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   .
The concept of semiopen sets and semi-continuity of function first appeared in a paper by Levine  in the realm of topological spaces and this idea was then generalized and extended to the frameworks of bitopological spaces by Maheshwari and Prasad , and also by Bose . In 1963, Berge  introduced the concepts of upper and lower continuous multifunctions and lately, Whyburn  also studied the general continuity of multifunctions in the realm of topological spaces. Smithson  and Popa  respectively then generalized these notions of multifunction continuity to the setting of bitopological spaces. In 2000, Popa and Noiri , 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  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  and also Berge .
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   and . 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  and then . 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 (          ).
By a bitopological space we mean a nonempty set X on which are defined two topologies and , this idea was first introduced by Kelly (  ). In the sequel, or in short hand X will mean a bitopological space unless stated. Given a bitopological space , ; , the interior and closure of with respect to the topology is denoted by and respectively.
Definition 2.1. Let , ; be a bitopological space, .
(i) A is said to be -open if . The complement of an -open set is a -closed set .
(ii) The -interior of A denoted by (or - ) is the union of all -open subsets of X contained in A. Clearly, A is -open provided .
(iii) The -closure of A denoted by defined to be the intersection of all -closed subsets of X containing A. And and .
(iv) A is said to be pairwise open if it is both -open and -open.
Definition 2.2. Let , ; be a bitopological space and .
(i) A is called -semiopen in X provided there is a -open subset O of X such that , equivalently . It’s complement is said to be -semiclosed .
(ii) The -semiinterior of A denoted by - is defined as the union of -semiopen subsets of X contained in A. The -semiclosure of A denoted by - , is the intersection of all -semiclosed sets of X containing A.
(iii) A is said to be pairwise semiopen if it is both -semiopen and -semiopen .
(iv) B is said to be a -semi-neighbourhood of provided there is a -semiopen subset O of X such that .
The family of all -semiopen and -semiclosed subsets of X are denote by and respectively.
Definition 2.3.  A point-to-set correspondence such that for each , is a nonempty subset of Y is said to be a multifunction .
Following Berge , the upper and lower inverse of with respect to a multifunction F, are denoted and defined respectively by:
Generally, for the lower inverse between Y and the power set of X, provided . Clearly for , and also,
For any , .
Definition 2.4.  A multifunction , ; is said to be
(i) Upper -semi continuous at a point provided for any -open subset such that , there exists an -semiopen set U in X with such that (or ).
(ii) Lower -semi continuous at a point provided for any -open set V in Y such that , there exists a -semiopen set U in X with such that for all or .
(iii) Upper (resp Lower) -semi continuous if it is Upper (resp Lower) -semi continuous at all points of X.
Clearly, and are -open in X for any -open set V in Y. F is -semicontinuous at provided it is both upper and lower— -continuous (resp., -semicontinuous) at . Generally, F is upper (resp lower)— -semi continuous if the multifunctions induced and are both upper (resp lower) semi continuous on X.
Definition 2.5.   A subfamily mX of a power set 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 is called an m-space and member of is said to be mX-open.
Definition 2.6.   A subset A of X in an m-space is said to be
(i) m-semiopen if .
(ii) m-semiclosed if .
Definition 2.7.  For a minimal space , mX is said to have property ( ) of Maki  if the union of any collection of mX-open subsets of X is also mX-open.
Lemma 2.8.  For an m-space , an arbitrary union of m-semiopen subsets of X is also m-semiopen.
Lemma 2.9. (cf.    ) For an m-space and subsets A, B and F of X the properties below holds:
(ii) F is mX-semiclosed if and only if .
(iii) A is mX-semiopen if and only if .
(iv) , and , .
(v) If , then and
(vi) and .
(vii) and .
Lemma 2.10. (cf.    ) Let be an m-space and . Then if and only if for every mX-semiopen set U containing x, .
Lemma 2.11.  For an m-space and subsets A and F of X, the following properties are equivalent:
(i) mX has property ,
(ii) provided ,
(iii) provided .
Lemma 2.12. (  ) Let be am m-space with mX satisfying property and .The properties below holds:
(i) if and only if .
(ii) if and only if
(iii) and 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 , ; be a botopological space and mX a minimal structure on X determined with respect to mi and mj. Then an ordered pair 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 and , ; , we shall denote it by mij and call the pair (or ) a minimal bitopological space unless explicitly stated.
Definition 3.2. Let , ; 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 . It is said to be mij-semiclosed in X if there exists a mi-open set U such that whenever .
We shall denote the collection of all mij-semiopen and mij-semiclosed sets in by and respectively.
Example 3.3. Let , and the minimal structures m1 and m2 be define by and respectively. The sets and are mij-open sets but and are not mij-open. Also, is neither mij-open nor mij-closed.
Remark 3.4. Let , ; be a minimal bitopological space.
(i) if and , the any mij-semiopen set is -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 on X by and respectively. Then the subset of X is mij-semiopen but not mij-open since , whence .
Proposition 3.6. Let , ; be am mij-space and . Then:
(i) if and only if .
(ii) if and only if .
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 . Since and , we have
On the other hand, let , then, referring to Lemma 2.9, we have
Consequently, and hence, . Therefore, . 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 . Since and , we have
and also from (i), . 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 , ; be an mij-space. Then, for a collection of subsets of X, the following properties hold:
(i) provided for all , .
(ii) provided for all , .
Proof. (i) Suppose that for all . Then, by definition 3.2 and Proposition 3.6, we have
which implies , so that, .
(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 , as illustrated in the next example.
Example 3.9. Define the minimal structures m1 and m2 on given set by and . We observer that, and are mij-semiopen sets, ; . But then, which is not mij-semiopen, indeed, .
Definition 3.10. Let , ; be an mij-space. A subset O of X is an mij-semineighborhood of a point if there exists an mij-semiopen subset U of X such that .
Definition 3.11. Let , ; be an mij-space and . Then, the mij-semiinterior and mij-semiclosure of A are respectively denoted and defined by:
Remark 3.12. For any bitopological spaces ;
(i) is a minimal structures of X.
(ii) In the following, we denote by a minimal structure on X a generalization of and . For , if , then by Definition 3.11;
Proposition 3.13. For any mij-space , ; and subsets A and B of X, the following mij-semiclosure and mij-semiinterior properties holds:
(i) and .
(ii) and provided .
(iii) , , and .
(iv) provided .
(v) provided .
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 , Definition 3.2 implies that, there exists an mi-open subset U of X such that . Since, and , we obtain following Proposition 3.6 that
And also . Hence,
(v) Suppose . Then by Definition 3.11 and Proposition 3.7 we have
Hence by (i), .
(vi) follows from (iii) and (iv). o
Proposition 3.14. Let , ; be an mij-space and . Then, for each such that , if and only if .
Proof. Suppose that . We need to show that for all mij-semiopen subset U of X containing x, . On the contrary, suppose for some mij-semiopen set U containing x, . Then and and Proposition 3.13 implies . Since , it follows that which contradicts our assertion. Consequently, .
Conversely, suppose that for every mij-semiopen subset U of X containing x, , then we need to show that . Suppose on the contrary, , then there exists an mij-semiclosed subset F of X such that . Proposition 3.13 then implies , so that . Since , then and so that, . Setting , we obtaining which contradicts our assertion that . Hence, . o
Proposition 3.15. For a minimal bitopological space , ; and any , the properties below holds:
(i) For each we obtain by Definition 3.11 and Proposition 3.13 that,
(ii) Let . Then, is some neighborhood of x for which . Because , . Proposition 3.14 then implies that, and as a consequence, . Therefore,
Convesely, if , the inclusion is obvious. Suppose , then by the complementation property implying . By Proposition 3.14, there exists an mij-semiopen neighborhood of x, V such that . Thus, giving . Hence,
Proposition 3.16. For an mij-space , ; and any subset , the properties below are satisfied:
(ii) provided . The converse to this assertion is not necessary true.
(i) Since by Proposition 3.6, is mij-semiclosed, we have
Thus, . Since , it holds that,
Conversely, let . We aim to show that B is an mij-semiclosed set. By definition,
Thus, is a mij-semiclosed set. Hence,
(ii) Let A be mij-open in X, then by Definition 3.2 and Proposition 3.6,
Consequently, by (i),
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 by and . One clearly see that, is only mi-open but not mii-open even thought .
Remark 3.18. For a bitopological space , ; the families and are all mij-structures of X satisfying property .
Proposition 3.19. For an mij-space , ; with mij satisfying property ( ) and subsets A and F of X, the properties below holds:
(i) provided .
(ii) If , then .
Proof. Suppose that mij satisfying property ( ):
(i) Let , then by Definition 3.11, . Hence, .
(ii) If , then Definition 3.11 and Proposition 3.15 gives
Thus, by part (i), we obtain . o
Proposition 3.20. For any mij-space , ; with mij satisfying property and any , the properties outlined below holds:
(i) if and only if A is mij-semiopen in X.
(ii) if and only if is mij-semiopen in X.
(iii) is mij-semiopen.
(iv) is mij-semiclosed.
Proof. (i) and (ii) follows mij-closed, mij-Interior, the property of 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 , ; be an mij-space with mij satisfying the property and let be an arbitrary collection of subsets of X. Then, provided for every .
Proof. Suppose that mij satisfies property and for all . By Definition 3.2, we can find some mi-open set such that for some . Consequently,
Since is a monotone operator, and is mi-open as mij satisfies property . As a consequence, . Therefore, . o
Proposition 3.22. Let , ; be an mij-space and . Provided mij satisfy property , then:
(i) , and
Proof. (i) Since mij satisfy property , then is an mij-semiclosed set. By Definition 3.2 and Proposition 3.6, we obtain that
Therefore, and it follows that, .
On the other hand, we observe that
Hence, Definition 3.2 and Proposition 3.6 implies,
Consequently, is an mij-semiclosed set and so . This completes the proof.
(ii) The proof follows from Theorems 3.13 and 3.20. o
If the property 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:
Then the collection of all mij-closed and mij-open subsets of are:
where . Take . Then , and . Further,
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 and , ; be minimal bitopological spaces. We shall call a multifunction to be:
(i) Upper -semi-continuous at some point provided for any -open subset satisfying , there is an -semiopen setU in X with for which (or ).
(ii) Lower -semi-continuous at some point provided for each -open subset satisfying , we can find an -semiopen set U in X with such that for all , .
(iii) Upper (resp Lower) -semi continuous if it is Upper (resp Lower) -semi continuous at each and every point of X.
Example 4.2. Let us define the two minimal structures and on by and . Also, let the minimal structure and on be defined by and . Defined a multifunction by:
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 , ; be a multifunction where, satisfies property . F is upper mij-semicontinuous at some point x of X if and only if for every -open set V in Y with , .
Proof. Let F be upper mij-semicontinuous at a point x of X and V be an -open set such that . By Definition 4.1, there is some -semiopen set U containing x for which . Therefore, . The -semiopeness of U implies, . Because satisfies property , we have from Definition 2.3 that, .
Conversely, if V is an -open set in Y with for which , then by definition, there is some -semiopen set U in X with satisfying . This implies, and so F is upper mij-semicontinuous at a point x of X. o
Theorem 4.4. Provided , ; satisfies property , the following properties are equivalent for a multifunction :
(i) F is upper mij-semicontinuous;
(ii) For every -open set V, the set is -semiopen;
(iii) For every -closed set K, the set is -semiclosed;
(iv) For every -open set, the set inclusion is true;
(v) For any subset B of Y, the set inclusion holds true;
(vi) For any subset B of Y, the set inclusion holds.
Proof. (i) (ii): let V be a -open set of Y and for all then, . From the hypothesis, we can find an -semi-open set U with for which . And so, one gets for every x belongning to . Consequently by Proposition 3.6,
This implies that . By Proposition 3.13, we have that, . This gives, and so, is semi-open in X.
(ii) (iii): Let . Since then, we have from , Proposition 3.13 and Proposition 3.19 that,
implying is -semiclosed.
(iii) (iv): If A is some subset of X, Definition 3.11 implies,
And so, .
(iv) (v): Proposition 3.13 and Proposition 3.20 implies is a -closed set for if B is any subset of Y. Proposition 3.13 and (iv) of this theorem gives:
and the implication follows.
(v) (vi): Using Definition 2.3, , Proposition 3.13 and (v), it follows for a subset B of Y that,
(vi) (ii): Since is a -closed set for any -open set V in Y, (vi) and Proposition 3.13 implies;
Consequently, we obtain , implying is an -semiopen set in X.
(ii) (i): For any point and -open set V in Y such that , part (ii) of this theorem implies, is a -semiopen set in X with . Setting , we have by the 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 ; , satisfies property , a multifunction is lower mij-semicontinuous at if and only if for every -open set V of Y such that .
Proof. Suppose F is lower mij-semicontinuous at . Let V be an -open set satisfying for . By Definition 4.1, there is an -semiopen set U with such that for all . Thus, from Definition 2.3, . Because U is -semiopen, . Since satisfies property , we get
Conversely, suppose V is an -open set with . From the assumption, . By Definition 2.3 and 4.1, there exists a -semiopen set O in X containing x such that for all , . Clearly, so that, by Definition 2.3. Consequently, F is lower mij-semicontinuous at a point x of X. o
Theorem 4.6. If , ; , satisfies property , then the following properties are equivalent for a multifunction :
(i) F is lower mij-semicontinuous;
(ii) For any -open set V in Y, the set is -semiopen in X
(iii) For any -closed set K in Y, the set is -semiclosed in X;
(iv)Given any subset B of Y, ;
(v) Given any every , ;
(vi) Given any subset , the inclusion holds true.
Proof. (i) (ii): let V be a -open set of Y such that for a point . By the hypothesis, we can find an -semiopen set O with such that for any , holds. Since O is -semiopen, . Because satisfies property , we then obtain;
implies . By Proposition 3.13, and so, is semiopen.
(ii) (iii): If K is any -closedset in Y then, . Thus,
Which shows that, implying is -semiclosed.
(iii) (iv): By Proposition 3.13 and Proposition 3.20, is -closed in Y for any subset B of Y. From (iii), we obtain that
(iv) (v): For we have for (iv) that,
(v) (vi): Because for a set , Proposition 3.13 and part (v), then implies:
(vi) (i): From (vi), we obtain for and -open set V in Y satisfying that,
Thus, there is an -semiopen set O with for which for all . This implies is -semiopen and so, F is a lower mij-semicontinuous multifunction. o
Lemma 4.7. For a minimal bitopological space , ; , and for a subset A of X, the properties below holds:
Proof. (i) For , Proposition 3.13, is an mij-semiclosed set. Hence from Proposition 3.6, we have
(ii) Similarly, by Proposition 3.13, is an mij-semiopen set and by Proposition 3.6, we have
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 , ; with satisfying property , the statements below are equivalent:
(i) F is lower mij semicontinuous;
(ii) for any -open set V in Y,
(iii) for any -closed set K in Y,
(iv) For , ;
(v) for any subset B of Y;
(vi) for any subset B of Y.
Proof. (i) (ii): From Theorem 4.6 and Proposition 3.6, we obtain
(ii) (iii): If K is a -closed subset in Y, then by Theorem 4.4 and Proposition 3.6,
(iii) (iv): By Theorem 4.4 and Lemma 4.7, we obtain for a subset A of X that,
This shows that, .
(iv) (v): From (iv) and Proposition 3.13, we obtain for any that,
By Proposition 3.6, .
(v) (vi): From Proposition 3.13 and (v), one obtain
(vi) (i): For an -open set V in Y we obtain from (vi) and Proposition 3.13 that,
As such, is -semiopen in X and by (ii), F is lower mij-semi-continuous. o
In the first part of this paper, we have investigated and then generalized Maki , Prasad  and Bose  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  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.
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.