# Onℳ𝒜(i,j)-Continuous Functions in Biminimal Structure Spaces

Authors: Chawalit Boonpok; Chokchai Viriyapong; Montri Thongmoon; Nongluk Viriyapong
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

We introduce the notion ofℳ𝒜(i,j)-continuous functions and some other forms of continuity in biminimal structure spaces. Some new characterizations and several fundamental properties of ℳ𝒜(i,j)-continuous functions are obtained.


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## Body

## 1. Introduction

Weak continuity due to Levine [1] is one of the most important weak forms of continuity in topological spaces. Rose [2] has introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. In [3], Baker obtained several properties of subweak continuity which are analogous to results in [4]. Njåstad [5] introduced a weak form of open sets called α-sets. In [6], the author showed that connectedness is preserved under weakly α-continuous surjections. Mashhour et al. [7] have called strongly semicontinuous α-continuous and obtained several properties of such functions. In [8], they stated without proofs that α-continuity implies θ-continuity and is independent of almost continuity in the sense of Singal [9]. On the other hand, in 1980 Maheshwari and Thakur [10] defined α-irresolute and obtained several properties of α-irresolute functions. Levine [11] defined the notions of semiopen sets and semicontinuity in topological spaces. Maheshwari and Prasad [12] extended the notions of semiopen sets and semicontinuity to the bitopological setting. Bose [13] further investigated many properties of semiopen sets and semicontinuity in bitopological spaces. Mashhour et al. [7] introduced the notions of preopen sets and precontinuity in topological spaces. Jelić [14] generalized the notions of preopen sets and precontinuity to the setting of bitopological spaces. The purpose of the present paper is to introduce the notion of ℳ𝒜(i,j)-continuous functions in biminimal structure spaces and investigate the properties of these functions.


## 2. Preliminaries

Definition 1 (see [15]).
LetX be a nonempty set and 𝒫(X) the power set of X. A subfamily mX of 𝒫(X) is called a minimal structure (briefly  m-structure) on X if ∅∈mX and X∈mX.By(X,mX), we denote a nonempty set X with an m-structure mX on X and it is called an m-space. Each member of mX is said to be mX-open, and the complement of an mX-open set is said to be mX-closed.Definition 2 (see [16]).
LetX be a nonempty set and mX an m-structure on X. For a subset A of X, the  mX-closure of A and the mX-interior of A are defined as follows:  (1)
mXCl(A)=∩{F:A⊆F,X-F∈mX};(2)
mXInt(A)=∪{U:U⊆A,U∈mX}.Lemma 3 (see [16]).
LetX be a nonempty set and mX a minimal structure on X. For subset A and B of X, the following properties hold:  (1)
mXCl(X-A)=X-mXInt(A) and mXInt(X-A)=X-mXCl(A).(2)
If(X-A)∈mX, then mXCl(A)=A and if A∈mX, then mXInt(A)=A.(3)
mXCl(∅)=∅, mXCl(X)=X, mXInt(∅)=∅, and mXInt(X)=X.(4)
IfA⊆B, then mXCl(A)⊆mXCl(B) and mXInt(A)⊆mXInt(B).(5)
A⊆mXCl(A) and mXInt(A)⊆A.(6)
mXCl(mXCl(A))=mXCl(A) and mXInt(mXInt(A))=mXInt(A).Definition 4 (see [16]).
Anm-structure mX on a nonempty set X is said to  have propertyℬ if the union of any family of subsets belonging to mX belongs to mX.Lemma 5 (see [17]).
LetX be a nonempty set and mX an m-structure on X satisfying property ℬ. For a subset A of X, the following properties hold:(1)
A∈mX if and only if mXInt(A)=A.(2)
A is mX-closed if and only if mXCl(A)=A.(3)
mXInt(A)∈mX and mXCl(A) is mX-closed.Definition 6.
LetX be a nonempty set and mX1,mX2 minimal structures on X. The triple (X,mX1,mX2) is called a bispace (briefly  bim-space) [18] or biminimal structure space (briefly bimspace) [19].Let(X,mX1,mX2) be a biminimal structure space and A a subset of X. The mX-closure of A and the mX-interior of A with respect to mXi are denoted by mXiInt(A) and mXiCl(A), respectively, for i=1,2. Also i,j=1,2 and i≠j.Definition 7 (see [20]).
A subsetA of a biminimal structure space (X,mX1,mX2) is said to be  (i,j)-mX-α-open (resp.,  (i,j)-mX-semiopen, (i,j)-mX-preopen) if A⊆mXiInt(mXjCl(mXiInt(A))) (resp., A⊆mXiCl(mXjInt(A)), A⊆mXiInt(mXjCl(A))).Lemma 8.
Let(X,mX1,mX2) be a biminimal structure space and {Ak:k∈𝒦} a family of subsets of X.  (1)
IfAk is (i,j)-mX-α-open for each k∈𝒦, then ∪k∈𝒦Ak is (i,j)-mX-α-open.(2)
IfAk is (i,j)-mX-α-closed for each k∈𝒦, then ∩k∈𝒦Ak is (i,j)-mX-α-closed.Definition 9.
Let(X,mX1,mX2) be a biminimal structure space and A a subset of X. Then the mXij-α-closure of A and the  mXij-α-interior of A are defined as follows:  (1)
mXijCl𝒜(A)=∩{F:A⊆F,Fis(i,j)-mX-α-closed};(2)
mXijInt𝒜(A)=∪{U:U⊆A,Uis(i,j)-mX-α-open}.Lemma 10.
Let(X,mX1,mX2) be a biminimal structure space. For a subset A of X, the following properties hold:  (1)
mXijCl𝒜(A) is (i,j)-mX-α-closed;(2)
mXijInt𝒜(A) is (i,j)-mX-α-open;(3)
A is (i,j)-mX-α-closed if and only if mXijCl𝒜(A)=A;(4)
A is (i,j)-mX-α-open if and only if mXijInt𝒜(A)=A.Lemma 11.
Let(X,mX1,mX2) be a biminimal structure space and A a subset of X. Then x∈mXijCl𝒜(A) if and only if U∩A≠∅ for every (i,j)-mX-α-open set U containing x.Lemma 12.
Let(X,mX1,mX2) be a biminimal structure space and A a subset A of X(1)
X-mXijInt𝒜(A)=mXijCl𝒜(X-A);(2)
X-mXijCl𝒜(A)=mXijInt𝒜(X-A).Proof.
(1) By Lemma10, mXijCl𝒜(A) is (i,j)-mX-α-closed. Then X-mXijCl𝒜(A) is (i,j)-mX-α-open. On the other hand, X-mXijCl𝒜(X-A)⊆A, and hence X-mXijCl𝒜(X-A)⊆X-mXijInt𝒜(A). Conversely, let x∈mXijInt𝒜(A). Then there exists an (i,j)-mX-α-open U such that x∈U⊆A. Then X-U is (i,j)-mX-α-closed and X-A⊆X-U. Since x∉X-U,x∉mXijCl𝒜(X-A), and hence mXijInt𝒜(A)⊆X-mXijCl𝒜(X-A). Therefore, X-mXijInt𝒜(A)=mXijCl𝒜(X-A).
(2) This follows from (1) immediately.


## 3. Characterizations ofℳ𝒜(i,j)-Continuous Functions

Definition 13.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. A function f:(X,mX1,mX2)→(Y,τ1,τ2) is said to be  ℳ𝒜(i,j)-continuous at pointx∈X if for each τi-open set V of Y containing f(x), there exists an (i,j)-mX-α-open set U containing x such that f(U)⊆V.
A functionf:(X,mX1,mX2)→(Y,τ1,τ2) is said to be ℳ𝒜(i,j)-continuous if f has this property at each point of X.Theorem 14.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. For a function f:(X,mX1,mX2)→(Y,τ1,τ2), the following properties are equivalent:  (1)
f is ℳ𝒜(i,j)-continuous at x∈X;(2)
x∈mXijInt𝒜(f-1(V)) for every τi-open set V containing f(x);(3)
x∈f-1(f(A)¯) for every subset A of X with x∈mXijCl𝒜(A);(4)
x∈f-1(B¯) for every subset B of Y with x∈mXijCl𝒜(f-1(B));(5)
x∈mXijInt𝒜(f-1(B)) for every subset B of Y with x∈f-1(Int(B));(6)
x∈f-1(F) for every τi-closed set F of Y with x∈mXijCl𝒜(f-1(F)).Proof.
(
1
)
⇒
(
2
): Let V∈τi containing f(x). Then there exists an (i,j)-mX-α-open set containing x such that f(U)⊆V. Thus, x∈U⊆f-1(V). Hence, x∈mXijInt𝒜(f-1(V)).
(
2
)
⇒
(
3
): Let A be any subset of X such that x∈mXij𝒜(A), and let V∈τi containing f(x). Then x∈mXijInt𝒜(f-1(V)). There exists an (i,j)-mX-α-open set  U such that x∈U⊆f-1(V). Since x∈mXijCl𝒜(A), by Lemma 11, U∩A≠∅ and ∅≠f(U∩A)⊆f(U)∩f(A)⊆V∩f(A). Since V∈τi containing f(x), f(x)∈f(A)¯, and hence x∈f-1(f(A)¯).
(
3
)
⇒
(
4
): Let B be any subset of Y and x∈mXijCl𝒜(f-1(B)). Then by (3), x∈f-1(f(f-1(B))¯⊆f-1(B¯). Hence, we have x∈f-1(B¯).
(
4
)
⇒
(
5
): Let B be any subset of Y such that x∉mXijInt𝒜(f-1(B)). Then x∈X-mXijInt𝒜(f-1(B))=mXijCl𝒜(X-f-1(B))=mXijCl𝒜(f-1(Y-B)). By (4), we have x∈f-1(Y-B¯)=f-1(Y-Int(B))=X-f-1(Int(B)). Hence, x∉f-1(Int(B)).
(
5
)
⇒
(
6
): Let F be any τi-closed set of Y such that x∉f-1(F). Then x∈X-f-1(F)=f-1(Y-F)=f-1(Int(Y-F)). By (5), x∈mXijInt𝒜(f-1(Y-F))=mXijInt𝒜(X-f-1(F))=X-mXijCl𝒜(f-1(F)). Hence, x∉mXijCl𝒜(f-1(F)).
(
6
)
⇒
(
2
): Let x∈X and V∈τi containing f(x). Suppose that x∉mXijInt𝒜(f-1(V)). Then x∈X-mXijInt𝒜(f-1(V))=mXijCl𝒜(X-f-1(V))=mXijCl𝒜(f-1(Y-V)). By (6), x∈f-1(Y-V)=X-f-1(V). Hence, x∉f-1(V). This contradicts the hypothesis.
(
2
)
⇒
(
1
): Let V∈τi containing f(x). Then by (2), x∈mXijInt𝒜(f-1(V)), and hence there exists an (i,j)-mX-α-open set U containing x such that x∈U⊆f-1(V). Therefore, f(U)⊆V, and hence f is ℳ𝒜(i,j)-continuous at x.Theorem 15.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. For a function f:(X,mX1,mX2)→(Y,τ1,τ2), the following properties are equivalent:  (1)
f is ℳ𝒜(i,j)-continuous;(2)
f-1(V) is (i,j)-mX-α-open for every τi-open set V in Y;(3)
f(mXijCl𝒜(A))⊆f(A)¯ for every subset A of X;(4)
mXijCl𝒜(f-1(B))⊆f-1(B¯) for every subset B of Y;(5)
f-1(Int(B))⊆mXijInt𝒜(f-1(B)) for every subset B of Y;(6)
f-1(F) is (i,j)-mX-α-closed for every τi-closed set F of Y.Proof.
(
1
)
⇒
(
2
): Let V be any τi-open set of Y and x∈f-1(V). Then f(x)∈V. There exists an (i,j)-mX-α-open set U containing x such that f(U)⊆V, and hence x∈U⊆f-1(V). Hence, we have x∈mXijInt𝒜(f-1(V)). Therefore, f-1(V)⊆mXijInt𝒜(f-1(V)). This shows that f-1(V) is (i,j)-mX-α-open.
(
2
)
⇒
(
3
): Let A be any subset of X. Since Y-f(A)¯ is τi-open and by (2), we have f-1(Y-f(A)¯)=mXijInt𝒜(f-1(Y-f(A)¯)). Hence, X-f-1(f(A)¯)=mXijInt𝒜(X-f-1(f(A)¯)=X-mXijCl𝒜(f-1(f(A)¯)). Therefore, mXijCl𝒜(f-1(f(A)¯))=f-1(f(A)¯). Since A⊆f-1(f(A))⊆f-1(f(A)¯) and hence mXijCl𝒜(A)⊆mXijCl𝒜(f-1(f(A)¯))=f-1(f(A)¯). Therefore, f(mXijCl𝒜(A))⊆f(A)¯.
(
3
)
⇒
(
4
): Let B be any subset of Y. Then by (3), we have f(mXijCl𝒜(f-1(B)))⊆f(f-1(B))¯⊆B¯. Hence, mXijCl𝒜(f-1(B))⊆f-1(B¯).
(
4
)
⇒
(
5
): Let B be any subset of Y and x∈f-1(Int(B)). Then f(x)∈Int(B), and hence f(x)∉Y-Int(B)=(Y-B)¯. Therefore, x∉f-1((Y-B)¯). By (4), we have x∉mXijCl𝒜(f-1(Y-B)), and hence x∉X-mXijInt𝒜(f-1(B)). Therefore, x∈mXijInt𝒜(f-1(B)).
(
5
)
⇒
(
6
): Let F be any τi-closed set of Y such that x∉f-1(F). Then x∈X-f-1(F)=f-1(Y-F)=f-1(Int(Y-F)). By (5), x∈mXijInt𝒜(f-1(Y-F))=mXijInt𝒜(X-f-1(F))=X-mXijCl𝒜(f-1(F)). Therefore, x∉mXijCl𝒜(f-1(F)), and hence mXijCl𝒜(f-1(F))⊆f-1(F). This shows that f-1(F) is (i,j)-mX-α-closed.
(
6
)
⇒
(
1
): Let F be any τi-closed set of Y such that x∈mXijCl𝒜(f-1(F)). Then by (6), we have x∈f-1(F). Hence, by Theorem 14(6), f is ℳ𝒜(i,j)-continuous.Definition 16.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. A function f:(X,mX1,mX2)→(Y,τ1,τ2) is said to be(1)
ℳ𝒮(i,j)-continuous (resp.,  ℳ𝒮(j,i)-continuous) if f-1(V) is (i,j)-mX-semiopen (resp., (j,i)-mX-semiopen) in X for each τi-open set V of Y;(2)
ℳ𝒫(i,j)-continuous or  (i,j)-m-precontinuous [21] if f-1(V) is (i,j)-mX-preopen in X for each τi-open set V of Y.Lemma 17.
LetA be a subset of a biminimal structure space (X,mX1,mX2). Then A is (i,j)-mX-α-open in (X,mX1,mX2) if and only if A is (j,i)-mX-semiopen and (i,j)-mX-preopen in (X,mX1,mX2).Proof.
LetA be (i,j)-mX-α-open in (X,mX1,mX2). By the definition of (i,j)-mX-α-open sets, we have A⊆mXiInt(mXjCl(A)) and A⊆mXjCl(mXiInt(A)). Therefore, we obtain A is (j,i)-mX-semiopen and (i,j)-mX-preopen in (X,mX1,mX2).
Conversely, letA be (j,i)-mX-semiopen and (i,j)-mX-preopen in (X,mX1,mX2). Since A is (j,i)-mX-semiopen, A⊆mXjCl(mXiInt(A)), and hence it follows from A is (i,j)-mX-preopen that A⊆mXiInt(mXjCl(A))⊆mXiInt(mXjCl(mXjCl(mXiInt(A))))=mXiInt(mXjCl(mXiInt(A))). Therefore, A is (i,j)-mX-α-open in (X,mX1,mX2).Theorem 18.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. A function f:(X,mX1,mX2)→(Y,τ1,τ2) is ℳ𝒜(i,j)-continuous if and only if f is ℳ𝒫(i,j)-continuous and ℳ𝒮(j,i)-continuous.Proof.
This is an immediate consequence of Lemma17.Definition 19.
Let(X,mX1,mX2) be a biminimal structure space and (Y,τ1,τ2) a bitopological space. A function f:(X,mX1,mX2)→(Y,τ1,τ2) is said to be  ℳ𝒫⋆(i,j)-continuous if f is ℳ𝒫(i,j)-continuous and for every U,V∈τi such that [mXiInt(mXjCl(f-1(U)))]∩[mXiInt(mXjCl(f-1(V)))]⊆mXiInt(mXjCl(f-1(U∩V))).Theorem 20.
Let(X,mX1,mX2) be a biminimal structure space, and let mX1,mX2 have property ℬ, and let (Y,τ1,τ2) be a bitopological space. A function f:(X,mX1,mX2)→(Y,τ1,τ2) is ℳ𝒫⋆(i,j)-continuous if and only if for every U,V∈τi such that  (1)
f-1(V)⊆mXiInt(mXjCl(f-1(V))) and(2)
mXiInt(mXjCl(f-1(U∩V)))=mXiInt(mXjCl(f-1(U)))∩mXiInt(mXjCl(f-1(V))).Proof.
It is obvious thatf is ℳ𝒫(i,j)-continuous if and only if f satisfies (1). We assume that f is ℳ𝒫⋆(i,j)-continuous and show equality (2). For any U,V∈τi, it follows from (1) that f-1(U∩V)⊆mXiInt(mXjCl(f-1(U)))∩mXiInt(mXjCl(f-1(V))). Since the intersection of two (i,j)-mX-regular open sets is (i,j)-mX-regular open, we obtain mXiInt(mXjCl(f-1(U∩V)))⊆mXiInt(mXjCl(f-1(U)))∩mXiInt(mXjCl(f-1(V))). Hence, equality (2) holds.



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