ON UPPER AND LOWER C-γ-CONTINUOUS MULTIFUNCTIONS N. Gowrisankar and N. Rajesh
Abstract. The aim of this paper is to introduce a new class of multifunctions namely C-γ-continuous multifunctions and we obtain some characterizations of it.
2000 Mathematics Subject Classification: 54C10, 54C08, 54C05.
Keywords: keywords, phrases. C-γ-continuous multifunctions,γ-open set.
1. Introduction
One of the most important and basic topics in the theory of classical point set topology and several branches of Mathematics, which has been investigated by many authors, is the continuity of functions. This concept has been extended to the setting of multifunctions. Multifunction or multivalued mapping has many applications in Mathematical Programming, Probability, Statistics, Differential Inclusions, Fixed point theorems and even in Economics. There are several various types of continuous functions and some of them have been extended to the multifunctions. The aim of this paper is to introduce a new class of multifunctions namely C-γ-continuous multifunctions and we obtain some characterizations and properties of it.
2. Preliminaries
Throughout this paper, (X, τ) and (Y, σ) (or simply X and Y) means topological spaces on which no separation axioms are assumed unless explicitly stated. For any subset A of X, the closure and the interior of A are denoted by Cl(A) and Int(A), respectively. A subsetAof (X, τ) is said to beα-open [5] (resp. γ-open [3]( =b-open [1]) if A⊂Int(Cl(Int(A))) (resp. A ⊂Int(Cl(A))∪Cl(Int(A)))). The complement of a γ-open set is calledγ-closed [3] (=b-closed [1]). While, the family of allγ-open (resp. γ-closed) subsets of (X, τ) is denoted by γO(X) (resp. γC(X)). We set γO(X, x) = {A : A ∈ γO(X) and x ∈ A}. The intersection (resp. union) of all γ-closed (resp. γ-open) sets of (X, τ) containing (resp. contained in) Ais called the
γ-closure [3] (resp. γ-interior [3]) ofAand is denoted byγCl(A) (resp. γInt(A)). By a multifunctionF :X→Y, we mean a point-to-set correspondence fromX intoY, also we always assume thatF(x)6=∅for allx∈X. For a multifunctionF :X→Y, the upper and lower inverse of any subsetAofY byF+(A) andF−(A), respectively, that is F+(A) = {x ∈X :F(x) ⊆A} and F−(A) = {x ∈X :F(x)∩A 6=∅}. In particular, F+(y) ={x∈X :y∈F(x)} for each point y∈Y.
Lemma 1. [1, 3] Let A be a subset of a topological space X. Then we have the following
1. A is γ-closed if and only if γCl(A) =A.
2. γCl(A) =A∪(Cl(Int(A))∩(Int(Cl(A)))).
3. γCl(γCl(A)) =γCl(A).
Definition 1. [2] A multifunction F :X →Y is said to be
1. upperγ-continuous if F+(V)∈γO(X) for each open set V of Y, 2. lower γ-continuous if F−(V)∈γO(X) for each open set V of Y.
3. C-γ-continuous Multifunctions Definition 2. A multifunction F :X→Y is said to be
1. upperC-γ-continuous at a pointx∈Xif each open setV containingF(x)and having compact complement, there exists U ∈γO(X, x) such that F(U)⊂V, 2. lower C-γ-continuous at a point x ∈ X if each open set V having compact
complement such that F(x)∩V 6= ∅, there exists U ∈ γO(X, x) such that F(u)∩V 6=∅ for each u∈U,
3. upper (lower) C-γ-continuous inX if it has this property at every point ofX.
Theorem 2. The following are equivalent for a multifunction F :X →Y: 1. F is upper C-γ-continuous;
2. F+(V)∈γO(X) for each open set V of Y having compact complement;
3. F−(K) is γ-closed in X for every compact closed setK of Y;
4. Cl(Int(F−(B)))∩Int(Cl(F−(B)))⊂F−(Cl(B))for every subsetB ofY having the compact closure;
5. γCl(F−(B))⊂F−(Cl(B))for every subsetB ofY having the compact closure;
6. F+(Int(B)) ⊂ γInt(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). There exists U ∈γO(X, x) such that F(U) ⊂V. Therefore, we have x ∈ U ⊂ Int(Cl(F+(V))) ∪Int(Cl(F+(V))). Thus F+(V) ⊂ Int(Cl(F+(V)))∪ Int(Cl(F+(V))) and hence F+(V)∈γO(X).
(2)⇒(3): The proof follows immediately from the fact thatF+(Y\B) =X\F−(B) for every subset B ofY.
(3)⇒(4): Let B be a subset of Y having the compact closure. Then F−(Cl(B)) is a γ-closed set of X. By Lemma 1, we have Cl(Int(F−(B)))∩Int(Cl(F−(B)))⊂ Cl(Int(F−(Cl(B))))∩Int(Cl(F−(Cl(B))))⊂γCl(F−(Cl(B))) =F−(Cl(B)). There- fore, we obtain Cl(Int(F−(B)))∩Int(Cl(F−(B))) =F−(Cl(B)).
(4)⇒(5): LetB be a subset ofY having the compact closure. It follows that Lemma 1 that γCl(F−(B)) =F−(B)∪(Cl(Int(F−(B)))∩Int(Cl(F−(B))))⊂F−(Cl(B)).
(5)⇒(6): Let B be a subset of Y such that Y\Int(B) is compact. Then we have X\γInt(F+(B)) = γCl(X\F+(B)) = γCl(F−(Y\B)) ⊂ F−(Cl(Y\B)) = F−(Y\Int(B)) =X\F+(Int(B)). Therefore, we obtainF+(Int(B))⊂γInt(F+(B)).
(6)⇒(1): Letx∈XandV be any open set ofY containingF(x) and having compact complement. Then F+(V) =F+(Int(V))⊂γInt(F+(V)). Put U =γInt(F+(V)), then U ∈γO(X, x) and F(U)⊂V. This shows that F is upperC-γ-continuous.
Theorem 3. The following are equivalent for a multifunction F :X →Y: 1. F is lower C-γ-continuous;
2. F−(V)∈γO(X) for each open set V of Y having compact complement;
3. F+(K) is γ-closed in X for every compact closed setK of Y;
4. Cl(Int(F+(B)))∩Int(Cl(F+(B)))⊂F+(Cl(B))for every subsetB ofY having the compact closure;
5. γCl(F+(B))⊂F+(Cl(B))for every subsetB ofY having the compact closure;
6. F−(Int(B)) ⊂ γInt(F−(B)) for every subset B of Y such that Y\Int(B) is compact.
Proof. The proof is similar to that of Theorem 2.
Corollary 4. A multifunction F : X → Y is upper C-γ-continuous (resp. lower C-γ-continuous) ifF−(G) (resp. F+(G)) is γ-closed inX for every compact set G of Y.
Proof. LetGbe an open set of Y having compact complement. ThenY\G is com- pact and F−(Y\G) is γ-closed in X. Therefore,F+(G)∈γO(X) and by Theorem 2 F is upper C-γ-continuous. The proof for F lower C-γ-continuous is entirely similar.
For a multifunction F : X → Y, by ClF : X → Y we denote a multifunction defined as follows: (ClF)(x) = Cl(F(x)) for each point x ∈ X. Similarly, we can define γClF :X →Y.
Definition 3. A subset A of a topological space (X, τ) is said to be:
(i) α-regular [4] if for each a∈ A and any open set U of X containing a, there exists an open set Gof X such that a∈G⊂Cl(G)⊂U;
(ii) α-paracompact [8] if every X-open cover A has an X-open refinement which covers A and is locally finite for each point of X.
Lemma 5. [4] IfA is an α-paracompact α-regular set of a topological space (X, τ) and U an open neighbourhood of A, then there exists an open set G of X such that A⊂G⊂Cl(G)⊂U.
Lemma 6. [2] IfF :X→Y be a multifunction such thatF(x) isα-paracompactα- regular for eachx∈X, then for each open setV ofY (ClF)+(V) = (γClF)+(V) = F+(V).
Theorem 7. If F : X → Y be a multifunction such that F(x) is α-regular α- paracompact for each x∈X. Then the following are equivalent:
1. F is upper C-γ-continuous.
2. γClF is upper C-γ-continuous;
3. ClF is upper C-γ-continuous.
Proof. We setG=γClF or ClF. Suppose thatF is upperC-γ-continuous. LetV be any open set ofY containingG(x) and having compact complement By Lemma 6, we haveG+(V) =F+(V) and hence there existsU ∈γO(X, x) such thatF(U)⊂V. SinceF(u) isα-paracompact andα-regular for eachu∈U, by Lemma 5 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 upper C-γ-continuous. Conversely, suppose that G is upper C-γ-continuous. Let x ∈ X and V be any open set of Y containing F(x) and having compact complement.
By Lemma 6, we have x ∈ F+(V) = G+(V) and hence G(x) ⊂ V. There exists U ∈ γO(X, x) such that G(U) ⊂ V. Therefore, we obtain U ⊂ G+(V) = F+(V) and hence F(U)⊂V. This shows that F is upperC-γ-continuous.
Lemma 8. [2] IfF :X →Y is a multifunction, then(ClF)−(V) = (γClF)−(V) = F−(V) for each open set V of Y.
Theorem 9. If F :X→Y, the following are equivalent:
1. F is lower C-γ-continuous;
2. γClF is lower C-γ-continuous;
3. ClF is lower C-γ-continuous.
Proof. By using Lemma 8 this is shown similarly as in Theorem 7.
Lemma 10. [1, 3] LetU and X0 be subsets of a topological space(X, τ).
1. If U ∈γO(X) and X0∈αO(X), then U ∩X0∈γO(X0).
2. If U ⊂X0 ⊂X,U ∈γO(X0) and X0 ∈γO(X), then U ∈γO(X).
Theorem 11. If a multifunction F : X → Y is upper C-γ-continuous and X0 ∈ αO(X), then the restriction F|X0 :X0 →Y is upper C-γ-continuous.
Proof. Let x ∈ X0 and V be an open set of Y having compact complement such that (F|X0)(x)⊂V. Since F is upperC-γ-continuous and (F|X0)(x) =F(x), there exists U ∈γO(X, x) such thatF(U)⊂V. SetU0 =U ∩X0, then by Lemma 10 we have U0 ∈γO(X0, x) and (F|X0)(U0) =F(U0)⊂V. This shows that F|X0 is upper C-γ-continuous.
Theorem 12. A multifunction F : X → Y is upper C-γ-continuous if for each x∈X there existsX0∈γO(X, x) such that the restriction F|X0 :X0 →Y is upper C-γ-continuous.
Proof. Let x ∈ X and V be an open set containing F(x) and having compact complement. There exists X0 ∈ γO(X, x) such that F|X0 :X0 → Y is upper C-γ- continuous. Therefore, there exists U0 ∈γO(X0, x) such that (F|X0)(U0)⊂V. By Lemma 10, U0 ∈ γO(X, x) andF(u) =F|X0(u) for every u∈U0. This shows that F is upperC-γ-continuous.
Theorem 13. If a multifunction F : X → Y is lower C-γ-continuous and X0 ∈ αO(X), then the restriction F|X0 :X0 →Y is lower C-γ-continuous.
Theorem 14. A multifunction F : X → Y is lower C-γ-continuous if for each x∈X there exists X0 ∈γO(X, x) such that the restriction F|X0 :X0 →Y is lower C-γ-continuous.
Corollary 15. Let {Ui : i ∈ Γ} be an α-open cover of X. A multifunction F : X → Y is upper C-γ-continuous (resp. lower C-γ-continuous) if and only if the restriction F|Ui :Ui →Y is upper C-γ-continuous (resp. lower C-γ-continuous) for each i∈Γ.
Proof. This is immediate consequence of Theorems 11 and 12 (resp. Theorems 13 and 14).
Theorem 16. If F : X → Y is lower C-γ-continuous multifunction and F(A) is compact for every subset A of X, then F is lower γ-continuous.
Proof. LetAbe any subset ofX. Since Cl(F(A)) is closed and compact by Theorem 3 F+(Cl(F(A))) is γ-closed in X and A ⊂ F+(F(A)) ⊂F+(Cl(F(A))). Thus, we have γCl(A) ⊂ F+(Cl(F(A))) and F(γCl(A)) ⊂ Cl(F(A)). It follows that, F is lower γ-continuous (see [2]).
For a multifunction F : X → Y, the graph multifunction GF : X → X×Y is defined GF(x) ={x} ×F(x) for each x∈X.
Lemma 17. [6] The following hold for a multifunctionF :X→Y: 1. G+F(A×B) =A∩F+(B) and
2. G−F(A×B) =A∩F−(B) for every subsets A⊂X and B⊂Y.
Theorem 18. Let F : X → Y be a multifunction and X be compact. If GF : X →X×Y is upperC-γ-continuous (resp. lower C-γ-continuous), thenF is upper C-γ-continuous (resp. lowerC-γ-continuous).
Proof. Suppose that GF is upper C-γ-continuous. Let x ∈ X and V be an open set of Y containing F(x) and having the compact complement. Then X×V is an open set of X ×Y and has compact complement. Since GF(x) ⊂ X ×V, there existsU ∈γO(X, x) such thatGF(U)⊂X×V. Therefore, by Lemma 17 we obtain U ⊂ G+F(X ×V) = F+(V) and hence F(U) ⊂ V. This shows that F is upper C-γ-continuous. The case for lower C-γ-continuous is similar.
For a multifunction F :X → Y, the graph G(F) = {(x, F(x)) :x∈ X} is said to be strongly γ-closed if for each (x, y)∈(X×Y)\G(F), there existU ∈γO(X, x) and an open set V ofY containingy such that (U ×Cl(V))∩G(F) =∅.
Lemma 19. A multifunction F :X →Y has a strongly γ-closed graph if and only if for each (x, y)∈(X×Y)\G(F), there exist U ∈γO(X, x) and an open set V of Y containing y such thatF(U)∩Cl(V) =∅.
Proof. This proof is obvious.
Theorem 20. Let Y be a regular locally compact space. If F :X→Y is an upper C-γ-continuous multifunction such that F(x) is closed for each x ∈X, then G(F) is strongly γ-closed.
Proof. Let (x, y) ∈ (X×Y)\G(F), then y ∈ Y\F(x). Since Y is regular, there exist disjoint open sets V1 and V2 ofY such that F(x)⊂V1 and y∈V2. Moreover, since Y is locally compact and regular, there exists an open compact set V such that y∈Cl(V)⊂V2. SinceF is upperC-γ-continuous andY\Cl(V) is an open set having compact complement, there existsU ∈γO(X, x) such thatF(U)⊂Y\Cl(V).
Therefore, we haveF(U)∩Cl(V) =∅and by Lemma 19G(F) is stronglyγ-closed.
For a multifunctionF : (X, τ)→(Y, σ), we defineDcγ+(F) andD−cγ(F) as follows:
D+cγ(F) ={x∈X :F is not upperC-γ-continuous at x}.
D−cγ(F) ={x∈X :F is not lowerC-γ-continuous atx}.
Theorem 21. For a multifunctionF : (X, τ)→(Y, σ), the following properties hold:
D+cγ = ∪
G∈σcc{F+(G)\γInt(F+(G))}
= ∪
B∈icc{F+(Int(B))\γInt(F+(B))}
= ∪
B∈cc{γCl(F−(B))\F−(Cl(B))}
= ∪
H∈F{γCl(F−(H))\F−(H)}, where σcc is the family of all σ-open sets of Y having compact complement, icc is the family of all subsetsB of Y such thatY\Int(B) is compact, cc is the family of all subsets B of Y having the compact closure, F is the family of all closed and compact sets of (Y, σ).
Proof. We shall only the first equality and the last equality since the proofs of other are similar to the first.
Let x ∈ Dcγ+(F). Then, by Theorem 2, there exists an open set V of Y contain- ing F(x) and having compact complement such that x ∈ γInt(F+(V)). There- fore, x∈F+(V)\γInt(F+(V))⊂ ∪
G∈σcc{F+(G)\γInt(F+(G))}. Conversely, let x∈
G∈σcc∪ {F+(G)\γInt(F+(G))}. Then there exists an open setV ofY having compact complement such that x ∈F+(V)\γInt(F+(V)). By Theorem 2, x ∈D+cγ(F). We prove the last equality. ∪
H∈F{γCl(F−(H))\F−(H)} ⊂ ∪
B∈cc{γCl(F−(B))\F−(Cl(B))}= D+cγ(F). Conversely, we haveD+cγ(F) = ∪
B∈cc{γCl(F−(B))S
∪
H∈F{γCl(F−(H))\F−(H)}.
Theorem 22. For a multifunctionF : (X, τ)→(Y, σ), the following properties hold:
Dcγ− = ∪
G∈σcc{F−(G)\γInt(F−(G))}
= ∪
B∈icc{F−(Int(B))\γInt(F−(B))}
= ∪
B∈cc{γCl(F+(B))\F+(Cl(B))}
= ∪
H∈F{γCl(F+(H))\F+(H)}.
Proof. The proof is similar to that of Theorem 21
Definition 4. A topological space (X, σ) is called a KC-space [7] if every compact set of Y is closed.
Definition 5. A multifunction F : (X, τ)→(Y, σ)is said to be bounded at the point p ∈X if there exists a γ-open set U containing p and a compact set C of Y such that F(x)⊂C for each x∈U.
Theorem 23. If a multifunction F : (X, τ) → (Y, σ) is upper γ-continuous (resp.
lower γ-continuous) at a point p∈X and (Y, σ) is a KC space, then F : (X, τ)→ (Y, σ) is bounded at p ∈X and upper c-γ-continuous (resp. lower c-γ-continuous) at p∈X.
Proof. We prove only the first case, the proof of the second being entirely similar.
LetU be aγ-open set containingpandCa compact set ofY such thatF(x)⊂Cfor each x∈U. LetV be any open set ofY such thatF(p)⊂V. Put G=V ∪(Y\C).
Then G is open and Y\G is compact. By hypothesis, there exists a γ-open set W containing p such that F(x) ⊂ G for every x ∈ W. Put H = W ∩U, thenH is a γ-open set containingpsuch thatF(x)⊂G∩C for anyx∈H. ThenF(x)⊂V for any x∈H. Therefore,F : (X, τ)→(Y, σ) is upperc-γ-continuous atp∈X.
Definition 6. A topological space (X, τ) is said to be γ-saturated if for any x∈X the intersection of all γ-open sets containingx is γ-open.
Theorem 24. Let (X, τ) be a γ-saturated topological space and (Y, σ) a T1-space.
If is upper C-γ-continuous, then F is upperγ-continuous.
Theorem 25. Let (X, τ) be a γ-saturated space and (Y, σ) a locally compact Haus- dorff space. If is an upper C-γ-continuous and closed valued multifunction, then F is upper c-γ-continuous.
Proof. Suppose that F is not upper γ-continuous at x0 ∈X. Then, there exists an open setV ofY such thatF(x0)⊂V andF(U)∩(Y\V)6=∅for everyγ-open setU containing x0. LetU0 be the intersection of all γ-open sets containingx0. ThenU0
isγ-open and there existsz1 ∈U0 such thatF(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 andy /∈F(x0), there exists an open setW containing y such that Cl(W) is a compact set and Cl(W)⊂Y\F(x0). SinceF(x0)⊂Y\Cl(W) and F is upperC-γ-continuous atx0, there exists aγ-open setGcontainingx0 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 thatF(z1)∩Cl(W)6=∅.
Theorem 26. Let (X, τ) be a γ-saturated space and (Y, σ) a KC-space. If F : (X, τ) →(Y, σ) is lower C-γ-continuous and for each x ∈X there exits a compact set Cx such that F(x)⊂Cx, then F is lower c-γ-continuous.
Proof. Suppose thatF is not lowerc-γ-continuous at x0 ∈X. Then, there exists an open set V of Y such that F(x0)∩V 6=∅ and for eachγ-open set U containing x0 there exists u∈U such thatF(u)∩V =∅. LetU0 be the intersection of allγ-open sets containingx0. ThenU0isγ-open and there existsx∈U0such thatF(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=∅. SinceF is lowerC-γ-continuous atx0, there exists a γ-open setGcontainingx0such that for anyz∈Gwe haveF(z)∩(Y\(Cx\V))6=∅.
This is a contradiction because x∈U0⊂G and F(x)⊂Cx\V.
Definition 7. Theγ-frontier [2] of a subsetAof a space X, denoted by γF r(A), is defined by γF r(A) =γCl(A)∩γCl(X\A) =γCl(A)\γInt(A).
Theorem 27. The set of all points x of X at which a multifunction F :X→Y is not upper C-γ-continuous (resp. lower C-γ-continuous) is identical with the union of the γ-frontier of the upper (resp. lower) inverse images of open sets containing (resp. meeting) F(x) and havening compact complement.
Proof. Letx be a point ofX at which F is not upper C-γ-continuous. Then, there exists an open set V of Y containing F(x) and having the compact complement such that U ∩(X\F+(V)) 6= ∅ for every U ∈ γO(X, x). Therefore, we have x ∈ X\γInt(F+(V)) and hence x ∈ γF r(F+(V)) since x ∈ F+(V) ⊂ γCl(F+(V)).
Conversely, suppose that V is an open set of Y containing F(x) and having the compact complement such that x ∈γCl(F+(V)). If F is upper C-γ-continuous at x, then there exists U ∈γO(X, x) such that U ⊂F+(V); hence x∈γInt(F+(V)).
This is a contradiction and henceF is not upper C-γ-continuous atx. The case for lower C-γ-continuous is similar.
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N. Gowrisankar
70/232 Kollupettai Street, M. Chavady, Thanjavur-613001, Tamilnadu, India.
email: [email protected] N. Rajesh
Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005,
Tamilnadu, India.
email: nrajesh [email protected]
