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**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 m_{X} - 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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

Minimal structure; m_{X} -αψ -closed set; m_{X} -αψ -continuous functions in minimal structure spaces

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

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 m_{X} -open set and m_{X} -closed set and characterize those sets using m_{X} -cl and m_{X} -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 m_{X} -αψ -closed set and also we study some of the upper/lowerm_{X} -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.

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 m_{X} ⊆ P(X) , where P(X) denote the power set of X . Where m_{X} is an m-structure (or a minimal structure) on X , if φ and X belong to m_{X}

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

**Definition**

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

1. m_{X} - Cl(A) =∩{F : A ⊆ F,X − F∈m_{X}}

2. m_{X} - Int(A) =∪{F :U ⊆ A,U ∈m_{X}} .

**Lemma**

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

1. m_{X} - Cl(X − A) = X −m_{X} - Int(A) and m_{X} - Int(X − A) = X −m_{X} ) - Cl(A) .

2. If (X − A)∈m_{X} , then m_{X} - cl(A) = A and if A∈m_{X} then m_{X} - int(A) = A .

3. m_{X} - Cl(φ ) =φ , m_{X} - Cl(X) = X , m_{X} - int(φ ) =φ and m_{X} - int(X) = X .

4. If A ⊆ B then m_{X} - Cl(A)⊆m_{X} - Cl(B) and m_{X} - int(A)⊆m_{X} - int(B) .

5. A ⊆m_{X} - Cl(A) and m_{X} - Int(A)⊆ A .

6. m_{X} - Cl(m_{X} - Cl(A)) =m_{X} - Cl(A) and m_{X} - Int(m_{X} - Int(A)) =m_{X} - Int(A) .

**Lemma**

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

**Definition**

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

**Remark**

[8] A minimal structure m_{X} with the property β coincides with a generalized topology on the sense of Lugojan.

**Lemma**

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

1. A∈m_{X} iff m_{X} - int(A) = A

2. A∈m_{X} iff m_{X} - cl(A) = A

3. m_{X} - int(A)∈m_{X} and m_{X} - cl(A)∈m_{X}

**Definition**

A subset A of an m -space (X,m_{X} ) is called

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

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

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

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

**Definition**

A subset A of an m -space (X,m_{X} ) is called an

1. m_{X} -α -open set if A ⊆m_{X} - int(m_{X} - cl(m_{X} - int(A))) and an m_{X} -α -closed set if m_{X} - cl(m_{X} - int(m_{X} - cl(A)))⊆ A ,

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

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

**Notation**

For an m -space *(X,m _{X} ) , O(X,m_{X} ) (resp. SO(X,m_{X} ) , PO(X,m_{X} ) , αO(X,m_{X} ) , SGO(X,m_{X} ) , ψ O(X,m_{X} ) , αψ - O(X,m_{X} ) )* denotes the class of all open (resp. m

**Definition**

Let *(X,m _{X} )* be an m -space and let A be a subset of X . Then

1.* the intersection of all m _{X} -m_{X} -closed sets containing A is called the m_{X} -αψ -closure of A and is denoted by m_{X} -αψ - cl(A) .*

2. *the union of all m _{X} -αψ -open sets that are contained in A is called the m_{X} -αψ -interior of A and is denoted by m_{X} -αψ - int(A)*

**Example (1)**

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

Then *SO(X,m _{X} )* = {φ ,X,{a},{b},{a,c}, {a,d},{b,d},{a,c,d}} ,

*αO(X,m _{X} ) *= {φ ,X,{a},{b},{a,b},{a,c},{a,b,c}} and

*αψ -O(X,m _{X} )* = {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: m_{X} = {φ ,X,{a},{b}} .

Then *SO(X,m _{X} )* = {φ ,X,{a},{b},{a,c},{b,c}} ,

*αO(X,m _{X} )* = {φ ,X,{a},{b}} and

**Example (3)**

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

Then

and *αψ*

**Definition**

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

**Theorem (1)**

*Let A be a subset of (X,m _{X} ) , then A is m_{X} - (α ,ψ ) -closed if and only if m_{X} -ψ cl(A)⊆m_{X} -αker(A) .*

**Proof**

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

Conversely, if *m _{X} -ψ cl(A)⊆m_{X} - aker(A) , take S∈αO(X,m_{X} )* such that A ⊆ S then by hypothesis,

*m _{X} -ψ cl(A)⊆m_{X} -αker(A)⊆ S .*

This shows that A is m_{X} -αψ -closed.

**Theorem (2)**

*For subsets A and B of (X,m _{X} ) , the following properties hold:*

1. *If A is m _{X} -ψ -closed, then A is m_{X} -αψ -closed.*

2. *If m _{X} has the property β and A is m_{X} -αψ -closed and m_{X} - α -open then A is m_{X} -ψ -closed.*

3. *If A is m _{X} -αψ -closed and A ⊆ B ⊆ψ cl(A) then B is m_{X} - αψ -closed.*

**Proof**

1. Let A be an m_{X} -ψ -closed set in *(X,m _{X} )* . Let

2. Since A is m_{X} -α -open and m_{X} -αψ -closed, we have m_{X} - *ψ cl(A)⊆ A* . Therefore, A is m_{X} -ψ -closed

3. Let U be an m_{X} -α -open set of (X,m_{X} ) such that B ⊆U , then *A ⊆U *. Since A is m_{X} -αψ -closed, *m _{X} -ψ cl(A)⊆U* . Now

**Theorem (3)**

*Union of two m _{X} -αψ -closed sets is m_{X} -αψ -closed.*

**Proof**

Let A and B be two m_{X} -αψ -closed sets in (X,m_{X} ) . Let *A∪B ⊆U* , U is m_{X} -α -open. Since A and B are m_{X} -αψ -closed sets, *m _{X} -ψ cl(A)⊆U *and

**Theorem (4)**

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

**Proof**

Suppose that A is an m_{X} -αψ -closed set and let F ⊆ X be an m_{X} -α -closed set such that *F ⊆m _{X} -ψ cl(A)− A* . It follows that,

Conversely, if A ⊆U and U is an m_{X} -α -open set, then m_{X} - *ψ cl(A)∩(X −U)⊆m _{X} -ψ cl(A)∩(X − A) =m_{X} -ψ cl(A)− A *. Since

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

and *αψ *

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

**Theorem (5)**

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

**Proof**

Let A be an m_{X} -αψ -open, F be m_{X} -α -closed set such that F ⊂ A . Then *X − A ⊂ X − F* , but X − F is m_{X} -α -closed and X − A is m_{X} -αψ -closed implies that *m _{X} -ψ cl(X − A)⊂ X − F *. Follows that

Conversely, if F is m_{X} -α -closed, F ⊂ A and F ⊂m_{X} -ψ int(A) . Let *X − A ⊂U* where U is m_{X} -α -open, then *X −U ⊂ A* and X −U is m_{X} -α -closed. By hypothesis, *X −U ⊂m _{X} -ψ int(A)* . Follows

**Definition (1)**

Let (X,m_{X} ) be an m -space and (Y,σ ) a topological space. A multifunction F : (X,m_{X} )→(Y,σ ) is said to be

1. *upper m _{X} -αψ -continuous (resp. upper almost m_{X} -αψ -continuous, upper weakly m_{X} -αψ -continuous) at a point x if for each open set V of Y containing F(x) , there exists an m_{X} - αψ -open set U of m_{X} containing x such that F(U)⊂V (resp. F(U)⊂int(cl(V)),F(U)⊂ cl(V)) ,*

2. *lower m _{X} -αψ -continuous (resp. lower almost m_{X} -αψ -continuous, lower weakly m_{X} -αψ -continuous) at a point x∈X if for each open set V of Y such that F(x)∩V ≠φ , there exists an m_{X} -αψ -open set U of m_{X} 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 m _{X} -αψ -continuous (resp. almost m_{X} -αψ -continuous, weakly m_{X} -αψ -continuous) if it has this property at each point x∈X .*

**Definition (2)**

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

**Theorem (1)**

*If a multifunction F : (X,m _{X} )→(Y,σ ) is upper weakly m_{X} -αψ -continuous and almost m_{X} -αψ -open, then F is upper almost m_{X} -αψ -continuous.*

**Proof**

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

**Theorem (2)**

*Let F : (X,m _{X} )→(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 m _{X} -αψ -continuous;*

*2. F is lower almost m _{X} -αψ -continuous;*

*3. F is lower weakly m _{X} -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 m_{X} -αψ -open set U of m_{X} 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 m_{X} -αψ -continuous.

**Definition**

*Let (X,m _{X} ) be an m -space and (Y,σ ) a topological space. A multifunction F : (X,m_{X} )→(Y,σ ) is said to be*

1. *upper slightly m _{X} -αψ -continuous if for each A∈m_{X} and each clopen set V of Y containing F(x) , there exists an m_{X} -αψ -open set U of m_{X} containing x such that F(U)⊂V ,*

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

**Theorem (1)**

*For a multifunction F : (X,m _{X} )→(Y,σ ) , the following are equivalent:*

1. *F is upper slightly m _{X} -αψ -continuous;*

2. F^{+}(V) =m_{X} -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 m_{X} -αψ -open set U of m_{X} containing x such that *F(U)⊂V* . Thus *x∈U ⊂ F ^{+}(V)* and hence x∈m

(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) =m_{X} - αψ - int(F^{+}(Y −K)) = X −[m_{X} -αψ - cl(F−(K))]* . Therefore, we obtain

(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) =m

**Theorem (2)**

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

1.* F is lower slightly m _{X} -αψ -continuous;*

2. *F ^{-}(V) =m_{X} -αψ - int(F^{-}(V)) for each V ∈CO(Y) ;*

3. *F ^{+}(V) =m_{X} -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 m

(ii) ⇒ (iii): Let V ∈CO(Y) . Then Y −V ∈CO(Y) and by (ii) we have X − F* ^{+}*(V) = F

(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 . By (iii), we have -αψ - cl(F+(Y −V)) . By Lemma 2.4., there exists an m_{X} -αψ -open set of m_{X} containing x such that U ∩F* ^{+}*(Y −V) =φ , hence U ⊂ F

**Corollary (1)**

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

1. *F is upper slightly m _{X} -αψ -continuous;*

2. *F ^{+}(V) is m_{X} -αψ -open in (X,m_{X} ) for each V ∈CO(Y) ;*

3. *F ^{-}(V) is m_{X} -αψ -closed in (X,m_{X} ) for each V ∈CO(Y) .*

**Corollary (2)**

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

1. *F is lower slightly m _{X} -αψ -continuous;*

2. *F ^{-}(V) is m_{X} -αψ -open in (X,m_{X} ) for each V ∈CO(Y) ;*

3.* F ^{+}(V) is m_{X} -αψ -closed in (X,m_{X} ) for each V ∈CO(Y) .*

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