We introduce the notion of ℳ𝒜(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.

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. PreliminariesDefinition 1 (see [<xref ref-type="bibr" rid="B19">15</xref>]).

Let X 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 [<xref ref-type="bibr" rid="B11">16</xref>]).

Let X 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:

mXCl(A)=∩{F:A⊆F,X-F∈mX};

mXInt(A)=∪{U:U⊆A,U∈mX}.

Lemma 3 (see [<xref ref-type="bibr" rid="B11">16</xref>]).

Let X be a nonempty set and mX a minimal structure on X. For subset A and B of X, the following properties hold:

mXCl(X-A)=X-mX
Int
(A) and mX
Int
(X-A)=X-mXCl(A).

If (X-A)∈mX, then mXCl(A)=A and if A∈mX, then mX
Int
(A)=A.

mXCl(∅)=∅, mXCl(X)=X, mX
Int
(∅)=∅, and mX
Int
(X)=X.

If A⊆B, then mXCl(A)⊆mXCl(B) and mX
Int
(A)⊆mX
Int
(B).

A⊆mXCl(A) and mX
Int
(A)⊆A.

mXCl(mXCl(A))=mXCl(A) and mX
Int
(mX
Int
(A))=mX
Int
(A).

Definition 4 (see [<xref ref-type="bibr" rid="B11">16</xref>]).

An m-structure mX on a nonempty set X is said to have property ℬ if the union of any family of subsets belonging to mX belongs to mX.

Lemma 5 (see [<xref ref-type="bibr" rid="B18">17</xref>]).

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

A∈mX if and only if mX
Int
(A)=A.

A is mX-closed if and only if mXCl(A)=A.

mX
Int
(A)∈mX and mXCl(A) is mX-closed.

Definition 6.

Let X be a nonempty set and mX1,mX2 minimal structures on X. The triple (X,mX1,mX2) is called a bispace (briefly bi m-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 mXi
Int
(A) and mXiCl(A), respectively, for i=1,2. Also i,j=1,2 and i≠j.

Definition 7 (see [<xref ref-type="bibr" rid="B3">20</xref>]).

A subset A 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⊆mXi
Int
(mXjCl(mXiInt(A))) (resp., A⊆mXiCl(mXj
Int
(A)), A⊆mXi
Int
(mXjCl(A))).

Lemma 8.

Let (X,mX1,mX2) be a biminimal structure space and {Ak:k∈𝒦} a family of subsets of X.

If Ak is (i,j)-mX-α-open for each k∈𝒦, then ∪k∈𝒦Ak is (i,j)-mX-α-open.

If Ak 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:

mXijCl𝒜(A)=∩{F:A⊆F,Fis(i,j)-mX-α-closed};

mXij
Int
𝒜(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:

mXijCl𝒜(A) is (i,j)-mX-α-closed;

mXij
Int
𝒜(A) is (i,j)-mX-α-open;

A is (i,j)-mX-α-closed if and only if mXijCl𝒜(A)=A;

A is (i,j)-mX-α-open if and only if mXij
Int
𝒜(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

X-mXij
Int
𝒜(A)=mXijCl𝒜(X-A);

X-mXijCl𝒜(A)=mXij
Int
𝒜(X-A).

Proof.

(1) By Lemma 10, 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-mXij
Int
𝒜(A). Conversely, let x∈mXij
Int
𝒜(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 mXij
Int
𝒜(A)⊆X-mXijCl𝒜(X-A). Therefore, X-mXij
Int
𝒜(A)=mXijCl𝒜(X-A).

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 point x∈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 function f:(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:

f is ℳ𝒜(i,j)-continuous at x∈X;

x∈mXij
Int
𝒜(f-1(V)) for every τi-open set V containing f(x);

x∈f-1(f(A)¯) for every subset A of X with x∈mXijCl𝒜(A);

x∈f-1(B¯) for every subset B of Y with x∈mXijCl𝒜(f-1(B));

x∈mXij
Int
𝒜(f-1(B)) for every subset B of Y with x∈f-1(
Int
(B));

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∈mXij
Int
𝒜(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∈mXij
Int
𝒜(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∉mXij
Int
𝒜(f-1(B)). Then x∈X-mXij
Int
𝒜(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∈mXij
Int
𝒜(f-1(Y-F))=mXij
Int
𝒜(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∉mXij
Int
𝒜(f-1(V)). Then x∈X-mXij
Int
𝒜(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∈mXij
Int
𝒜(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:

f is ℳ𝒜(i,j)-continuous;

f-1(V) is (i,j)-mX-α-open for every τi-open set V in Y;

f(mXijCl𝒜(A))⊆f(A)¯ for every subset A of X;

mXijCl𝒜(f-1(B))⊆f-1(B¯) for every subset B of Y;

f-1(
Int
(B))⊆mXij
Int
𝒜(f-1(B)) for every subset B of Y;

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∈mXij
Int
𝒜(f-1(V)). Therefore, f-1(V)⊆mXij
Int
𝒜(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)¯)=mXij
Int
𝒜(f-1(Y-f(A)¯)). Hence, X-f-1(f(A)¯)=mXij
Int
𝒜(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, mXij
Cl
𝒜(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-mXij
Int
𝒜(f-1(B)). Therefore, x∈mXij
Int
𝒜(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∈mXij
Int
𝒜(f-1(Y-F))=mXij
Int
𝒜(X-f-1(F))=X-mXijCl𝒜(f-1(F)). Therefore, x∉mXij
Cl
𝒜(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∈mXij
Cl
𝒜(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

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;

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.

Let A 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.

Let A 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, let A 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 Lemma 17.

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)))]∩[mXi
Int
(mXjCl(f-1(V)))]⊆mXi
Int
(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

f-1(V)⊆mXi
Int
(mXjCl(f-1(V))) and

mXi
Int
(mXjCl(f-1(U∩V)))=mXi
Int
(mXjCl(f-1(U)))∩mXi
Int
(mXjCl(f-1(V))).

Proof.

It is obvious that f 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.

Acknowledgment

This research was financially supported by the Faculty of Science, Mahasarakham University.

LevineN.A decomposition of continuity in topological spacesRoseD. A.Weak continuity and almost continuityBakerC. W.Properties of subweakly continuous functionsNoiriT.On weakly continuous mappingsNjåstadO.On some classes of nearly open setsNoiriT.Weakly α-continuous functionsMashhourA. S.Abd El-MonsefM. E.El-DeebS. N.On precontinuous and week precontinuous mappingsMashhourA. S.HasaneinI. A.El-DeebS. N.α-continuous and α-open mappingsSingalM. K.SingalA. R.Almost-continuous mappingsMaheshwariS. N.ThakurS. S.On α-irresolute mappingsLevineN.Semi-open sets and semi-continuity in topological spacesMaheshwariS. N.PrasadR.Semi-open sets and semicontinuous functions in bitopological spacesBoseS.Semi-open sets, semicontinuity and semi-open mappings in bitopological spacesJelićM.A decomposition of pairwise continuityPopaV.NoiriT.On M-continuous functions,MakiH.Chandrasekhara RaoK.Nagoor GaniA.On generalizing semi-open sets and preopen setsPopaV.NoiriT.A unified theory of weak continuity for functionsNoiriT.The further unified theory for modifications of g-closed setsBoonpokC.Biminimal structure spacesBoonpokC.M-Continuous functions on biminimal structure spacesCarpinteroC.RajeshN.RosasE.m-preopen sets in biminimal spaces