] M. Akdag, On Upper and Lower I continuous Multifunctions, Far East J. Math.,25(1) (2007), [17] M. Akdag, Upper and Lower semi-i continuous Multifunctions, accepted at JARPM. (2013).

11 Upper and lower α I continuous multifunctions 235 [18] N. Levine, Semi open sets and semi continuity in topological spaces, Amer. Math. Monthly, 70 (1963), [19] P. Samuels; A Topology Formed From a Given Topological Space, J. London Math. Soc., (2) 10 (1975), [20] R. L. Newcamb; Topologies Which are Compact Modulo an Ideal, Ph. D. Dissertation, Univ. of California at Santa Barbara (1967). [21] R. Vaisyanathasuamy; The Localization in Set Theory, Proc. Indian Acad. Sci., 20 (1945), [22] V. I. Ponomarev; Properties of Topological Spaces Preserved Under Multivalued Continuous Mappings, Amer. Math. Soc. Trans., 38 (2) (1964), [23] V. Popa; Some Properties of H Almost Continuous Multifunctions, Problemy Mat., Slovaca, 10 (1988), Received: November 15, 2013