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 APM  Vol.11 No.4 , April 2021
On M-Asymmetric Semi-Open Sets and Semicontinuous Multifunctions in Bitopological Spaces
Abstract: The purpose of this paper is to introduce the notions of m-asymmetric semiopen sets and M-asymmetric semicontinuous multifunctions defined between asymmetric sets satisfying certain minimal conditions in the framework of bitopological spaces. Some new characterizations of m-asymmetric semiopen sets and M-asymmetric semicontinuous multifunctions will be investigated and several fundamental properties will be obtained.

1. Introduction

The concept of topology is an important tool that has received considerable attention from many scholars in many fields of applied sciences and other branches of pure mathematics. Continuity and multifunctions on the other hand, which are basic notions in the theory of classical point set topology also play a vital role not only in the field of pure mathematics but also in applied mathematics. In this regard, several scholars have generalizations the notion of continuity in topological spaces using the weaker forms of open and closed sets the semiopen and semiclosed sets [1] [2] [3].

The concept of semiopen sets and semi-continuity of function first appeared in a paper by Levine [1] in the realm of topological spaces and this idea was then generalized and extended to the frameworks of bitopological spaces by Maheshwari and Prasad [2], and also by Bose [4]. In 1963, Berge [5] introduced the concepts of upper and lower continuous multifunctions and lately, Whyburn [6] also studied the general continuity of multifunctions in the realm of topological spaces. Smithson [7] and Popa [8] respectively then generalized these notions of multifunction continuity to the setting of bitopological spaces. In 2000, Popa and Noiri [9], introduced and studied the concepts of m-structures and M-continuous function as a function defined between topologies satisfying certain minimal conditions. They showed that the M-continuous functions have properties similar to those of continuous functions between topological spaces. With these ideas of multifunction continuity and M-continuous functions, Noiri and Popa [10] then introduced and studied upper and lower M-continuous multifunctions and showed how these functions have properties similar to those of upper and lower continuous functions and continuous multifunctions between topological spaces.

In this present paper, we introduce and investigate some basic characterizations and properties of m-asymmetric semiopen sets, m-asymmetric semiclosed sets and, upper and lower M-asymmetric semicontinuous multifunctions in the realm of bitopological spaces satisfying a minimal structure, which is a generalization of results of Noiri and Popa [10] and also Berge [5].

This paper is organized as follows. Section 2 presents some necessary preliminaries concerning semiopen sets, continuity and multifunctions on sets satisfying certain minimal structures. In Section 3, we investigate and study the concept of m-asymmetric semiopen and semiclosed sets in the realm of bitopological spaces, a generalition of semiopen and semiclose closed sets in [1] [2] and [4]. Section 4 presents and discusses some results of upper and lower M-asymmetric semicontinuous multifunctions as a generalized idea of upper and lower upper M-continuous multifunctions first introduced in [8] and then [10]. Section 5 gives some concluding remarks.

2. Preliminaries and Basic Properties

Throughout the section, we recall some basic notations and properties, which we refer the reader to ( [1] [2] [3] [4] [7] [8] [9] [10] [11] ).

By a bitopological space ( X , T 1 , T 2 ) we mean a nonempty set X on which are defined two topologies T 1 and T 2 , this idea was first introduced by Kelly ( [12] ). In the sequel, ( X , T 1 , T 2 ) or in short hand X will mean a bitopological space unless stated. Given a bitopological space ( X , T i , T j ) , i , j = 1 , 2 ; i j , the interior and closure of A X with respect to the topology T i = T j is denoted by I n t T i ( B ) and C l T i ( A ) respectively.

Definition 2.1. Let ( X , T i , T j ) , i , j = 1 , 2 ; i j be a bitopological space, A X .

(i) A is said to be T i T j -open if A T i T j . The complement of an T i T j -open set is a T i T j -closed set [12].

(ii) The T i T j -interior of A denoted by I n t T i ( I n t T j ( A ) ) (or T i T j - I n t ( A ) ) is the union of all T i T j -open subsets of X contained in A. Clearly, A is T i T j -open provided A = I n t T i ( I n t T j ( A ) ) .

(iii) The T i T j -closure of A denoted by C l T i ( C l T j ( B ) ) defined to be the intersection of all T i T j -closed subsets of X containing A. And C l T i ( C l T j ( B ) ) C l T i ( B ) and C l T i ( C l T j ( B ) ) C l T j ( B ) .

(iv) A is said to be pairwise open if it is both T 1 T 2 -open and T 2 T 1 -open.

Definition 2.2. Let ( X , T i , T j ) , i , j = 1 , 2 ; i j be a bitopological space and A , B X .

(i) A is called T i T j -semiopen in X provided there is a T i -open subset O of X such that O A C l T j ( O ) , equivalently A C l T j ( I n t T i ( A ) ) . It’s complement is said to be T i T j -semiclosed [2].

(ii) The T i T j -semiinterior of A denoted by T i T j - s I n t ( A ) is defined as the union of T i T j -semiopen subsets of X contained in A. The T i T j -semiclosure of A denoted by T i T j - s C l ( A ) , is the intersection of all T i T j -semiclosed sets of X containing A.

(iii) A is said to be pairwise semiopen if it is both T 1 T 2 -semiopen and T 2 T 1 -semiopen [11].

(iv) B is said to be a T i T j -semi-neighbourhood of x X provided there is a T i T j -semiopen subset O of X such that x O B .

The family of all T i T j -semiopen and T i T j -semiclosed subsets of X are denote by T i T j s O ( X ) and T i T j s C ( X ) respectively.

Definition 2.3. [10] A point-to-set correspondence F : X Y such that for each x X , F ( x ) is a nonempty subset of Y is said to be a multifunction .

Following Berge [5], the upper and lower inverse of G Y with respect to a multifunction F, are denoted and defined respectively by:

F + ( G ) = { x X : F ( x ) G } and F ( G ) = { x X : F ( x ) G } .

Generally, for the lower inverse F between Y and the power set P ( X ) of X, F ( y ) = { x X : y F ( x ) } provided y Y . Clearly for G Y , F ( G ) = { F ( y ) : y G } and also,

F + ( G ) = X \ F ( Y \ G ) and F ( G ) = X \ F + ( Y \ G )

For any A X , F ( A ) = x A F ( x ) .

Definition 2.4. [10] A multifunction F : ( X , T i , T j ) ( Y , Q i , Q j ) , i , j = 1 , 2 ; i j is said to be

(i) Upper T i T j -semi continuous at a point x o X provided for any Q 1 Q 2 -open subset V Y such that F ( x o ) V , there exists an T i T j -semiopen set U in X with x o U such that F ( U ) V (or U F + ( V ) ).

(ii) Lower T i T j -semi continuous at a point x o X provided for any Q i Q j -open set V in Y such that F ( x o ) V , there exists a T i T j -semiopen set U in X with x o U such that F ( z ) V for all z U or U F ( V ) .

(iii) Upper (resp Lower) T i T j -semi continuous if it is Upper (resp Lower) T i T j -semi continuous at all points of X.

Clearly, F + ( V ) and F ( V ) are T i T j -open in X for any Q i Q j -open set V in Y. F is T i T j -semicontinuous at x o X provided it is both upper and lower— T i T j -continuous (resp., T i T j -semicontinuous) at x o . Generally, F is upper (resp lower)— T i T j -semi continuous if the multifunctions induced F 1 : ( X , T 1 ) ( Y , Q 1 ) and F 2 : ( X , T 2 ) ( Y , Q 2 ) are both upper (resp lower) semi continuous on X.

Definition 2.5. [10] [13] A subfamily mX of a power set P ( X ) of a nonempty set X is said to be a minimal structure (briefly m-structure) on X provided both and X belongs to mX. The pair ( X , m X ) is called an m-space and member of ( X , m X ) is said to be mX-open.

Definition 2.6. [3] [13] A subset A of X in an m-space ( X , m X ) is said to be

(i) m-semiopen if A m C l ( m I n t ( A ) ) .

(ii) m-semiclosed if m I n t ( m C l ( A ) ) A .

Definition 2.7. [10] For a minimal space ( X , m X ) , mX is said to have property ( B ) of Maki [3] if the union of any collection of mX-open subsets of X is also mX-open.

Lemma 2.8. [13] For an m-space ( X , m X ) , an arbitrary union of m-semiopen subsets of X is also m-semiopen.

Lemma 2.9. (cf. [3] [10] [13] ) For an m-space ( X , m X ) and subsets A, B and F of X the properties below holds:

(i) m X s C l ( X \ A ) = X \ ( m X s I n t ( A ) ) and m X s I n t ( X \ A ) = X \ ( m s C l ( A ) )

(ii) F is mX-semiclosed if and only if m X s C l ( F ) = F .

(iii) A is mX-semiopen if and only if m X s I n t ( A ) = A .

(iv) m X s C l ( ) = , m X s C l ( X ) = X and m X s I n t ( ) = , m X s I n t ( X ) = X .

(v) If A B , then m X s C l ( A ) m X s C l ( B ) and m X s I n t ( A ) m X s I n t ( B )

(vi) m X s C l ( m X s C l ( A ) ) = m X s C l ( A ) and m X s I n t ( m X s I n t ( A ) ) = m X s I n t ( A ) .

(vii) A m X s C l ( A ) and m X s I n t ( A ) A .

Lemma 2.10. (cf. [3] [10] [13] ) Let ( X , m X ) be an m-space and A X . Then x m X s C l ( A ) if and only if for every mX-semiopen set U containing x, A U .

Lemma 2.11. [10] For an m-space ( X , m X ) and subsets A and F of X, the following properties are equivalent:

(i) mX has property B ,

(ii) A m X provided m X I n t ( A ) = A ,

(iii) X \ F m X provided m X C l ( F ) = F .

Lemma 2.12. ( [10] ) Let ( X , m X ) be am m-space with mX satisfying property B and A X .The properties below holds:

(i) m X I n t ( A ) = A if and only if A m X .

(ii) m X C l ( A ) = A if and only if X \ A m X

(iii) m X I n t ( A ) m X and m X C l ( A ) is mX-closed.

3. On m-Asymmetric Semi Open and Closed Sets

This section introduces the notions of m-asymmetric semiopen and semiclosed sets in the framework of bitopological spaces satisfying certain minimal conditions.

Definition 3.1. Let ( X , T i , T j ) , i , j = 1 , 2 ; i j be a botopological space and mX a minimal structure on X determined with respect to mi and mj. Then an ordered pair ( ( X , T i , T j ) , m X ) shall be called a minimal bitopological space (or mij-spaces).

Since in our sense the minimal structure mX on X is determined by the left and right minimal structures mi and mj which are as a generalization of the two topologies T i and T j , i , j = 1 , 2 ; i j , we shall denote it by mij and call the pair ( ( X , T i , T j ) , m i j ) (or ( X , m i j ) ) a minimal bitopological space unless explicitly stated.

Definition 3.2. Let ( X , m i j ) , i , j = 1 , 2 ; i j be a minimal bitopological space and A a subset of X. Then A is said to be mij-semiopen in X if there exists an mi-open set U such that U A C l m j ( U ) . It is said to be mij-semiclosed in X if there exists a mi-open set U such that C l m j ( A ) U whenever A U .

We shall denote the collection of all mij-semiopen and mij-semiclosed sets in ( X , m i j ) by m i j s O ( X ) and m i j s C ( X ) respectively.

Example 3.3. Let X = { 2, 1,0,1,2 } , and the minimal structures m1 and m2 be define by m 1 = { , { 1 } , { 0 } , { 0,1 } , { 2,0,1 } , X } and m 2 = { , { 0 } , { 2 } , { 0,1 } , X } respectively. The sets and { 0 } are mij-open sets but { 1 } and { 1,1 } are not mij-open. Also, { 1 } is neither mij-open nor mij-closed.

Remark 3.4. Let ( X , m i j ) , i , j = 1 , 2 ; i j be a minimal bitopological space.

(i) if m i = T i and m j = T j , the any mij-semiopen set is T i T j -semiopen.

(ii) every mij-open (resp. mij-closed) set is mij-semiopen (resp. mij-semiclosed), but the converse is not generally true as our next example illustrates.

Example 3.5. Let us define mi and m2 on X = { 2, 1,0,1,2,3 } on X by m 1 = { , { 1 } , { 0 } , { 2 } , X } and m 2 = { , { 2 } , { 1 } , { 1,0 } , X } respectively. Then the subset A = { 1,3 } of X is mij-semiopen but not mij-open since { 1 } A C l m 2 ( { 1 } ) , whence A m i j .

Proposition 3.6. Let ( X , m i j ) , i , j = 1 , 2 ; i j be am mij-space and A X . Then:

(i) A m i j s O ( X ) if and only if A C l m j ( I n t m i ( A ) ) .

(ii) A m i j s C ( X ) if and only if I n t m j ( C l m i ( A ) ) A .

Proof. (i) Suppose A is a mij-semiopen subset of X. By Definition 3.2 we can always find an mi-open subset U of X such that U A C l m j ( U ) . Since U I n t m i ( A ) and U = I n t m i ( U ) , we have

A C l m j ( U ) C l m j ( I n t m i ( A ) ) .

On the other hand, let U = C l m j ( I n t m i ( A ) ) , then, referring to Lemma 2.9, we have

X \ U = X \ C l m j ( I n t m i ( A ) ) = I n t m j ( X \ I n t m i ( A ) ) = I n t m j ( C l m i ( X \ A ) ) .

Thus,

C l m j ( X \ U ) = C l m j ( I n t m j ( C l m i ( X \ A ) ) ) = I n t m j ( C l m i ( X \ A ) ) = X \ U .

Consequently, X \ ( X \ U ) m i j s O ( X ) and hence, U m i j s O ( X ) . Therefore, U A C l m j ( U ) . This implies that A is a mij-semiopen set.

(ii) Suppose A is a mij-semiclosed set. By definition 3.2, there is some mi-open set U such that C l m j ( A ) U . Since C l m j ( A ) C l m j ( U ) and U = I n t m i ( U ) , we have

I n t m j ( C l m i ( A ) ) I n t m j ( A ) A .

and also from (i), C l m j ( A ) U . This implies that A is an mij-semiclosed set. o

Generally, the mij-open sets and the mij-semiopen sets are not stable for the union. Nevertheless, for certain mij-structure, the class of mij-semiopen sets are stable under union of sets, like it is demonstrated in the following proposition.

Proposition 3.7. Let ( X , m i j ) , i , j = 1 , 2 ; i j be an mij-space. Then, for a collection { A γ : γ Γ } of subsets of X, the following properties hold:

(i) γ Γ A γ m i j s O ( X ) provided for all γ Γ , A γ m i j s O ( X ) .

(ii) γ Γ A γ m i j s C ( X ) provided for all γ Γ , A γ m i j s C ( X ) .

Proof. (i) Suppose that A γ m i j s O ( X ) for all γ Γ . Then, by definition 3.2 and Proposition 3.6, we have

A γ C l m j I n t m i ( A γ ) C l m j I n t m i ( γ Γ A γ ) ,

which implies γ Γ A γ C l m j I n t m i ( γ Γ A γ ) , so that, γ Γ A γ m i j s O ( X ) .

(ii) It follows from Proposition 3.7 (i) and De-Morgan’s laws. o

Remark 3.8. It should generally be noted that, the intersection of any two mij-semiopen sets may not be mij-semiopen in a minimal bitopological spaces ( X , m i j ) , as illustrated in the next example.

Example 3.9. Define the minimal structures m1 and m2 on given set X = { 3, 2, 1,0,1,2,3 } by m 1 = { , { 0 } , { 2 } , { 0,2 } , X } and m 2 = { , { 2 } , { 0 } , { 2 } , { 0,2 } , X } . We observer that, { 1,0 } and { 1,2 } are mij-semiopen sets, i , j = 1 , 2 ; i j . But then, { 1,0 } { 1,2 } = { 1 } which is not mij-semiopen, indeed, { 1 } C l m i ( I n t m j ( { 1 } ) ) = .

Definition 3.10. Let ( X , m i j ) , i , j = 1 , 2 ; i j be an mij-space. A subset O of X is an mij-semineighborhood of a point x X if there exists an mij-semiopen subset U of X such that x U O .

Definition 3.11. Let ( X , m i j ) , i , j = 1 , 2 ; i j be an mij-space and A X . Then, the mij-semiinterior and mij-semiclosure of A are respectively denoted and defined by:

(i) m i j s I n t ( A ) = { U : U A and U m i j s O ( X ) } ,

(ii) m i j s C l ( A ) = { F : A F and F m i j s C ( X ) } ,

Remark 3.12. For any bitopological spaces ( X , T 1 , T 2 ) ;

(i) T i T j s O ( X ) is a minimal structures of X.

(ii) In the following, we denote by m i j ( X ) a minimal structure on X a generalization of T i and T j . For A X , if m i j ( X ) = T i T j s O ( X ) , then by Definition 3.11;

(a) m i j I n t ( A ) = T i T j s I n t ( A ) ,

(b) m i j C l ( A ) = T i T j s C l ( A ) .

Proposition 3.13. For any mij-space ( X , m i j ) , i , j = 1 , 2 ; i j and subsets A and B of X, the following mij-semiclosure and mij-semiinterior properties holds:

(i) m i j s I n t ( A ) A and m i j s C l ( A ) A A .

(ii) m i j s I n t ( A ) m i j s I n t ( B ) and m i j s C l ( A ) m i j s C ( B ) provided A B .

(iii) m i j s I n t ( ) = , m i j s I n t ( X ) = X , m i j s C l ( ) = and m i j s C l ( X ) = X .

(iv) A = m i j s I n t ( A ) provided A m i j s O ( X ) .

(v) A = m i j s C l ( A ) provided X \ A m i j s O ( X ) .

(vi) m i j s I n t ( m i j s I n t ( A ) ) = m i j s I n t ( A ) and m i j s C l ( m i j s C l ( A ) ) = m i j s C l ( A )

Proof. (i) and (ii) are obvious by the interior and closure property and using Proposition 3.6.

(iii) Follows from the interior and closure properties and also part (i) and (ii).

(iv) If A m i j s O ( X ) , Definition 3.2 implies that, there exists an mi-open subset U of X such that U A C l m j ( U ) . Since, U I n t m i ( A ) and U = I n t m i ( U ) , we obtain following Proposition 3.6 that

A C l m j ( U ) C l m j ( I n t m i ( A ) ) .

And also A C l m j ( I n t m i ( A ) ) . Hence, A = C l m j ( I n t m i ( A ) )

(v) Suppose m i j s C l ( A ) = A . Then by Definition 3.11 and Proposition 3.7 we have

X \ A = X \ m i j s C l ( A ) = m i j s I n t ( X \ A )

Hence by (i), X \ A m i j s O ( X ) .

(vi) follows from (iii) and (iv). o

Proposition 3.14. Let ( X , m i j ) , i , j = 1 , 2 ; i j be an mij-space and A X . Then, for each U m i j s O ( X ) such that x U , U A if and only if x m i j s C l ( A ) .

Proof. Suppose that x m i j s C l ( A ) . We need to show that for all mij-semiopen subset U of X containing x, U A . On the contrary, suppose for some mij-semiopen set U containing x, U A = . Then A X \ U and X \ ( X \ U ) m i j s O ( X ) and Proposition 3.13 implies m i j s C l ( A ) m i j s C l ( X \ U ) = X \ U . Since x m i j s C l ( A ) X \ U , it follows that x U which contradicts our assertion. Consequently, U A .

Conversely, suppose that for every mij-semiopen subset U of X containing x, U A , then we need to show that x m i j s C l ( A ) . Suppose on the contrary, x m i j s C l ( A ) , then there exists an mij-semiclosed subset F of X such that A F . Proposition 3.13 then implies x m i j s C l ( A ) m i j s C l ( F ) = F , so that x F . Since F m i j s C ( X ) , then X \ F m i j s O ( X ) and x X \ F so that, ( X \ F ) A = . Setting U = X \ F , we obtaining U A = which contradicts our assertion that U A . Hence, x m i i s C l ( A ) . o

Proposition 3.15. For a minimal bitopological space ( X , m i j ) , i , j = 1 , 2 ; i j and any A X , the properties below holds:

(i) m i j s C l ( X \ A ) = X \ ( m i j s I n t ( A ) ) ,

(ii) m i j s I n t ( X \ A ) = X \ ( m i j s C l ( A ) ) .

Proof.

(i) For each A X we obtain by Definition 3.11 and Proposition 3.13 that,

m i j s C l ( X \ A ) = { X \ V : X \ V m i j s C ( X ) , X \ A X V } = { X \ V : V m i j s O ( X ) , V A } = X \ { V : V A , V m i j s O ( X ) } by De-Morgans Law = X \ ( m i j s I n t ( A ) )

(ii) Let x m i j s I n t ( X \ A ) . Then, is some neighborhood U x m i j s O ( X ) of x for which U X \ A . Because x A , U A = . Proposition 3.14 then implies that, x m i j s C l ( A ) and as a consequence, x X \ m i j s C l ( A ) . Therefore,

m i j s I n t ( X \ A ) X \ ( m i j s C l ( A ) )

Convesely, if X \ m i j s C l ( A ) = , the inclusion is obvious. Suppose x X \ m i j s C l ( A ) , then by the complementation property x m i j s C l ( A ) implying x A . By Proposition 3.14, there exists an mij-semiopen neighborhood of x, V such that U A = . Thus, V X \ A giving x m i j s C l ( X \ A ) . Hence,

X \ ( m i j s I n t ( A ) ) m i j s C l ( X \ A ) . o

Proposition 3.16. For an mij-space ( X , m i j ) , i , j = 1 , 2 ; i j and any subset A X , the properties below are satisfied:

(i) m i j s C l ( A ) = I n t m j ( C l m i ( A ) ) A .

(ii) m i j s C l ( A ) = I n t m j ( C l m i ( A ) ) provided A m i j O ( X ) . The converse to this assertion is not necessary true.

Proof.

(i) Since by Proposition 3.6, m i j s C l ( A ) is mij-semiclosed, we have

m i j s C l ( A ) I n t m j ( C l m i ( m i j s C l ( A ) ) ) .

Thus, I n t m i ( C l m j ( A ) ) m i j s C l ( A ) . Since A m i j s C l ( A ) , it holds that,

I n t m j ( C l m i ( A ) ) A m i j s C l ( A ) A = m i j s C l ( A ) .

Conversely, let B = I n t m j ( C l m i ( A ) ) A . We aim to show that B is an mij-semiclosed set. By definition,

I n t m j ( C l m i ( B ) ) = I n t m j ( C l m i ( I n t m j ( C l m i ( A ) ) A ) ) I n t m j ( C l m i ( C l m i ( A ) A ) ) = I n t m j ( C l m i ( C l m i ( A ) ) ) = I n t m j ( C l m i ( A ) ) I n t m j ( C l m i ( A ) ) A = B

Thus, B = I n t m j ( C l m i ( A ) ) A is a mij-semiclosed set. Hence,

I n t m j ( C l m i ( A ) ) A m i j s C l ( A ) .

Therefore, m i j s C l ( A ) = I n t m j ( C l m i ( A ) ) A .

(ii) Let A be mij-open in X, then by Definition 3.2 and Proposition 3.6,

A = I n t m j ( I n t m i ( A ) ) I n t m j ( C l m i ( A ) )

Consequently, by (i),

m i j s C l ( A ) = I n t m j ( C l m i ( A ) ) . o

Example 3.17. In this example we aim to show that, the converse to part (ii) of Proposition 3.16 is not necessary true: Define the two minimal structures m1 and m2 on X = { 3 , 2 , 1 , 0 , 1 , 2 } by m 1 = { , { 2,1 } , X } and m 2 = { , { 3 } , { 2 } , X } . One clearly see that, { 2,1 } is only mi-open but not mii-open even thought m i j s C l ( { 2 , 1 } ) = I n t m j ( C l m i ( { 2 , 1 } ) ) .

Remark 3.18. For a bitopological space ( X , T i , T j ) , i , j = 1 , 2 ; i j the families T i T j O ( X ) and m i j s O ( X ) are all mij-structures of X satisfying property B .

Proposition 3.19. For an mij-space ( X , m i j ) , i , j = 1 , 2 ; i j with mij satisfying property ( B ) and subsets A and F of X, the properties below holds:

(i) m i j s I n t ( A ) = A provided A m i j s O ( X ) .

(ii) If X \ F m i j s O ( X ) , then m i j s C l ( F ) = F .

Proof. Suppose that mij satisfying property ( B ):

(i) Let A m i j s O ( X ) , then by Definition 3.11, m i j s I n t ( A ) m i j s O ( X ) . Hence, m i j s I n t ( A ) = A .

(ii) If m i j s C l ( F ) = F , then Definition 3.11 and Proposition 3.15 gives

X \ F = X \ m i j s C l ( F ) = m i j s I n t ( X \ F )

Thus, by part (i), we obtain X \ F m i j s O ( X ) . o

Proposition 3.20. For any mij-space ( X , m i j ) , i , j = 1 , 2 ; i j with mij satisfying property B and any A X , the properties outlined below holds:

(i) A = m i j s I n t ( A ) if and only if A is mij-semiopen in X.

(ii) A = m i j s C l ( A ) if and only if X \ A is mij-semiopen in X.

(iii) m i j s I n t ( A ) is mij-semiopen.

(iv) m i j s C l ( A ) is mij-semiclosed.

Proof. (i) and (ii) follows mij-closed, mij-Interior, the property of B and also Proposition 3.13 and Proposition 3.19.

(iii) and (iv) follows from (i) and (ii) with the aid of Proposition 3.7, Proposition 3.13 and Proposition 3.19. o

Proposition 3.21. Let ( X , m i j ) , i , j = 1 , 2 ; i j be an mij-space with mij satisfying the property B and let { A γ : γ Γ } be an arbitrary collection of subsets of X. Then, γ Γ A γ m i j s O ( X ) provided A γ m i j s O ( X ) for every γ Γ .

Proof. Suppose that mij satisfies property B and A γ m i j s O ( X ) for all γ Γ . By Definition 3.2, we can find some mi-open set U γ such that U γ A γ C l m j U γ for some γ Γ . Consequently,

γ Γ ( U γ ) γ Γ ( A γ ) γ Γ ( C l m j ( U γ ) ) .

Since C l m j is a monotone operator, γ Γ ( C l m j ( U γ ) ) C l m j ( γ Γ ( U γ ) ) and γ Γ ( U γ ) is mi-open as mij satisfies property B . As a consequence, γ Γ ( U γ ) γ Γ ( A γ ) C l m j ( γ Γ ( U γ ) ) . Therefore, γ Γ A γ m i j s O ( X ) . o

Proposition 3.22. Let ( ( X , T i , T j ) , m i j ) , i , j = 1 , 2 ; i j be an mij-space and A X . Provided mij satisfy property B , then:

(i) m i j s C l ( A ) = A I n t m i ( C l m i ( A ) ) , and

(ii) m i j s I n t ( A ) = A C l m i ( I n t m i ( A ) ) holds.

Proof. (i) Since mij satisfy property B , then m i j s C l ( A ) is an mij-semiclosed set. By Definition 3.2 and Proposition 3.6, we obtain that

I n t m j ( C l m i ( m i j s C l ( A ) ) ) m i j s C l ( A ) .

Therefore, I n t m j ( C l m i ( A ) ) m i j s C l ( A ) and it follows that, A I n t m j ( C l m i ( A ) ) m i j s C l ( A ) .

On the other hand, we observe that

I n t m j ( C l m i ( A I n t m j ( C l m i ( A ) ) ) ) = I n t m i ( C l m i ( A ) ) I n t m j ( C l m i ( I n t m j ( C l m i ( A ) ) ) ) ( C l m i ( A ) ) I n t m j ( C l m i ( I n t m j ( C l m i ( A ) ) ) ) = C l m i ( A ) I n t m j ( C l m i ( A ) ) = C l m i ( A ) .

Hence, Definition 3.2 and Proposition 3.6 implies,

I n t m j ( C l m i ( A I n t m j ( C l m i ( A ) ) ) ) I n t m j ( C l m i ( A ) ) A I n t m j ( C l m i ( A ) ) .

and so,

I n t m j ( C l m i ( A I n t m j ( C l m j ( A ) ) ) ) A I n t m j ( C l m j ( A ) ) .

Consequently, A I n t m j ( C l m j ( A ) ) is an mij-semiclosed set and so m i j s C l ( A ) A I n t m j ( C l m j ( A ) ) . This completes the proof.

(ii) The proof follows from Theorems 3.13 and 3.20. o

If the property B of Make is removed in the previous theorem, the equality does not necessarily hold as show in the example below:

Example 3.23. Defined two minimal structures m1 and m2 on as:

m 1 = { , P ( { 2 n 1 : n } ) , { 2 } , } and m 2 = { , P ( { 2 n 1 : n } ) , { 2 } , } .

Then the collection of all mij-closed and mij-open subsets of are:

m i j C ( ) = { , P ( { 2 n 1 : n } ) c , { 2 } , } and m i j O ( ) = { , P ( { 2 n 1 : n } ) , { 2 } , V , } ,

where V { 2 n 1 : n } . Take A = { 4 } . Then m i j s C l ( A ) = { 4 } , m i j C l ( A ) = { 2 n : n } and m i j I n t ( A ) = { 2 n : n } = { 2 } . Further,

A i n t m 2 ( C l m i ( A ) ) = { 2,4 } .

Consequently, m i j s O ( A ) A i n t m 2 ( C l m i ( A ) ) .

4. OnM-Asymmetric Semi Continuous Multifunctions

In this part of our paper, we introduce and investigate some properties of the notion of upper and lower M asymmetric semi-continuous multifunctions.

Definition 4.1. Let ( ( X , T i , T j ) , m i j ( X ) ) and ( ( Y , T 1 , T 2 ) , m i j ( Y ) ) , i , j = 1 , 2 ; i j be minimal bitopological spaces. We shall call a multifunction F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) to be:

(i) Upper m i j ( X ) -semi-continuous at some point x o X provided for any m i j ( Y ) -open subset V Y satisfying F ( x o ) V , there is an m i j ( X ) -semiopen setU in X with x o U for which F ( U ) V (or U F + ( V ) ).

(ii) Lower m i j ( X ) -semi-continuous at some point x o X provided for each m i j ( Y ) -open subset V Y satisfying F ( x o ) V , we can find an m i j ( Y ) -semiopen set U in X with x o U such that for all z U , F ( z ) V .

(iii) Upper (resp Lower) m i j ( X ) -semi continuous if it is Upper (resp Lower) m i j ( X ) -semi continuous at each and every point of X.

Example 4.2. Let us define the two minimal structures m 1 ( X ) and m 2 ( X ) on X = { 2 , 1 , 0 , 1 , 2 , 3 } by m 1 ( X ) = { , { 1 } , { 2 } , { 1 , 2 } , X } and m 2 ( X ) = { , { 1 } , { 2 } , { 1,2 } , X } . Also, let the minimal structure m 1 ( Y ) and m 2 ( Y ) on Y = { 3 , 2 , 1 , 0 , 1 , 2 , 3 } be defined by m 1 ( Y ) = { , { 3 } , { 2,0,1,3 } , X } and m 2 ( Y ) = { , { 2 } , { 0 } , { 2,0 } , { 2,0,1,3 } , X } . Defined a multifunction F : ( ( X , T 1 , T 2 ) , m i j ) ( ( Y , Q 1 , Q 2 ) , m i j ) by:

f ( x ) = { { 2 , 0 } x = 1 { 3 , 1 , 3 } x = 1 { 3 , 2 } x = 2 (1)

Clearly, F is upper and lower M-asymmetric semi-continuous.

In this first part of these sections we discuss some characterizations of upper mij-semicontinuous multifunctions.

Theorem 4.3. Let F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) , i , j = 1 , 2 ; i j be a multifunction where, ( ( Y , Q j , Q i ) , m i j ( Y ) ) satisfies property B . F is upper mij-semicontinuous at some point x of X if and only if for every m i j ( Y ) -open set V in Y with F ( x ) V , x m i j s I n t ( F + ( V ) ) .

Proof. Let F be upper mij-semicontinuous at a point x of X and V be an m i j ( Y ) -open set such that F ( x ) V . By Definition 4.1, there is some m i j ( X ) -semiopen set U containing x for which F ( U ) V . Therefore, x U F + ( V ) . The m i j ( X ) -semiopeness of U implies, U = m i j ( X ) s I n t ( U ) . Because ( ( Y , Q j , Q i ) , m i j ( Y ) ) satisfies property B , we have from Definition 2.3 that, x m i j ( X ) s I n t ( F + ( V ) ) .

Conversely, if V is an m i j ( Y ) -open set in Y with F ( x ) V for which x m i j ( X ) s I n t ( F + ( V ) ) , then by definition, there is some m i j ( X ) -semiopen set U in X with x U satisfying U F + ( V ) . This implies, F ( U ) V and so F is upper mij-semicontinuous at a point x of X. o

Theorem 4.4. Provided ( ( Y , Q j , Q i ) , m i j ( Y ) ) , i , j = 1 , 2 ; i j satisfies property B , the following properties are equivalent for a multifunction F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) :

(i) F is upper mij-semicontinuous;

(ii) For every m i j ( Y ) -open set V, the set F + ( V ) is m i j ( X ) -semiopen;

(iii) For every m i j ( Y ) -closed set K, the set F ( K ) is m i j ( X ) -semiclosed;

(iv) For every m i j ( X ) -open set, the set inclusion F ( m i j ( X ) s C l ( A ) ) m i j ( Y ) C l ( F ( A ) ) is true;

(v) For any subset B of Y, the set inclusion m i j ( X ) s C l ( F ( B ) ) F ( m i j ( Y ) C l ( B ) ) holds true;

(vi) For any subset B of Y, the set inclusion F + ( m i j ( Y ) I n t ( B ) ) m i j ( X ) s I n t ( F + ( B ) ) holds.

Proof. (i) (ii): let V be a m i j ( Y ) -open set of Y and x F + ( V ) for all x X then, F ( x ) V . From the hypothesis, we can find an m i j ( X ) -semi-open set U with x U for which F ( U ) V . And so, one gets x U F + ( V ) for every x belongning to F + ( V ) . Consequently by Proposition 3.6,

x U C l m j ( I n t m i ( U ) ) C l m j ( I n t m i ( F + ( V ) ) ) .

This implies that F + ( V ) m i j ( X ) s I n t ( F + ( V ) ) . By Proposition 3.13, we have that, F + ( V ) m i j s I n t ( F + ( V ) ) . This gives, F + ( V ) = m i j s I n t ( F + ( V ) ) and so, F + ( V ) is m i j ( X ) semi-open in X.

(ii) (iii): Let K m i j s C ( Y ) . Since Y \ K m i j ( Y ) O ( Y ) then, we have from F + ( Y \ K ) = X \ F ( K ) , Proposition 3.13 and Proposition 3.19 that,

X \ F ( K ) = F + ( Y \ K ) = m i j ( X ) s I n t ( F + ( Y \ K ) ) = m i j ( X ) s I n t ( X \ F ( K ) ) = X \ m i j ( X ) s C l ( F ( K ) )

X \ m i j ( X ) s C l ( F ( K ) ) = m i j ( X ) s I n t ( X \ F ( K ) ) = m i j ( X ) s I n t ( F + ( Y \ K ) ) = F + ( Y \ K ) = X \ F ( K )

Hence,

F ( K ) = m i j ( X ) s C l ( F ( K ) ) ,

implying F ( K ) is m i j ( X ) -semiclosed.

(iii) (iv): If A is some subset of X, Definition 3.11 implies,

F ( m i j ( Y ) C l ( F ( A ) ) ) = F ( { K Y : F ( A ) K and K m i j C ( Y ) } ) { F ( K ) X : A F ( K ) and F ( K ) m i j s C ( X ) } = { L X : A L and L m i j s C ( X ) } = m i j ( X ) s C l ( A ) .

And so, m i j ( Y ) C l ( F ( A ) ) F ( m i j ( X ) s C l ( A ) ) .

(iv) (v): Proposition 3.13 and Proposition 3.20 implies m i j ( Y ) C l ( B ) is a m i j ( Y ) -closed set for if B is any subset of Y. Proposition 3.13 and (iv) of this theorem gives:

F ( m i j ( Y ) C l ( B ) ) = F ( { Q Y : B Q and Q m i j C ( Y ) } ) = { F ( Q ) X : F ( B ) F ( Q ) and F ( Q ) m i j s C ( X ) } { R X : F ( B ) R and R m i j s C ( X ) } = m i j ( X ) s C l ( F ( B ) ) .

and the implication follows.

(v) (vi): Using Definition 2.3, m i j ( Y ) I n t ( B ) = Y \ m i j ( Y ) C l ( Y \ B ) , Proposition 3.13 and (v), it follows for a subset B of Y that,

X \ m i j ( X ) s I n t ( F + ( B ) ) = m i j ( X ) s C l ( X \ F + ( B ) ) = m i j ( X ) s C l ( F ( Y \ B ) ) F ( m i j ( Y ) C l ( Y \ B ) ) = F ( Y \ m i j ( Y ) I n t ( B ) ) = X \ F + ( m i j ( Y ) I n t ( B ) ) .

Consequently, F + ( m i j ( Y ) I n t ( B ) ) m i j ( X ) s I n t ( F + ( B ) ) .

(vi) (ii): Since Y \ V is a m i j ( Y ) -closed set for any m i j ( Y ) -open set V in Y, (vi) and Proposition 3.13 implies;

X \ F + ( V ) X \ F + ( m i j ( Y ) I n t ( V ) ) = F ( Y \ m i j ( Y ) I n t ( V ) ) = F ( m i j ( Y ) C l ( Y \ V ) ) m i j ( X ) s C l ( F ( Y \ V ) ) by ( iv ) = m i j ( X ) s C l ( X \ F + ( V ) ) = X \ m i j ( X ) s I n t ( F + ( V ) ) .

Consequently, we obtain F + ( V ) F + ( m i j ( Y ) I n t ( V ) ) m i j ( X ) s I n t ( F + ( V ) ) , implying F + ( V ) is an m i j ( X ) -semiopen set in X.

(ii) (i): For any point x X and m i j ( Y ) -open set V in Y such that F + ( x ) V , part (ii) of this theorem implies, F + ( V ) is a m i j ( X ) -semiopen set in X with x F + ( V ) . Setting x F + ( V ) , we have by the F ( U ) V and consequently, by Definition 4.1, F is upper mij-semicontinuous. o

This second part of the sections gives some characterizations of lower mij-semicontinuous multifunctions.

Theorem 4.5. Provided ( ( Y , Q j , Q i ) , m i j ( Y ) ) i , j = 1 , 2 ; i j , satisfies property B , a multifunction F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) is lower mij-semicontinuous at x X if and only if x m i j s I n t ( F ( V ) ) for every m i j ( Y ) -open set V of Y such that F ( x ) V .

Proof. Suppose F is lower mij-semicontinuous at x o X . Let V be an m i j ( Y ) -open set satisfying F ( x ) V for x o X . By Definition 4.1, there is an m i j ( X ) -semiopen set U with x o U such that F ( x ) V for all x U . Thus, from Definition 2.3, x U F ( V ) . Because U is m i j ( X ) -semiopen, U = m i j ( X ) s I n t ( U ) . Since ( ( Y , Q j , Q i ) , m i j ( Y ) ) satisfies property B , we get

x m i j ( X ) s I n t ( U ) m i j ( X ) s I n t ( F + ( V ) ) .

Conversely, suppose V is an m i j ( Y ) -open set with F ( x ) V . From the assumption, x m i j ( X ) s I n t ( F + ( V ) ) . By Definition 2.3 and 4.1, there exists a m i j ( X ) -semiopen set O in X containing x such that for all z O , F ( z ) V . Clearly, O F ( V ) so that, F ( O ) V by Definition 2.3. Consequently, F is lower mij-semicontinuous at a point x of X. o

Theorem 4.6. If ( ( Y , Q j , Q i ) , m i j ( Y ) ) , i , j = 1 , 2 ; i j , satisfies property B , then the following properties are equivalent for a multifunction F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) :

(i) F is lower mij-semicontinuous;

(ii) For any m i j ( Y ) -open set V in Y, the set F ( V ) is m i j ( X ) -semiopen in X

(iii) For any m i j ( Y ) -closed set K in Y, the set F + ( K ) is m i j ( X ) -semiclosed in X;

(iv)Given any subset B of Y, m i j ( X ) s C l ( F + ( B ) ) F + ( m i j ( Y ) C l ( B ) ) ;

(v) Given any every A m i j ( X ) , F ( m i j s C l ( A ) ) C l ( F ( A ) ) ;

(vi) Given any subset B Y , the inclusion F ( m i j ( Y ) I n t ( B ) ) m i j ( X ) s I n t ( F ( B ) ) holds true.

Proof. (i) (ii): let V be a m i j ( Y ) -open set of Y such that F ( x ) V for a point x X . By the hypothesis, we can find an m i j ( X ) -semiopen set O with x O such that for any z O , F ( z ) V holds. Since O is m i j ( X ) -semiopen, O = m i j ( X ) s I n t ( U ) . Because ( ( Y , Q j , Q i ) , m i j ( Y ) ) satisfies property B , we then obtain;

x m i j ( X ) s I n t ( U ) m i j ( X ) s I n t ( F ( V ) ) ,

implies F ( V ) m i j ( X ) s I n t ( F ( V ) ) . By Proposition 3.13, F ( V ) m i j s I n t ( F ( V ) ) and so, F ( V ) is m i j ( X ) semiopen.

(ii) (iii): If K is any m i j ( Y ) -closedset in Y then, Y K m i j ( Y ) O ( Y ) . Thus,

X \ F + ( K ) = F ( Y \ K ) = m i j ( X ) s I n t ( F ( Y \ K ) ) = m i j ( X ) s I n t ( X \ F + ( K ) ) = X \ m i j ( X ) s C l ( F + ( K ) ) .

Which shows that, F + ( K ) = m i j ( X ) s C l ( F + ( K ) ) implying F + ( K ) is m i j ( X ) -semiclosed.

(iii) (iv): By Proposition 3.13 and Proposition 3.20, m i j ( Y ) C l ( B ) is m i j ( Y ) -closed in Y for any subset B of Y. From (iii), we obtain that

m i j ( X ) s C l ( F + ( B ) ) = { F + ( K ) X : F + ( B ) F + ( K ) and F + ( K ) m i j s C ( X ) } F + ( { K Y : B K and K m i j C ( Y ) } ) = F + ( m i j ( Y ) C l ( B ) )

Thus, m i j ( X ) s C l ( F + ( B ) ) F + ( m i j ( Y ) C l ( B ) ) .

(iv) (v): For A X we have for (iv) that,

F + ( m i j ( Y ) C l ( F ( A ) ) ) = F + ( { K Y : F ( A ) K and K m i j C ( Y ) } ) = { F + ( K ) X : A F + ( K ) and F + ( K ) m i j s C ( X ) } { D X : A D and D m i j s C ( X ) } = m i j ( X ) s C l ( A ) .

Consequently, F ( m i j ( X ) s C l ( A ) ) m i j ( Y ) C l ( F ( A ) ) .

(v) (vi): Because m i j ( Y ) I n t ( B ) = Y \ m i j ( Y ) C l ( Y \ B ) for a set B Y , Proposition 3.13 and part (v), then implies:

F ( m i j ( Y ) I n t ( B ) ) = F ( Y \ m i j ( Y ) C l ( Y \ B ) ) = X \ F + ( m i j ( Y ) C l ( Y \ B ) ) X \ m i j ( X ) s C l ( F + ( Y \ B ) ) = X \ m i j ( X ) s C l ( X \ F ( B ) ) = m i j ( X ) s I n t ( F ( B ) ) .

Consequently, F + ( m i j ( Y ) I n t ( B ) ) m i j ( X ) s I n t ( F + ( B ) ) .

(vi) (i): From (vi), we obtain for x X and m i j ( Y ) -open set V in Y satisfying F + ( x ) V that,

x F ( V ) = F ( m i j ( Y ) I n t ( V ) ) m i j ( X ) s I n t ( F ( V ) ) .

Thus, there is an m i j ( X ) -semiopen set O with x O for which F + ( z ) V for all z O . This implies F ( V ) is m i j ( X ) -semiopen and so, F is a lower mij-semicontinuous multifunction. o

Lemma 4.7. For a minimal bitopological space ( ( X , T i , T j ) , m i j ( X ) ) , i , j = 1 , 2 ; i j , and for a subset A of X, the properties below holds:

(i) C l m j ( I n t m i ( A ) ) C l m j ( I n t m i ( m i j s C l ( A ) ) ) m i j s C l ( A ) .

(ii) m i j s I n t ( A ) I n t m j ( C l m i ( m i j s I n t ( A ) ) ) I n t m j ( C l m i ( A ) ) .

Proof. (i) For A X , Proposition 3.13, m i j s C l ( A ) is an mij-semiclosed set. Hence from Proposition 3.6, we have

C l m j ( I n t m i ( A ) ) C l m j ( I n t m i ( m i j s C l ( A ) ) ) m i j s C l ( A ) .

(ii) Similarly, by Proposition 3.13, m i j s I n t ( A ) is an mij-semiopen set and by Proposition 3.6, we have

m i j s I n t ( A ) I n t m j ( C l m i ( m i j s I n t ( A ) ) ) I n t m j ( C l m i ( A ) ) . o

The concepts of Theorem 4.4 and 4.6 and, Lemma 4.7, leads us the the theorem that follows:

Theorem 4.8. For a multifunction F : ( ( X , T i , T j ) , m i j ( X ) ) ( ( Y , Q j , Q i ) , m i j ( Y ) ) , i , j = 1 , 2 ; i j with ( ( Y , Q j , Q i ) , m i j ( Y ) ) satisfying property B , the statements below are equivalent:

(i) F is lower mij semicontinuous;

(ii) F ( V ) C l m j ( I n t m i ( F ( V ) ) ) for any m i j ( Y ) -open set V in Y,

(iii) I n t m j ( C l m i ( F ( K ) ) ) F ( K ) for any m i j ( Y ) -closed set K in Y,

(iv) For A X , F ( I n t m j ( C l m i ( A ) ) ) m i j ( Y ) C l ( F ( A ) ) ;

(v) I n t m j ( C l m i ( F ( B ) ) ) F ( m i j ( Y ) C l ( B ) ) for any subset B of Y;

(vi) F ( m i j ( Y ) I n t ( B ) ) C l m j ( I n t m j ( F ( B ) ) ) for any subset B of Y.

Proof. (i) (ii): From Theorem 4.6 and Proposition 3.6, we obtain

x m i j ( X ) s I n t ( U ) m i j ( X ) s I n t ( F ( V ) ) .

Consequently, F ( V ) C l m j ( I n t m i ( F ( V ) ) ) .

(ii) (iii): If K is a m i j ( Y ) -closed subset in Y, then by Theorem 4.4 and Proposition 3.6,

X \ F ( K ) = F + ( Y \ K ) m i j ( X ) s I n t ( F + ( Y \ K ) ) = m i j ( X ) s I n t ( X \ F ( K ) ) = X \ m i j ( X ) s C l ( F ( K ) ) .

Consequently, F ( K ) C l m j ( I n t m i ( F ( K ) ) ) .

(iii) (iv): By Theorem 4.4 and Lemma 4.7, we obtain for a subset A of X that,

I n t m j ( C l m i ( A ) ) m i j ( X ) s C l ( A ) F ( C l m i ( F ( A ) ) ) .

This shows that, F ( I n t m j ( C l m i ( A ) ) ) C l m i ( F ( A ) ) .

(iv) (v): From (iv) and Proposition 3.13, we obtain for any B Y that,

F ( m i j ( Y ) C l ( B ) ) = F ( { K Y : B K and K m i j C ( Y ) } ) = { F ( K ) X : F ( B ) F ( K ) and F ( K ) m i j s C ( X ) } { Q X : F ( B ) Q and Q m i j s C ( X ) } = m i j ( X ) s C l ( F ( B ) ) .

By Proposition 3.6, I n t m j ( C l m i j ( F ( B ) ) ) F ( m i j ( Y ) C l ( B ) ) .

(v) (vi): From Proposition 3.13 and (v), one obtain

F ( m i j ( Y ) I n t ( B ) ) = F ( Y \ m i j ( Y ) C l ( Y \ B ) ) = X \ F + ( m i j ( Y ) C l ( Y \ B ) ) X \ m i j ( X ) s C l ( F + ( Y \ B ) ) = X \ m i j ( X ) s C l ( X \ F ( B ) ) = m i j ( X ) s I n t ( F ( B ) ) .

Thus, F ( m i j ( Y ) I n t ( B ) ) C l m j ( I n t m j ( F ( B ) ) ) .

(vi) (i): For an m i j ( Y ) -open set V in Y we obtain from (vi) and Proposition 3.13 that,

F ( V ) = F ( m i j ( Y ) I n t ( V ) ) C l m j ( I n t m i ( F ( V ) ) ) .

As such, F ( V ) is m i j ( X ) -semiopen in X and by (ii), F is lower mij-semi-continuous. o

5. Conclusion

In the first part of this paper, we have investigated and then generalized Maki [3], Prasad [2] and Bose [4] extensions of the weak forms of sets: the semiopen sets, to asymmetric topologies with sets satisfying certain minimal conditions. In the second part of our work, we have to use Noiri and Popa’s [10] ideas to successfully introduce and investigate several fundamental properties of upper and lower M-asymmetric semicontinuous multifunctions defined between sets satisfying some minimal conditions in the realm of bitopological spaces. We have observed that the M-asymmetric semicontinuous multifunctions have properties similar to those of upper and lower continuous functions and M-continuous multifunctions between topological spaces, with the only difference that, the semiopen sets use in our results belonging to two topologies.

Acknowledgements

The authors wish to acknowledge the support of Mulungushi University and the refereed authors for their helpful work towards this paper. They also thank the anonymous referees for their helpful comments towards the improvement of the paper.

Cite this paper: Matindih, L. and Moyo, E. (2021) On M-Asymmetric Semi-Open Sets and Semicontinuous Multifunctions in Bitopological Spaces. Advances in Pure Mathematics, 11, 218-236. doi: 10.4236/apm.2021.114016.
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[9]   Popa, V. and Noiri, T. (2000) On m-Continuous Functions. Annals of the University Dunarea de Jos of Galati, 18, 31-41.

[10]   Noiri, T. and Popa, V. (2000) On Upper and Lower m-Continuous Multifunctions. Filomat, 14, 73-86.

[11]   Banerjee, K.A. and Saha, K.P. (2015) Semi Open Sets in Bispaces. Cubo (Temuco), 17, 99-106.
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[12]   Kelly, J.C. (1963) Bitopological Spaces. Proceedings of the London Mathematical Society, 3, 71-89.
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[13]   Min, W.K. (2009) m-Semiopen Sets and m-Semicontinuous Functions on Spaces with Minimal Structures. Honam Mathematical Journal, 31, 239-245.
https://doi.org/10.5831/HMJ.2009.31.2.239

 
 
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