On ℳ_𝒜^(𝑖,𝑗)-Continuous Functions in Biminimal Structure Spaces

Abstract

We introduce the notion of ℳ_𝒜^(𝑖,𝑗)-continuous functions and some other forms of continuity in biminimal structure spaces. Some new characterizations and several fundamental properties of ℳ_𝒜^(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 ℳ_𝒜^(𝑖,𝑗)-continuous functions in biminimal structure spaces and investigate the properties of these functions.

2. Preliminaries

Definition 1 (see [15]). Let 𝑋 be a nonempty set and 𝒫(𝑋) the power set of 𝑋. A subfamily 𝑚_𝑋 of 𝒫(𝑋) is called a minimal structure (briefly m-structure) on 𝑋 if ∅∈𝑚_𝑋 and 𝑋∈𝑚_𝑋.

By (𝑋,𝑚_𝑋), we denote a nonempty set 𝑋 with an 𝑚-structure 𝑚_𝑋 on 𝑋 and it is called an 𝑚-space. Each member of 𝑚_𝑋 is said to be 𝑚_𝑋-open, and the complement of an 𝑚_𝑋-open set is said to be 𝑚_𝑋-closed.

Definition 2 (see [16]). Let 𝑋 be a nonempty set and 𝑚_𝑋 an 𝑚-structure on 𝑋. For a subset 𝐴 of 𝑋, the 𝑚_𝑋-closure of 𝐴 and the 𝑚_𝑋-interior of 𝐴 are defined as follows: (1)𝑚_𝑋Cl(𝐴)=∩{𝐹∶𝐴⊆𝐹,𝑋−𝐹∈𝑚_𝑋}; (2)𝑚_𝑋Int(𝐴)=∪{𝑈∶𝑈⊆𝐴,𝑈∈𝑚_𝑋}.

Lemma 3 (see [16]). Let 𝑋 be a nonempty set and 𝑚_𝑋 a minimal structure on 𝑋. For subset 𝐴 and 𝐵 of 𝑋, the following properties hold: (1)𝑚_𝑋𝐶𝑙(𝑋−𝐴)=𝑋−𝑚_𝑋Int(𝐴) and 𝑚_𝑋Int(𝑋−𝐴)=𝑋−𝑚_𝑋𝐶𝑙(𝐴). (2)If (𝑋−𝐴)∈𝑚_𝑋, then 𝑚_𝑋𝐶𝑙(𝐴)=𝐴 and if 𝐴∈𝑚_𝑋, then 𝑚_𝑋Int(𝐴)=𝐴. (3)𝑚_𝑋𝐶𝑙(∅)=∅, 𝑚_𝑋𝐶𝑙(𝑋)=𝑋, 𝑚_𝑋Int(∅)=∅, and 𝑚_𝑋Int(𝑋)=𝑋. (4)If 𝐴⊆𝐵, then 𝑚_𝑋𝐶𝑙(𝐴)⊆𝑚_𝑋𝐶𝑙(𝐵) and 𝑚_𝑋Int(𝐴)⊆𝑚_𝑋Int(𝐵). (5)𝐴⊆𝑚_𝑋𝐶𝑙(𝐴) and 𝑚_𝑋Int(𝐴)⊆𝐴. (6)𝑚_𝑋𝐶𝑙(𝑚_𝑋𝐶𝑙(𝐴))=𝑚_𝑋𝐶𝑙(𝐴) and 𝑚_𝑋Int(𝑚_𝑋Int(𝐴))=𝑚_𝑋Int(𝐴).

Definition 4 (see [16]). An 𝑚-structure 𝑚_𝑋 on a nonempty set 𝑋 is said to have property ℬ if the union of any family of subsets belonging to 𝑚_𝑋 belongs to 𝑚_𝑋.

Lemma 5 (see [17]). Let 𝑋 be a nonempty set and 𝑚_𝑋 an 𝑚-structure on 𝑋 satisfying property ℬ. For a subset 𝐴 of 𝑋, the following properties hold:(1)𝐴∈𝑚_𝑋 if and only if 𝑚_𝑋Int(𝐴)=𝐴. (2)𝐴 is 𝑚_𝑋-closed if and only if 𝑚_𝑋𝐶𝑙(𝐴)=𝐴. (3)𝑚_𝑋Int(𝐴)∈𝑚_𝑋 and 𝑚_𝑋𝐶𝑙(𝐴) is 𝑚_𝑋-closed.

Definition 6. Let 𝑋 be a nonempty set and 𝑚¹_𝑋,𝑚²_𝑋 minimal structures on 𝑋. The triple (𝑋,𝑚¹_𝑋,𝑚²_𝑋) is called a bispace (briefly bi m-space) [18] or biminimal structure space (briefly bimspace) [19].

Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and 𝐴 a subset of 𝑋. The 𝑚_𝑋-closure of 𝐴 and the 𝑚_𝑋-interior of 𝐴 with respect to 𝑚^𝑖_𝑋 are denoted by 𝑚^𝑖_𝑋Int(𝐴) and 𝑚^𝑖_𝑋𝐶𝑙(𝐴), respectively, for 𝑖=1,2. Also 𝑖,𝑗=1,2 and 𝑖≠𝑗.

Definition 7 (see [20]). A subset 𝐴 of a biminimal structure space (𝑋,𝑚¹_𝑋,𝑚²_𝑋) is said to be (𝑖,𝑗)-𝑚_𝑋-𝛼-open (resp., (𝑖,𝑗)-𝑚_𝑋-semiopen, (𝑖,𝑗)-𝑚_𝑋-preopen) if 𝐴⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋Cl(𝑚^𝑖_𝑋Int(𝐴))) (resp., 𝐴⊆𝑚^𝑖_𝑋𝐶𝑙(𝑚^𝑗_𝑋Int(𝐴)), 𝐴⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋Cl(𝐴))).

Lemma 8. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and {𝐴_𝑘∶𝑘∈𝒦} a family of subsets of 𝑋. (1)If 𝐴_𝑘 is (𝑖,𝑗)-𝑚_𝑋-𝛼-open for each 𝑘∈𝒦, then ∪_𝑘∈𝒦𝐴_𝑘 is (𝑖,𝑗)-𝑚_𝑋-𝛼-open. (2)If 𝐴_𝑘 is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed for each 𝑘∈𝒦, then ∩_𝑘∈𝒦𝐴_𝑘 is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed.

Definition 9. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and 𝐴 a subset of 𝑋. Then the 𝑚_𝑋^𝑖𝑗-α-closure of 𝐴 and the 𝑚_𝑋^𝑖𝑗-α-interior of 𝐴 are defined as follows: (1)𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴)=∩{𝐹∶𝐴⊆𝐹,𝐹is(𝑖,𝑗)-𝑚_𝑋-𝛼-closed}; (2)𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴)=∪{𝑈∶𝑈⊆𝐴,𝑈is(𝑖,𝑗)-𝑚_𝑋-𝛼-open}.

Lemma 10. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space. For a subset 𝐴 of 𝑋, the following properties hold: (1)𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴) is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed; (2)𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴) is (𝑖,𝑗)-m_𝑋-𝛼-open; (3)𝐴 is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed if and only if 𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴)=𝐴; (4)𝐴 is (𝑖,𝑗)-𝑚_𝑋-𝛼-open if and only if 𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴)=𝐴.

Lemma 11. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and 𝐴 a subset of 𝑋. Then 𝑥∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴) if and only if 𝑈∩𝐴≠∅ for every (𝑖,𝑗)-𝑚_𝑋-𝛼-open set 𝑈 containing 𝑥.

Lemma 12. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and 𝐴 a subset A of 𝑋(1)𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴)=𝑚_𝑋^𝑖𝑗Cl_𝒜(𝑋−𝐴); (2)𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴)=𝑚_𝑋^𝑖𝑗Int_𝒜(𝑋−𝐴).

Proof. (1) By Lemma 10, 𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴) is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed. Then 𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴) is (𝑖,𝑗)-𝑚_𝑋-𝛼-open. On the other hand, 𝑋−𝑚_𝑋^𝑖𝑗Cl_𝒜(𝑋−𝐴)⊆𝐴, and hence 𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑋−𝐴)⊆𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴). Conversely, let 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴). Then there exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open 𝑈 such that 𝑥∈𝑈⊆𝐴. Then 𝑋−𝑈 is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed and 𝑋−𝐴⊆𝑋−𝑈. Since 𝑥∉𝑋−𝑈,𝑥∉𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑋−𝐴), and hence 𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴)⊆𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑋−𝐴). Therefore, 𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝐴)=𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑋−𝐴).
(2) This follows from (1) immediately.

3. Characterizations of ℳ_𝒜^(𝑖,𝑗)-Continuous Functions

Definition 13. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is said to be ℳ_𝒜^(𝑖,𝑗)-continuous at point 𝑥∈𝑋 if for each 𝜏_𝑖-open set 𝑉 of 𝑌 containing 𝑓(𝑥), there exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open set 𝑈 containing 𝑥 such that 𝑓(𝑈)⊆𝑉.
A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is said to be ℳ_𝒜^(𝑖,𝑗)-continuous if 𝑓 has this property at each point of 𝑋.

Theorem 14. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. For a function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂), the following properties are equivalent: (1)𝑓 is ℳ_𝒜^(𝑖,𝑗)-continuous at 𝑥∈𝑋; (2)𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)) for every 𝜏_𝑖-open set 𝑉 containing 𝑓(𝑥); (3)𝑥∈𝑓⁻¹(𝑓(𝐴)) for every subset 𝐴 of 𝑋 with 𝑥∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴); (4)𝑥∈𝑓⁻¹(𝐵) for every subset 𝐵 of 𝑌 with 𝑥∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐵)); (5)𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵)) for every subset 𝐵 of 𝑌 with 𝑥∈𝑓⁻¹(Int(𝐵)); (6)𝑥∈𝑓⁻¹(𝐹) for every 𝜏_𝑖-closed set 𝐹 of 𝑌 with 𝑥∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐹)).

Proof. (1)⇒(2): Let 𝑉∈𝜏_𝑖 containing 𝑓(𝑥). Then there exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open set containing 𝑥 such that 𝑓(𝑈)⊆𝑉. Thus, 𝑥∈𝑈⊆𝑓⁻¹(𝑉). Hence, 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)).
(2)⇒(3): Let 𝐴 be any subset of 𝑋 such that 𝑥∈𝑚_𝑋^𝑖𝑗𝒜(𝐴), and let 𝑉∈𝜏_𝑖 containing 𝑓(𝑥). Then 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)). There exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open set 𝑈 such that 𝑥∈𝑈⊆𝑓⁻¹(𝑉). Since 𝑥∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴), by Lemma 11, 𝑈∩𝐴≠∅ and ∅≠𝑓(𝑈∩𝐴)⊆𝑓(𝑈)∩𝑓(𝐴)⊆𝑉∩𝑓(𝐴). Since 𝑉∈𝜏_𝑖 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∈𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(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∉𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵)). Then 𝑥∈𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵))=𝑚_𝑋^𝑖𝑗Cl_𝒜(𝑋−𝑓⁻¹(𝐵))=𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(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 𝜏_𝑖-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∈𝑚^𝑖𝑗_XInt_𝒜(𝑓⁻¹(𝑌−𝐹))=𝑚_𝑋^𝑖𝑗Int_𝒜(𝑋−𝑓⁻¹(𝐹))=𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐹)). Hence, 𝑥∉𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐹)).
(6)⇒(2): Let 𝑥∈𝑋 and 𝑉∈𝜏_𝑖 containing 𝑓(𝑥). Suppose that 𝑥∉𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)). Then 𝑥∈𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉))=𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑋−𝑓⁻¹(𝑉))=𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(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∈𝜏_𝑖 containing 𝑓(𝑥). Then by (2), 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)), and hence there exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open set 𝑈 containing 𝑥 such that 𝑥∈𝑈⊆𝑓⁻¹(𝑉). Therefore, 𝑓(𝑈)⊆𝑉, and hence 𝑓 is ℳ_𝒜^(𝑖,𝑗)-continuous at 𝑥.

Theorem 15. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. For a function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂), the following properties are equivalent: (1)𝑓 is ℳ_𝒜^(𝑖,𝑗)-continuous; (2)𝑓⁻¹(𝑉) is (𝑖,𝑗)-𝑚_𝑋-𝛼-open for every 𝜏_𝑖-open set 𝑉 in 𝑌; (3)𝑓(𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴))⊆𝑓(𝐴) for every subset 𝐴 of 𝑋; (4)𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐵))⊆𝑓⁻¹(𝐵) for every subset 𝐵 of 𝑌; (5)𝑓⁻¹(Int(𝐵))⊆𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵)) for every subset 𝐵 of 𝑌;(6)𝑓⁻¹(𝐹) is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed for every 𝜏_𝑖-closed set 𝐹 of 𝑌.

Proof. (1)⇒(2): Let 𝑉 be any 𝜏_𝑖-open set of 𝑌 and 𝑥∈𝑓⁻¹(𝑉). Then 𝑓(𝑥)∈𝑉. There exists an (𝑖,𝑗)-𝑚_𝑋-𝛼-open set 𝑈 containing 𝑥 such that 𝑓(𝑈)⊆𝑉, and hence 𝑥∈𝑈⊆𝑓⁻¹(𝑉). Hence, we have 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)). Therefore, 𝑓⁻¹(𝑉)⊆𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑉)). This shows that 𝑓⁻¹(𝑉) is (𝑖,𝑗)-𝑚_𝑋-𝛼-open.
(2)⇒(3): Let 𝐴 be any subset of 𝑋. Since 𝑌−𝑓(𝐴) is 𝜏_𝑖-open and by (2), we have 𝑓⁻¹(𝑌−𝑓(𝐴))=𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑌−𝑓(𝐴))). Hence, 𝑋−𝑓⁻¹(𝑓(𝐴))=𝑚_𝑋^𝑖𝑗Int_𝒜(𝑋−𝑓⁻¹(𝑓(𝐴))=𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝑓(𝐴))). Therefore, 𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝑓(𝐴)))=𝑓⁻¹(𝑓(𝐴)). Since 𝐴⊆𝑓⁻¹(𝑓(𝐴))⊆𝑓⁻¹(𝑓(𝐴)) and hence 𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴)⊆𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝑓(𝐴)))=𝑓⁻¹(𝑓(𝐴)). Therefore, 𝑓(𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝐴))⊆𝑓(𝐴).
(3)⇒(4): Let 𝐵 be any subset of 𝑌. Then by (3), we have 𝑓(𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐵)))⊆𝑓(𝑓⁻¹(𝐵))⊆𝐵. Hence, 𝑚_𝑋^𝑖𝑗Cl_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∉𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝑌−𝐵)), and hence 𝑥∉𝑋−𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵)). Therefore, 𝑥∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝐵)).
(5)⇒(6): Let 𝐹 be any 𝜏_𝑖-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∈𝑚_𝑋^𝑖𝑗Int_𝒜(𝑓⁻¹(𝑌−𝐹))=𝑚_𝑋^𝑖𝑗Int_𝒜(𝑋−𝑓⁻¹(𝐹))=𝑋−𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐹)). Therefore, 𝑥∉𝑚_𝑋^𝑖𝑗Cl_𝒜(𝑓⁻¹(𝐹)), and hence 𝑚_𝑋^𝑖𝑗𝐶𝑙_𝒜(𝑓⁻¹(𝐹))⊆𝑓⁻¹(𝐹). This shows that 𝑓⁻¹(𝐹) is (𝑖,𝑗)-𝑚_𝑋-𝛼-closed.
(6)⇒(1): Let 𝐹 be any 𝜏_𝑖-closed set of 𝑌 such that 𝑥∈𝑚_𝑋^𝑖𝑗Cl_𝒜(𝑓⁻¹(𝐹)). Then by (6), we have 𝑥∈𝑓⁻¹(𝐹). Hence, by Theorem 14(6), 𝑓 is ℳ_𝒜^(𝑖,𝑗)-continuous.

Definition 16. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is said to be(1)ℳ_𝒮^(𝑖,𝑗)-continuous (resp., ℳ_𝒮^(𝑗,𝑖)-continuous) if 𝑓⁻¹(𝑉) is (𝑖,𝑗)-𝑚_𝑋-semiopen (resp., (𝑗,𝑖)-𝑚_𝑋-semiopen) in 𝑋 for each 𝜏_𝑖-open set 𝑉 of 𝑌; (2)ℳ_𝒫^(𝑖,𝑗)-continuous or (𝑖,𝑗)-𝑚-precontinuous [21] if 𝑓⁻¹(𝑉) is (𝑖,𝑗)-𝑚_𝑋-preopen in 𝑋 for each 𝜏_𝑖-open set 𝑉 of 𝑌.

Lemma 17. Let 𝐴 be a subset of a biminimal structure space (𝑋,𝑚¹_𝑋,𝑚²_𝑋). Then 𝐴 is (𝑖,𝑗)-𝑚_𝑋-𝛼-open in (𝑋,𝑚¹_𝑋,𝑚²_𝑋) if and only if 𝐴 is (𝑗,𝑖)-𝑚_𝑋-semiopen and (𝑖,𝑗)-𝑚_𝑋-preopen in (𝑋,𝑚¹_𝑋,𝑚²_𝑋).

Proof. Let 𝐴 be (𝑖,𝑗)-𝑚_𝑋-𝛼-open in (𝑋,𝑚¹_𝑋,𝑚²_𝑋). By the definition of (𝑖,𝑗)-𝑚_𝑋-𝛼-open sets, we have 𝐴⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝐴)) and 𝐴⊆𝑚^𝑗_𝑋𝐶𝑙(𝑚^𝑖_𝑋Int(𝐴)). Therefore, we obtain 𝐴 is (𝑗,𝑖)-𝑚_𝑋-semiopen and (𝑖,𝑗)-𝑚_𝑋-preopen in (𝑋,𝑚¹_𝑋,𝑚²_𝑋).
Conversely, let 𝐴 be (𝑗,𝑖)-𝑚_𝑋-semiopen and (𝑖,𝑗)-𝑚_𝑋-preopen in (𝑋,𝑚¹_𝑋,𝑚²_𝑋). Since 𝐴 is (𝑗,𝑖)-𝑚_𝑋-semiopen, 𝐴⊆𝑚^𝑗_𝑋𝐶𝑙(𝑚^𝑖_𝑋Int(𝐴)), and hence it follows from 𝐴 is (𝑖,𝑗)-𝑚_𝑋-preopen that 𝐴⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝐴))⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑚^𝑗_𝑋𝐶𝑙(𝑚^𝑖_𝑋Int(𝐴))))=𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑚^𝑖_𝑋Int(𝐴))). Therefore, 𝐴 is (𝑖,𝑗)-𝑚_𝑋-𝛼-open in (𝑋,𝑚¹_𝑋,𝑚²_𝑋).

Theorem 18. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is ℳ_𝒜^(𝑖,𝑗)-continuous if and only if 𝑓 is ℳ_𝒫^(𝑖,𝑗)-continuous and ℳ_𝒮^(𝑗,𝑖)-continuous.

Proof. This is an immediate consequence of Lemma 17.

Definition 19. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space and (𝑌,𝜏₁,𝜏₂) a bitopological space. A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is said to be ℳ_𝒫^(𝑖,𝑗)_⋆-continuous if 𝑓 is ℳ_𝒫^(𝑖,𝑗)-continuous and for every 𝑈,𝑉∈𝜏_𝑖 such that [𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈)))]∩[𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑉)))]⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈∩𝑉))).

Theorem 20. Let (𝑋,𝑚¹_𝑋,𝑚²_𝑋) be a biminimal structure space, and let 𝑚¹_𝑋,𝑚²_𝑋 have property ℬ, and let (𝑌,𝜏₁,𝜏₂) be a bitopological space. A function 𝑓∶(𝑋,𝑚¹_𝑋,𝑚²_𝑋)→(𝑌,𝜏₁,𝜏₂) is ℳ_𝒫^(𝑖,𝑗)_⋆-continuous if and only if for every 𝑈,𝑉∈𝜏_𝑖 such that (1)𝑓⁻¹(𝑉)⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑉))) and (2)𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈∩𝑉)))=𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈)))∩𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑉))).

Proof. It is obvious that 𝑓 is ℳ_𝒫^(𝑖,𝑗)-continuous if and only if 𝑓 satisfies (1). We assume that 𝑓 is ℳ_𝒫^(𝑖,𝑗)_⋆-continuous and show equality (2). For any 𝑈,𝑉∈𝜏_𝑖, it follows from (1) that 𝑓⁻¹(𝑈∩𝑉)⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈)))∩𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑉))). Since the intersection of two (𝑖,𝑗)-𝑚_𝑋-regular open sets is (𝑖,𝑗)-𝑚_𝑋-regular open, we obtain 𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈∩𝑉)))⊆𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑈)))∩𝑚^𝑖_𝑋Int(𝑚^𝑗_𝑋𝐶𝑙(𝑓⁻¹(𝑉))). Hence, equality (2) holds.

Acknowledgment

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