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ISSN: 2168-9679
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Upper and Lower Weaky mX - αψ - Continuous Multifunctions

M. Parimala*

Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam-638401, Tamil Nadu, India

*Corresponding Author:
M. Parimala
Department of Mathematics, Bannari Amman Institute of Technology
Sathyamangalam-638401, Tamil Nadu, India
E-mail: [email protected]

Received Date: March 16, 2012; Accepted Date: May 14, 2012; Published Date: May 18, 2012

Citation: Parimala M (2012) Upper and Lower Weaky mX - αψ - Continuous Multifunctions. J Appl Computat Math 1:107. doi: 10.4172/2168-9679.1000107

Copyright: © 2012 Parimala M. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Keywords

Minimal structure; mX -αψ -closed set; mX -αψ -continuous functions in minimal structure spaces

AMS (2000) Subject Classification

54A05, 54A20, 54C08, 54D10, 54C60.

Introduction

In 1961, Levine [1] introduced the notion of weakly continuous functions. Popa and Smithson [2,3] independently introduced the concept of weakly continuous multifunctions. Noiri [4] introduced the concept of minimal structure on a nonempty set. Also they introduced the notion of mX -open set and mX -closed set and characterize those sets using mX -cl and mX -int operators respectively. Further they introduced m-continuous functions [5] and studied some of its basic properties. Noiri and Popa [6] introduced and studied other forms of continuous multifunctions namely, slightly m-continuous multifunctions.

In this paper, we introduce mX -αψ -closed set and also we study some of the upper/lowermX -αψ -continuous multifunctions as the multifunctions are defined between a set satisfying certain minimal condition into a topological space. We obtain some characterizations and some properties of such multifunctions.

Preliminaries

In this section, we introduce the m-structure and define some important subsets associated to the m-structure and the relation between them.

Definition

Let X be a nonempty set and let mX ⊆ P(X) , where P(X) denote the power set of X . Where mX is an m-structure (or a minimal structure) on X , if φ and X belong to mX

The members of the minimal structure mX are called mX -open sets, and the pair (X,mX ) is called an m -space. The complement of mX -open set is said to be mX -closed.

Definition

[7] Let X be a nonempty set and mX an m -structure on X . For a subset A of X , mX -closure of A and mX -interior of A are defined as follows:

1. mX - Cl(A) =∩{F : A ⊆ F,X − F∈mX}

2. mX - Int(A) =∪{F :U ⊆ A,U ∈mX} .

Lemma

[7] Let X be a nonempty set and mX an m -structure on X . For subsets A and B of X , the following properties hold:

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(φ ) =φ , mX - Cl(X) = X , mX - int(φ ) =φ 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

[5] Let (X,mX ) be an m -space and A a subset of X . Then x∈mX - cl(A) if and only if U ∩A ≠φ for every U ∈mX containing x .

Definition

[7] A minimal structure mX on a nonempty set X is said to have the property β if the union of any family of subsets belonging to mX belongs to mX .

Remark

[8] A minimal structure mX with the property β coincides with a generalized topology on the sense of Lugojan.

Lemma

[9] Let X be a nonempty set and mX an m -structure on X satisfying the property β . For a subset A of X , the following property hold:

1. A∈mX iff mX - int(A) = A

2. A∈mX iff mX - cl(A) = A

3. mX - int(A)∈mX and mX - cl(A)∈mX

Definition

A subset A of an m -space (X,mX ) is called

1. an mX -preopen set [10] if A ⊆U - int(mX - cl(A)) and a mX -preclosed set if mX - cl(mX - int(A))⊆ A ,

2. an mX -semiopen set [10] if A ⊆mX - cl(mX - int(A)) and a mX -semiclosed set if mX - int(mX - cl(A))⊆ A ,

3. an mX -semi generalized-closed [10] (briefly mX - sg -closed) set if mX - scl(A)⊆U whenever A ⊆U and U is mX -semi-open in (X,mX ) . The complement of an mX - sg -closed set is called an mX - sg -open set.

The mX -pre closure (resp. mX -semi closure, mX -α -closure) of a subset A of an m-space (X,mX ) is the intersection of all mX -pre closed (resp. mX -semi closed, mX -α -closed) sets that contain A and is denoted by mX - pcl(A) (resp. mX - scl(A) , mX -mX ).

mX -αψ -closed and mX -αψ -open sets

Definition

A subset A of an m -space (X,mX ) is called an

1. mX -α -open set if A ⊆mX - int(mX - cl(mX - int(A))) and an mX -α -closed set if mX - cl(mX - int(mX - cl(A)))⊆ A ,

2. mX -ψ -closed set if mX - scl(A)⊆U whenever A ⊆U and U is mX - sg -open in (X,mX ) .The complement of an mX -ψ -closed set is called an mX -ψ -open set.

3. mX -αψ -closed set if mX -ψ cl(A)⊆U whenever A ⊆U and U is mX -α -open in (X,mX ) .The complement of an mX -αψ -closed set is called an mX -αψ -open set.

Notation

For an m -space (X,mX ) , O(X,mX ) (resp. SO(X,mX ) , PO(X,mX ) , αO(X,mX ) , SGO(X,mX ) , ψ O(X,mX ) , αψ - O(X,mX ) ) denotes the class of all open (resp. mX -semiopen, mX -preopen, mX -α -open, mX -sg-open, mX -ψ -open, mX -αψ -open) subsets of (X,mX ) .

Definition

Let (X,mX ) be an m -space and let A be a subset of X . Then

1. the intersection of all mX -mX -closed sets containing A is called the mX -αψ -closure of A and is denoted by mX -αψ - cl(A) .

2. the union of all mX -αψ -open sets that are contained in A is called the mX -αψ -interior of A and is denoted by mX -αψ - int(A)

Example (1)

Let X = {a,b,c,d} . Define the m -structure on X as follows: mX = {φ ,X,{a},{b},{a,b} .

Then SO(X,mX ) = {φ ,X,{a},{b},{a,c}, {a,d},{b,d},{a,c,d}} ,

αO(X,mX ) = {φ ,X,{a},{b},{a,b},{a,c},{a,b,c}} and

αψ -O(X,mX ) = {φ ,X,{a},{b},{c},{a,b},{a,c},{a,d},{b,c},{b,d},{a,b,c},{a,c,d}} .

Example (2)

Let X = {a,b,c} . Define the m -structure on X as follows: mX = {φ ,X,{a},{b}} .

Then SO(X,mX ) = {φ ,X,{a},{b},{a,c},{b,c}} ,

αO(X,mX ) = {φ ,X,{a},{b}} and αψ -O(X,mX ) = P(X) .

Example (3)

Let X = {a,b,c,d} . Define the m -structure on X as follows: mX = {φ ,X,{a},{b},{a,b,c},{a,b,d}} .

Then equation

equation and αψ

equation

Definition

The intersection of all mX -α -open subsets of (X,mX ) containing A is called the mX -α -kernel of A (briefly, mX -αker(A) ) i.e. mX -α ker(A) =∩{G∈mX -αO(X) : A ⊆G} . And mX - s ker(A) , mX - sgker(A) , an mX -ψ ker(A) are defined similarly.

Theorem (1)

Let A be a subset of (X,mX ) , then A is mX - (α ,ψ ) -closed if and only if mX -ψ cl(A)⊆mX -αker(A) .

Proof

Suppose that A is mX -αψ -closed and let D = {S : S ⊆ X,A⊆ S : Sis anmX -α -open}. Then mX -equation Observe that S∈D implies that A ⊆ S follows mX -ψ cl(A)⊆ S for all S∈D .

Conversely, if mX -ψ cl(A)⊆mX - aker(A) , take S∈αO(X,mX ) such that A ⊆ S then by hypothesis,

mX -ψ cl(A)⊆mX -αker(A)⊆ S .

This shows that A is mX -αψ -closed.

Theorem (2)

For subsets A and B of (X,mX ) , the following properties hold:

1. If A is mX -ψ -closed, then A is mX -αψ -closed.

2. If mX has the property β and A is mX -αψ -closed and mX - α -open then A is mX -ψ -closed.

3. If A is mX -αψ -closed and A ⊆ B ⊆ψ cl(A) then B is mX - αψ -closed.

Proof

1. Let A be an mX -ψ -closed set in (X,mX ) . Let A ⊆U , where U is mX -α -open in (X,mX ) . Since A is mX -ψ -closed, mX - ψ cl(A) = A , mX -ψ cl(A)⊆U . Therefore, A is mX -αψ -closed.

2. Since A is mX -α -open and mX -αψ -closed, we have mX - ψ cl(A)⊆ A . Therefore, A is mX -ψ -closed

3. Let U be an mX -α -open set of (X,mX ) such that B ⊆U , then A ⊆U . Since A is mX -αψ -closed, mX -ψ cl(A)⊆U . Now mX -ψ cl(B)⊆mX -ψ cl(mX -ψ cl(A))⊆U . Therefore, B is also an mX -αψ -closed set of (X,mX ) .

Theorem (3)

Union of two mX -αψ -closed sets is mX -αψ -closed.

Proof

Let A and B be two mX -αψ -closed sets in (X,mX ) . Let A∪B ⊆U , U is mX -α -open. Since A and B are mX -αψ -closed sets, mX -ψ cl(A)⊆U and mX -ψ cl(B)⊆U . This implies that mX - ψ cl(A∪B)⊆mX -ψ cl(A) -ψ cl(B)⊆U and so mX -ψ cl(A∪B)⊆U . Therefore A∪B is mX -αψ -closed.

Theorem (4)

Let mX be an m -structure on X satisfying the property β and A ⊆ X . Then A is an mX -αψ -closed set if and only if there does not exist a nonempty mX -α -closed set F such that F ≠φ and F ⊆mX - ψ cl(A)− A .

Proof

Suppose that A is an mX -αψ -closed set and let F ⊆ X be an mX -α -closed set such that F ⊆mX -ψ cl(A)− A . It follows that, A ⊆ X − F and X − F is an mX -α -open set. Since A is an mX - αψ -closed set, we have that mX -ψ cl(A)⊆ X − F and F ⊆ X −mX -ψ cl(A) . Follows that, F+(V)⊂mX -ψ cl(A)∩(X −mX -ψ cl(A)) =φ , implying that F =φ .

Conversely, if A ⊆U and U is an mX -α -open set, then mX - ψ cl(A)∩(X −U)⊆mX -ψ cl(A)∩(X − A) =mX -ψ cl(A)− A . Since mX -ψ cl(A)− A does not contain subsets mX -α -closed sets different from the empty set, we obtain that mX -ψ cl(A)∩(X −U) =φ and this implies that mX -ψ cl(A)⊆U in consequence A is mX -αψ -closed.

We can observe that if in Theorem 3.11., the property β is omitted then the result can be false as we can see in the following example.

Example

Let X = {a,b,c,d} . The m -structure on X is defined as equation

equation and αψ

equation

The set {a} is not an mX -αψ -closed set and there does not exist mX -α -closed set F such that F ≠φ and F ⊆mX -ψ cl(A)− A .

Theorem (5)

Let (X,mX ) be an m -space and A ⊆ X , then A is mX -αψ -open if and only if F ⊂mX -ψ int(A) where F is mX -α -closed and F ⊂ A .

Proof

Let A be an mX -αψ -open, F be mX -α -closed set such that F ⊂ A . Then X − A ⊂ X − F , but X − F is mX -α -closed and X − A is mX -αψ -closed implies that mX -ψ cl(X − A)⊂ X − F . Follows that X −mX -ψ int(A)⊂ X − F . In consequence F ⊂mX -ψ int(A) .

Conversely, if F is mX -α -closed, F ⊂ A and F ⊂mX -ψ int(A) . Let X − A ⊂U where U is mX -α -open, then X −U ⊂ A and X −U is mX -α -closed. By hypothesis, X −U ⊂mX -ψ int(A) . Follows X −mX -ψ int(A)⊂U but it is equivalent to mX -ψ cl(X − A)⊂U . Therefore, X − A is mX -αψ -closed and hence A is mX -αψ -open.

Weak mX -αψ -continuous and almost mX -αψ-continuous multifunctions

Definition (1)

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 mX -αψ -continuous (resp. upper almost mX -αψ -continuous, upper weakly mX -αψ -continuous) at a point x if for each open set V of Y containing F(x) , there exists an mX - αψ -open set U of mX containing x such that F(U)⊂V (resp. F(U)⊂int(cl(V)),F(U)⊂ cl(V)) ,

2. lower mX -αψ -continuous (resp. lower almost mX -αψ -continuous, lower weakly mX -αψ -continuous) at a point x∈X if for each open set V of Y such that F(x)∩V ≠φ , there exists an mX -αψ -open set U of mX containing x such that F(u)∩V ≠φ (resp. F(u)∩int(cl(V)) ≠φ ,F(u)∩cl(V) ≠φ ) for each u∈U ,

3. upper/lower mX -αψ -continuous (resp. almost mX -αψ -continuous, weakly mX -αψ -continuous) if it has this property at each point x∈X .

Definition (2)

multifunction F : (X,mX )→(Y,σ ) is said to be almost mX -αψ -open if F(U)⊂int(cl(F(U))) for every mX -αψ -open set U of mX .

Theorem (1)

If a multifunction F : (X,mX )→(Y,σ ) is upper weakly mX -αψ -continuous and almost mX -αψ -open, then F is upper almost mX -αψ -continuous.

Proof

Let V be any open set in Y containing F(x) . Then there exists an mX -αψ -open set U of mX containing x such that F(U)⊂ cl(V) . Since F(x) is almost mX -αψ -open, F(U)⊂int(cl(F(U)))⊂int(cl(V)) . Therefore, F is upper almost mX -αψ -continuous.

Theorem (2)

Let F : (X,mX )→(Y,σ ) be a multifunction such that F(x) is open in Y for each x∈X . Then, the following properties are equivalent:

1. F is lower mX -αψ -continuous;

2. F is lower almost mX -αψ -continuous;

3. F is lower weakly mX -αψ -continuous.

Proof

(i) ⇒ (ii) and (ii) ⇒ (iii): The proofs of these implications are obvious.

iii) ⇒ (i): Let x∈X and V be any open set such that F(x)∩V ≠φ . There exists an mX -αψ -open set U of mX such that F(u)∩cl(V) ≠φ for each u∈U . Since F(u) is open, F(u)∩V ≠φ for each u∈U and hence F is lower mX -αψ -continuous.

Slightly mX -αψ -continuous Multifunctions

Definition

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 slightly mX -αψ -continuous if for each A∈mX and each clopen set V of Y containing F(x) , there exists an mX -αψ -open set U of mX containing x such that F(U)⊂V ,

2. lower slightly mX -αψ -continuous if for each x∈X and each clopen set V of Y such that F(x)∩V ≠φ , there exists an mX -αψ -open set U of mX containing x such that F(u)∩V ≠φ for each u∈U .

Theorem (1)

For a multifunction F : (X,mX )→(Y,σ ) , the following are equivalent:

1. F is upper slightly mX -αψ -continuous;

2. F+(V) =mX -αψ - int(F(V)) for each V ∈CO(Y) ;

3. F(V)) -αψ - cl(F−(V)) for each V ∈CO(Y) .

Proof

(i) ⇒ (ii): Let V be any clopen set of Y and x∈F+(V) . Then F(x)∈V . There exists an mX -αψ -open set U of mX containing x such that F(U)⊂V . Thus x∈U ⊂ F+(V) and hence x∈mX - αψ - int(F+(V)) . Therefore, we have F+(V)⊂mX -αψ - int(F+(V)) . But , mX -αψ - int(F+(V))⊂ Fv(V) , we obtain F+(V) =mX -αψ - int(F+(V)) .

(ii) ⇒ (iii): Let K be any clopen set of Y . Then Y −K is clopen in Y . By (ii) and Lemma 2.3., we have X − F+(K) = (Y −K) =mX - αψ - int(F+(Y −K)) = X −[mX -αψ - cl(F−(K))] . Therefore, we obtain F-(K) =mX -αψ - cl(F-(K)) .

(iii) ⇒ (ii): This follows from the fact that F-(Y − B) = F+(B) for every subset B of Y .

(ii) ⇒ (i): Let x∈X and V be any clopen set of Y containing F(x) . Then x∈F+(V) =mX -αψ - int(F+(V)) . There exists an mX - αψ -open set U of mX containing x such that x∈U ⊂ F+(V) . Therefore, we have x∈U , U is an mX -αψ -open set of mX and f (U)⊂V. Hence F is upper slightly mX -αψ -continuous.

Theorem (2)

For a multifunction F : (X,mX )→(Y,σ ) , the following are equivalent:

1. F is lower slightly mX -αψ -continuous;

2. F-(V) =mX -αψ - int(F-(V)) for each V ∈CO(Y) ;

3. F+(V) =mX -αψ - cl(F+(V)) for each V ∈CO(Y) .

Proof

(i) ⇒ (ii): Let V ∈CO(Y) and x∈F-(V) . Then F(x)∩V ≠φ and by (i) there exists an mX -αψ -open set U of mX containing x such that F(u)∩V ≠φ for each u∈U . Therefore, we have U ⊂ F-(V) and hence x∈U ⊂mX -αψ - int(F-(V)) . Thus, we obtain F-(V)⊂mX - αψ - int(F-(V)) and by Lemma 2.3., F-(V) =mX -αψ - int(F-(V)) .

(ii) ⇒ (iii): Let V ∈CO(Y) . Then Y −V ∈CO(Y) and by (ii) we have X − F+(V) = F-(Y −V) =mX -αψ - int(F-(Y −V)) = X −mX - αψ - cl(F+(V)) . Hence we obtain F+(V) =mX -αψ - cl(F+(V)) .

(iii) ⇒ (i): Let x be any point of X and V any clopen set of Y such that F(x)∩V ≠φ . Then x∈F−(V) and equation . By (iii), we have equation -αψ - cl(F+(Y −V)) . By Lemma 2.4., there exists an mX -αψ -open set of mX containing x such that U ∩F+(Y −V) =φ , hence U ⊂ F-(V) . Therefore, F(u)∩V ≠φ for each u∈U and F is lower slightly mX -αψ -continuous.

Corollary (1)

For a multifunction F : (X,mX )→(Y,σ ) , where mX has the property β , the following are equivalent:

1. F is upper slightly mX -αψ -continuous;

2. F+(V) is mX -αψ -open in (X,mX ) for each V ∈CO(Y) ;

3. F-(V) is mX -αψ -closed in (X,mX ) for each V ∈CO(Y) .

Corollary (2)

For a multifunction F : (X,mX )→(Y,σ ) , where mX has the property β , the following are equivalent:

1. F is lower slightly mX -αψ -continuous;

2. F-(V) is mX -αψ -open in (X,mX ) for each V ∈CO(Y) ;

3. F+(V) is mX -αψ -closed in (X,mX ) for each V ∈CO(Y) .

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