EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6568
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Almost τ ⋆(σ1, σ2)-Continuity and Weak
τ ⋆(σ1, σ2)-Continuity

Nongluk Viriyapong1, Areeyuth Sama-Ae2, Chawalit Boonpok1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand
2 Department of Mathematics and Computer Science, Faculty of Science and Technology,
Prince of Songkla University, Pattani Campus, Pattani, 94000, Thailand

Abstract. This paper is concerned with the concepts of almost τ⋆(σ1, σ2)-continuous functions
and weakly τ⋆(σ1, σ2)-continuous functions. Moreover, some characterizations of almost τ⋆(σ1, σ2)-
continuous functions and weakly τ⋆(σ1, σ2)-continuous functions are investigated. Furthermore, the
relationships between almost τ⋆(σ1, σ2)-continuity and weak τ⋆(σ1, σ2)-continuity are considered.

2020 Mathematics Subject Classifications: 54C05, 54C08

Key Words and Phrases: Almost τ⋆(σ1, σ2)-continuous function, weakly τ⋆(σ1, σ2)-continuous
function

1. Introduction

In 1968, Singal and Singal [1] introduced the concept of almost continuous functions
as a generalization of continuity. Munshi and Bassan [2] studied the notion of almost
semi-continuous functions. Noiri [3] introduced and investigated the concept of almost
α-continuous functions. Nasef and Noiri [4] introduced two classes of functions, namely
almost precontinuous functions and almost β-continuous functions. The class of almost
precontinuity is a generalization of almost α-continuity. The class of almost β-continuity
is a generalization of almost semi-continuity. Levine [5] introduced and investigated the
concept of weakly continuous functions. Husain [6] introduced and studied the notion of
almost continuous functions. Janković [7] introduced almost weak continuity as a gen-
eralization of both weak continuity and almost continuity. Noiri [8] investigated several
characterizations of almost weakly continuous functions. Rose [9] introduced the notion
of subweakly continuous functions and investigated the relationships between subweak
continuity and weak continuity. Popa and Noiri [10] introduced the concept of weakly

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6568

Email addresses: nongluk.h@msu.ac.th (N. Viriyapong), areeyuth.s@psu.ac.th (A. Sama-Ae),
chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 2 of 18

(τ,m)-continuous functions as functions from a topological space into a set satisfying some
minimal conditions and investigated several characterizations of weakly (τ,m)-continuous
functions. Ekici et al. [11] introduced and studied the concept of weakly λ-continuous
functions. In 1992, Abd El-Monsef et al. [12] introduced and studied the notions of
I -closed sets and I -continuous functions. Semi-I -open sets, pre-I -open sets, α-I -
open sets, β-I -open sets and δ-I -open sets play an important role in the research of
generalizations of continuity. Using these notions many authors introduced and studied
various types of generalizations of continuity for functions and multifunctions. Hatir and
Noiri [13] introduced and investigated the notions of weakly pre-I -open sets and weakly
pre-I -continuous functions. Moreover, Hatir and Noiri [14] investigated further prop-
erties of semi-I -open sets and semi-I -continuous functions [15]. On the other hand,
the present author introduced and studied the concepts of ⋆-continuous functions [16],
θ-I -continuous functions [17], weakly ⋆-continuous functions [18], θ(⋆)-continuous func-
tions [18], almost ⋆-precontinuous functions [19], weakly ⋆-precontinuous functions [19],
pı-continuous functions [20] and weakly pı-continuous functions [20]. Recently, Boonpok
and Srisarakham [21] introduced and investigated the notion of (τ1, τ2)-continuous func-
tions. Furthermore, some characterizations of almost (τ1, τ2)-continuous functions and
weakly (τ1, τ2)-continuous functions were established in [22] and [23], respectively. In this
paper, we introduce the concepts of almost τ⋆(σ1, σ2)-continuous functions and weakly
τ⋆(σ1, σ2)-continuous functions. We also investigate several characterizations of almost
τ⋆(σ1, σ2)-continuous functions and weakly τ⋆(σ1, σ2)-continuous functions.

2. Preliminaries

Throughout the present paper, spaces (X, τ1, τ2) and (Y, σ1, σ2) (or simply X and
Y ) always mean bitopological spaces on which no separation axioms are assumed unless
explicitly stated. Let A be a subset of a bitopological space (X, τ1, τ2). The closure of A
and the interior of A with respect to τi are denoted by τi-Cl(A) and τi-Int(A), respectively,
for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed [24]
if A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. The
intersection of all τ1τ2-closed sets of X containing A is called the τ1τ2-closure [24] of A
and is denoted by τ1τ2-Cl(A). The union of all τ1τ2-open sets of X contained in A is called
the τ1τ2-interior [24] of A and is denoted by τ1τ2-Int(A).

Lemma 1. [24] Let A and B be subsets of a bitopological space (X, τ1, τ2). For the τ1τ2-
closure, the following properties hold:

(1) A ⊆ τ1τ2-Cl(A) and τ1τ2-Cl(τ1τ2-Cl(A)) = τ1τ2-Cl(A).

(2) If A ⊆ B, then τ1τ2-Cl(A) ⊆ τ1τ2-Cl(B).

(3) τ1τ2-Cl(A) is τ1τ2-closed.

(4) A is τ1τ2-closed if and only if A = τ1τ2-Cl(A).



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 3 of 18

(5) τ1τ2-Cl(X −A) = X − τ1τ2-Int(A).

A subset A of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)r-open [25] (resp.
(τ1, τ2)s-open [26], (τ1, τ2)p-open [26], (τ1, τ2)β-open [26]) if A = τ1τ2-Int(τ1τ2-Cl(A))
(resp. A ⊆ τ1τ2-Cl(τ1τ2-Int(A)), A ⊆ τ1τ2-Int(τ1τ2-Cl(A)), A ⊆ τ1τ2-Cl(τ1τ2-Int(τ1τ2-Cl(A)))).
The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-open, (τ1, τ2)p-open, (τ1, τ2)β-open)
set is said to be (τ1, τ2)r-closed (resp. (τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed).
A subset A of a bitopological space (X, τ1, τ2) is said to be α(τ1, τ2)-open [27] if A ⊆
τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A))). The complement of an α(τ1, τ2)-open set is said to be
α(τ1, τ2)-closed. A subset A of a bitopological space (X, τ1, τ2) is said to be τ1τ2-δ-open
[16] if A is the union of (τ1, τ2)r-open sets of X. The complement of a τ1τ2-δ-open set is
called τ1τ2-δ-closed [16]. The union of all τ1τ2-δ-open sets of X contained in A is called
the τ1τ2-δ-interior [16] of A and is denoted by τ1τ2-δ-Int(A). The intersection of all τ1τ2-
δ-closed sets of X containing A is called the τ1τ2-δ-closure [16] of A and is denoted by
τ1τ2-δ-Cl(A). Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is
called a (τ1, τ2)θ-cluster point [25] of A if τ1τ2-Cl(U) ∩ A ̸= ∅ for every τ1τ2-open set U
containing x. The set of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure [25]
of A and is denoted by (τ1, τ2)θ-Cl(A). A subset A of a bitopological space (X, τ1, τ2) is
said to be (τ1, τ2)θ-closed [25] if (τ1, τ2)θ-Cl(A) = A. The complement of a (τ1, τ2)θ-closed
set is said to be (τ1, τ2)θ-open. The union of all (τ1, τ2)θ-open sets of X contained in A is
called the (τ1, τ2)θ-interior [25] of A and is denoted by (τ1, τ2)θ-Int(A).

An ideal I on a topological space (X, τ) is a nonempty collection of subsets of X
satisfying the following properties: (1) A ∈ I and B ⊆ A imply B ∈ I ; (2) A ∈ I and
B ∈ I imply A ∪ B ∈ I . A topological space (X, τ) with an ideal I on X is called
an ideal topological space and is denoted by (X, τ,I ). For an ideal topological space
(X, τ,I ) and a subset A of X, A⋆(I ) is defined as follows:

A⋆(I ) = {x ∈ X : U ∩A ̸∈ I for every open neighbourhood U of x}.

In case there is no chance for confusion, A⋆(I ) is simply written as A⋆. In [28], A⋆ is called
the local function of A with respect to I and τ and Cl⋆(A) = A⋆∪A defines a Kuratowski
closure operator for a topology τ⋆(I ) finer than τ . A subset A is said to be ⋆-closed [29]
if A⋆ ⊆ A. The interior of a subset A in (X, τ⋆(I )) is denoted by Int⋆(A). A subset A of
an ideal topological space (X, τ,I ) is said to be semi⋆-I -open [30] (resp. semi-I -open
[15]) if A ⊆ Cl(Int⋆(A)) (resp. A ⊆ Cl⋆(Int(A))). The complement of a semi⋆-I -open
(resp. semi-I -open) set is said to be semi⋆-I -closed [30] (resp. semi-I -closed [15]).

3. On almost τ ⋆(σ1, σ2)-continuous functions

In this section, we introduce the concept of almost τ⋆(σ1, σ2)-continuous functions.
Moreover, some characterizations of almost τ⋆(σ1, σ2)-continuous functions are discussed.

Definition 1. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost τ⋆(σ1, σ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 4 of 18

exists a ⋆-open set U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A
function f : (X, τ,I ) → (Y, σ1, σ2) is said to be almost τ⋆(σ1, σ2)-continuous if f is
almost τ⋆(σ1, σ2)-continuous at each point x of X.

Lemma 2. [31] Let A be a subset of a bitopological space (X, τ1, τ2). If A is τ1τ2-open in
X, then (τ1, τ2)-sCl(A) = τ1τ2-Int(τ1τ2-Cl(A)).

Theorem 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous at x ∈ X;

(2) x ∈ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing
f(x);

(3) x ∈ Int⋆(f−1((σ1, σ2)-sCl(V ))) for every σ1σ2-open set V of Y containing f(x);

(4) x ∈ Int⋆(f−1(V )) for every (σ1, σ2)r-open set V of Y containing f(x);

(5) for each (σ1, σ2)r-open set V of Y containing f(x), there exists a ⋆-open set U of X
containing x such that f(U) ⊆ V .

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing f(x). Thus by (1), there
exists a ⋆-open set U ofX containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Therefore,
x ∈ U ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V ))) and hence x ∈ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))).

(2) ⇒ (3): This follows from Lemma 2.
(3) ⇒ (4): Let V be any (σ1, σ2)r-open set of Y containing f(x). It follows from

Lemma 2 that V = σ1σ2-Int(σ1σ2-Cl(V )) = (σ1, σ2)-sCl(V ).
(4) ⇒ (5): Let V be any (σ1, σ2)r-open set of Y containing f(x). Then by (4), we have

x ∈ Int⋆(f−1(V )) and there exists a ⋆-open set U of X containing x such that U ⊆ f−1(V );
hence f(U) ⊆ V .

(5) ⇒ (1): Let V be any σ1σ2-open set of Y containing f(x). Since σ1σ2-Int(σ1σ2-Cl(V ))
is (σ1, σ2)r-open, there exists a ⋆-open set U of X containing x such that

f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )).

This shows that f is almost τ⋆(σ1, σ2)-continuous at x.

Theorem 2. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous;

(2) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ;

(3) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ;



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 5 of 18

(4) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of
Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every subset B
of Y ;

(6) f−1(V ) is ⋆-open in X for every (σ1, σ2)r-open set V of Y ;

(7) f−1(K) is ⋆-closed in X for every (σ1, σ2)r-closed set K of Y .

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V .
Thus by Theorem 1, we have x ∈ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) and hence

f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))).

(2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y − K is σ1σ2-open in Y
and by (2), X − f−1(K) = f−1(Y − K) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(Y − K)))) =
Int⋆(X − f−1(σ1σ2-Cl(σ1σ2-Int(K)))) = X − Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))). Thus,
Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K)))) ⊆ f−1(K).

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is a σ1σ2-closed set of Y and
by (3), Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)).

(4) ⇒ (5): Let B be any subset of Y . Then by (4),

f−1(σ1σ2-Int(B)) = X − f−1(σ1σ2-Cl(Y −B))

⊆ X − Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B)))))

= X − Cl⋆(f−1(Y − σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B)))))

= Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))).

(5) ⇒ (6): Let V be any (σ1, σ2)r-open set of Y . By (5), we have f−1(V ) ⊆
Int⋆(f−1(V )) and hence f−1(V ) is ⋆-open in X.

(6) ⇒ (7): The proof is obvious.
(7) ⇒ (1): Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing f(x). Since

Y − V is (σ1, σ2)r-closed and by (7), X − f−1(V ) = f−1(Y − V ) is ⋆-closed in X. Thus,
f−1(V ) is ⋆-open and hence x ∈ Int⋆(F+(V )). Then, there exists a ⋆-open set U of X
containing x such that f(U) ⊆ V . It follows from Theorem 1 that f is almost τ⋆(σ1, σ2)-
continuous.

Theorem 3. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ;

(3) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y .



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Proof. (1) ⇒ (2): Let V be any (σ1, σ2)β-open set of Y . Then, σ1σ2-Cl(V ) is a
(σ1, σ2)r-closed set of Y . Since f is almost τ⋆(σ1, σ2)-continuous, by Theorem 2 we have
f−1(σ1σ2-Cl(V )) is ⋆-closed in X. Thus, Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, K is (σ1, σ2)s-open in Y .

Then by (3), Cl⋆(f−1(K)) ⊆ f−1(σ1σ2-Cl(K)) = f−1(K) and hence f−1(K) is ⋆-closed in
X. By Theorem 2, f is almost τ⋆(σ1, σ2)-continuous.

Lemma 3. [32] For a bitopological space (X, τ1, τ2), the following properties hold:

(1) α(τ1, τ2)-Cl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)β-open set V of X;

(2) (τ1, τ2)-pCl(V ) = τ1τ2-Cl(V ) for every (τ1, τ2)s-open set V of X.

Corollary 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(V )) ⊆ f−1(α(σ1, σ2)-Cl(V )) for every (σ1, σ2)β-open set V of Y ;

(3) Cl⋆(f−1(V )) ⊆ f−1((σ1, σ2)-pCl(V )) for every (σ1, σ2)s-open set V of Y .

Theorem 4. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-
open set V of Y ;

(3) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V
of Y ;

(4) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Then, we have σ1σ2-Cl(V ) is
σ1σ2-closed in Y and by Theorem 2,

Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))) ⊆ f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2),

Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(V )))) ⊆ Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))))

⊆ f−1(σ1σ2-Cl(V )).

(3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus by (3), we have

X − Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) = Cl⋆(X − f−1(σ1σ2-Int(σ1σ2-Cl(V ))))



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 7 of 18

= Cl⋆(f−1(Y − σ1σ2-Int(σ1σ2-Cl(V ))))

= Cl⋆(f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V ))))

= Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(Y − σ1σ2-Cl(V )))))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

= f−1(Y − σ1σ2-Int(σ1σ2-Cl(V )))

⊆ f−1(Y − V )

= X − f−1(V )

and hence f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))).
(4) ⇒ (1): Let V be any (σ1, σ2)r-open set of Y . Then, V is (σ1, σ2)p-open in Y

and by (4), f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) = Int⋆(f−1(V )). Thus, f−1(V ) is
⋆-open in X. It follows from Theorem 2 that f is almost τ⋆(σ1, σ2)-continuous.

Lemma 4. [33] Let A be a subset of a bitopological space (X, τ1, τ2). Then, the following
properties hold:

(1) If A is τ1τ2-open in X, then τ1τ2-Cl(A) = τ1τ2-δ-Cl(A).

(2) τ1τ2-δ-Cl(A) is τ1τ2-closed.

Theorem 5. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ f−1(σ1σ2-δ-Cl(B)) for every subset B
of Y ;

(3) Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-δ-Cl(B)) for every subset B
of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 4, σ1σ2-δ-Cl(B) is σ1σ2-closed
in Y and by Theorem 2, Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-δ-Cl(B))))) ⊆ f−1(σ1σ2-δ-Cl(B)).

(2) ⇒ (3): This is obvious since σ1σ2-Cl(B) ⊆ σ1σ2-δ-Cl(B) for every subset B of Y .
(3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then by (3), we have

Cl⋆(f−1(K)) = Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(K))))

= Cl⋆(f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(K)))))

⊆ f−1(σ1σ2-δ-Cl(K)) = f−1(K)

and hence f−1(K) is ⋆-closed in X. By Theorem 2, f is almost τ⋆(σ1, σ2)-continuous.



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 8 of 18

4. On weakly τ ⋆(σ1, σ2)-continuous functions

In this section, we introduce the notion of weakly τ⋆(σ1, σ2)-continuous functions.
Moreover, some characterizations of weakly τ⋆(σ1, σ2)-continuous functions are discussed.

Definition 2. A function f : (X, τ,I ) → (Y, σ1, σ2) is said to be weakly τ⋆(σ1, σ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists
a ⋆-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function

f : (X, τ,I ) → (Y, σ1, σ2)

is said to be weakly τ⋆(σ1, σ2)-continuous if f is weakly τ⋆(σ1, σ2)-continuous at each point
x of X.

Remark 1. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following implication holds:

almost τ⋆(σ1, σ2)-continuity ⇒ weakly τ⋆(σ1, σ2)-continuity.

The converse of the implication is not true in general. We give an example for the
implication as follows.

Example 1. Let X = {1, 2, 3} with a topology τ = {∅, {2}, {1, 3}, X} and an ideal I =
{∅}. Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {a, b}, Y } and

σ2 = {∅, {a}, {b}, {a, b}, Y }.

A function f : (X, τ,I ) → (Y, σ1, σ2) is defined as follows: f(1) = a, f(2) = b and f(3) =
c. Then, f is weakly τ⋆(σ1, σ2)-continuous but f is not almost τ⋆(σ1, σ2)-continuous.

Theorem 6. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(3) Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ;

(4) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ;

(6) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(7) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(8) Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y .



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 9 of 18

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y such that x ∈ f−1(V ). Then,
we have f(x) ∈ V . By (1), there exists a ⋆-open set U of X containing x such that
f(U) ⊆ σ1σ2-Cl(V ). Therefore, U ⊆ f−1(σ1σ2-Cl(V )). Since U is ⋆-open, we have

x ∈ Int⋆(f−1(σ1σ2-Cl(V )))

and hence f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))).
(2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y . By (2),

X − f−1(K) = f−1(Y −K) ⊆ Int⋆(f−1(σ1σ2-Cl(Y −K))) = X −Cl⋆(f−1(σ1σ2-Int(K))).
Thus, Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K).

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is a σ1σ2-closed set of Y and
by (3), Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)).

(4) ⇒ (5): Let B be any subset of Y . Thus by (4), we have

X − Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))) = Cl⋆(X − f−1(σ1σ2-Cl(σ1σ2-Int(B))))

= Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(Y −B))))

⊆ f−1(σ1σ2-Cl(Y −B))

= X − f−1(σ1σ2-Int(B))

and hence f−1(σ1σ2-Int(B)) ⊆ Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))).
(5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (5),

x ∈ f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) and there exists a ⋆-open set U of X containing
x such that U ⊆ f−1(σ1σ2-Cl(V )). Thus, f(U) ⊆ σ1σ2-Cl(V ) and hence f is weakly
τ⋆(σ1, σ2)-continuous.

(4) ⇒ (6) and (6) ⇒ (7): The proofs are obvious.
(7) ⇒ (8): Let K be any (σ1, σ2)r-closed set of Y . Thus by (7),

Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) = f−1(K).

(8) ⇒ (3): LetK be any σ1σ2-closed set of Y . Then, σ1σ2-Cl(σ1σ2-Int(K)) is (σ1, σ2)r-
closed in Y and σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K))) = σ1σ2-Int(σ1σ2-Cl(K)) = σ1σ2-Int(K).
By (8),

Cl⋆(f−1(σ1σ2-Int(K))) = Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) ⊆ f−1(K).

Theorem 7. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V
of Y ;



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 10 of 18

(3) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V
of Y .

Proof. (1) ⇒ (2): This follows from (4) of Theorem 6.
(2) ⇒ (3): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open.
(3) ⇒ (1): Since every σ1σ2-open set is (σ1, σ2)s-open, the proof is obvious by (7) of

Theorem 6.

Theorem 8. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V
of Y ;

(3) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(4) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Since σ1σ2-Int(σ1σ2-Cl(V ))
is σ1σ2-open, by Theorem 6 (7)

Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))))

⊆ f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): Let V be any (σ1, σ2)p-open set of Y . By (2), we have

Cl⋆(f−1(V )) ⊆ Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Thus by (3),

X − Int⋆(f−1(Cl⋆(V ))) = Cl⋆(X − f−1(Cl⋆(V )))

= Cl⋆(f−1(Y − Cl⋆(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − f−1(V )

and so f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))).
(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y and by (4),

f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))). By Theorem 6 (2), f is weakly τ⋆(σ1, σ2)-continuous.

Lemma 5. [25] For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 11 of 18

(1) If A is τ2τ2-open in X, then τ1τ2-Cl(A) = (τ1, τ2)θ-Cl(A).

(2) (τ1, τ2)θ-Cl(A) is τ1τ2-closed in X.

Theorem 9. For a function f : (X, τ,I ) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of
Y ;

(3) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . By Lemma 5, (σ1, σ2)θ-Cl(B) is σ1σ2-
closed in Y and by Theorem 6, Cl⋆(f−1(σ1σ2-Int((σ1, σ2)θ-Cl(B)))) ⊆ f−1((σ1, σ2)θ-Cl(B)).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (1): Let K be any (σ1, σ2)r-closed set of Y . Then, we have

(σ1, σ2)θ-Cl(σ1σ2-Int(K)) = σ1σ2-Cl(σ1σ2-Int(K)) = K

and hence

Cl⋆(f−1(σ1σ2-Int(K))) = Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1((σ1, σ2)θ-Cl(σ1σ2-Int(K)))

= f−1(σ1σ2-Cl(σ1σ2-Int(K)))

= f−1(K).

Thus, by Theorem 6 f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 10. A function f : (X, τ,I ) → (Y, σ1, σ2) is weakly τ⋆(σ1, σ2)-continuous at
x ∈ X if and only if x ∈ Int⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y containing
f(x).

Proof. Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, there
exists a ⋆-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Thus, U ⊆
f−1(σ1σ2-Cl(V )) and hence x ∈ Int⋆(f−1(σ1σ2-Cl(V ))).

Conversely, let V be any σ1σ2-open set of Y containing f(x). By the hypothesis,
x ∈ Int⋆(f−1(σ1σ2-Cl(V ))). Then, there exists a ⋆-open set U of X such that

x ∈ U ⊆ f−1(σ1σ2-Cl(V )).

Thus, f(U) ⊆ σ1σ2-Cl(V ) and so f is weakly τ⋆(σ1, σ2)-continuous at x.

Theorem 11. A function f : (X, τ,I ) → (Y, σ1, σ2) is weakly τ⋆(σ1, σ2)-continuous if
and only if f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y .



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 12 of 18

Proof. Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Then, f(x) ∈ V . Since f
is weakly τ⋆(σ1, σ2)-continuous at x, by Theorem 10 we have x ∈ Int⋆(f−1(σ1σ2-Cl(V )))
and hence f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))).

Conversely, let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Then, we have
x ∈ f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))). By Theorem 10, f is weakly τ⋆(σ1, σ2)-continuous
at x. This shows that f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 12. A function f : (X, τ,I ) → (Y, σ1, σ2) is weakly τ⋆(σ1, σ2)-continuous if
and only if Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y .

Proof. Let V be any σ1σ2-open set of Y . Suppose that Cl⋆(f−1(V )) ⊈ f−1(σ1σ2-Cl(V )).
There exists x ∈ Cl⋆(f−1(V )), but x ̸∈ f−1(σ1σ2-Cl(V )). Then, f(x) ̸∈ σ1σ2-Cl(V )
and there exists a ⋆-open set W of Y containing f(x) such that W ∩ V = ∅. Thus,
σ1σ2-Cl(W ) ∩ V = ∅. Since f is weakly τ⋆(σ1, σ2)-continuous at x, there exists a ⋆-open
set U of X containing x such that f(U) ⊆ σ1σ2-Cl(W ). Therefore, f(U) ∩ V = ∅. Since
x ∈ Cl⋆(f−1(V )), U ∩f−1(V ) ̸= ∅ and f(U)∩V ̸= ∅, which is a contradiction. This shows
that Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

Conversely, let V be any σ1σ2-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open in
Y . By the hypothesis, Cl⋆(f−1(Y −σ1σ2-Cl(V ))) ⊆ f−1(σ1σ2-Cl(Y −σ1σ2-Cl(V ))). Thus,
X − Int⋆(f−1(σ1σ2-Cl(V ))) ⊆ X − f−1(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ X − f−1(V ) and hence
f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))). By Theorem 11, f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 13. For a function (X, τ,I ) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(3) Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every σ1σ2-closed set K of Y ;

(4) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of Y ;

(6) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y .

Proof. (1) ⇒ (2): It follows from Theorem 11.
(2) ⇒ (3): Let K be any σ1σ2-closed set of Y . Then, Y −K is σ1σ2-open in Y and by

(2),

X − f−1(K) = f−1(Y −K) ⊆ Int⋆(f−1(σ1σ2-Cl(Y −K)))

= Int⋆(f−1(Y − σ1σ2-Int(K)))

= X − Cl⋆(f−1(σ1σ2-Int(K))).

Thus, Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K).



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 13 of 18

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y . By (3),
Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)).

(4) ⇒ (5): Let B be any subset of Y . Thus by (4),

f−1(σ1σ2-Int(B)) = X − f−1(σ1σ2-Cl(Y −B))

⊆ X − Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(Y −B))))

= Int⋆(f−1(σ1σ2-Cl(σ1σ2-Int(B)))).

(5) ⇒ (6): Let V be any σ1σ2-open set of Y and x ̸∈ f−1(σ1σ2-Cl(V )). Then,
there exists a σ1σ2-open set U of Y containing f(x) such that U ∩ V = ∅. By (5),
x ∈ f−1(U) ⊆ Int⋆(f−1(σ1σ2-Cl(U))) and there exists a ⋆-open set G of X containing x
such that f(G) ⊆ σ1σ2-Cl(U). Thus, G∩f−1(V ) = ∅ and so x ̸∈ Cl⋆(f−1(V )). This shows
that Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

(6) ⇒ (1): Let x ∈ X and V be any be any σ1σ2-open set of Y containing f(x). Since
V = σ1σ2-Int(V ) ⊆ σ1σ2-Int(σ1σ2-Cl(V )), by (6) we have

x ∈ f−1(V ) ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − Cl⋆(f−1(Y − σ1σ2-Cl(V )))

= Int⋆(f−1(σ1σ2-Cl(V ))).

There exists a ⋆-open set U of X containing x such that U ⊆ f−1(σ1σ2-Cl(V )); hence
f(U) ⊆ σ1σ2-Cl(V ). Thus, f is weakly τ⋆(σ1, σ2)-continuous at x. This shows that f is
weakly τ⋆(σ1, σ2)-continuous.

Theorem 14. For a function (X, τ,I ) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ;

(3) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V
of Y ;

(4) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V
of Y .

Proof. (1) ⇒ (2): Let K be any (σ1, σ2)r-closed set of Y . Then, σ1σ2-Int(K) is
σ1σ2-open in Y , by Theorem 13 (6) we have

Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K))) = f−1(K).

(2) ⇒ (3): Let V be any (σ1, σ2)β-open set of Y . Then, we have

σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))) ⊆ σ1σ2-Cl(V )



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 14 of 18

and hence σ1σ2-Cl(V ) is (σ1, σ2)r-closed. By (2),

Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(3) ⇒ (4): This is obvious.
(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, we have V is (σ1, σ2)s-open in

Y . By (4), Cl⋆(f−1(V )) ⊆ Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) and by
Theorem 13 (6), f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 15. For a function (X, τ,I ) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V
of Y ;

(3) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(4) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let V be any (σ1, σ2)p-open set of Y . Then, we have

σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))

and hence σ1σ2-Cl(V ) is (σ1, σ2)r-closed in Y . Thus, by Theorem 14 (2) we have

Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(2) ⇒ (3): The proof is obvious.
(3) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . By (3),

f−1(V ) ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − Cl⋆(f−1(Y − σ1σ2-Cl(V )))

= Int⋆(f−1(σ1σ2-Cl(V ))).

(4) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y . Thus by
(4) and Theorem 13 (2), f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 16. For a function (X, τ,I ) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 15 of 18

(3) Cl⋆(f−1(σ1σ2-Int(K))) ⊆ f−1(K) for every (σ1, σ2)r-closed set K of Y ;

(4) Cl(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(5) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(6) Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(7) f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y and x ̸∈ f−1(σ1σ2-Cl(B)). Then,
we have f(x) ̸∈ σ1σ2-Cl(B) and there exists a σ1σ2-open set U of Y containing f(x)
such that U ∩ B = ∅. Therefore, σ1σ2-Cl(U) ∩ σ1σ2-Int(σ1σ2-Cl(B)) = ∅. Since f is
weakly τ⋆(σ1, σ2)-continuous at x, there exists a ⋆-open set W of X containing x such
that f(W ) ⊆ σ1σ2-Cl(U). Thus, W ∩ f−1(σ1σ2-Int(σ1σ2-Cl(B))) = ∅ and hence

x ̸∈ Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))).

This shows that Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(B)))) ⊆ f−1(σ1σ2-Cl(B)).
(2) ⇒ (3): Let K be any (σ1, σ2)r-closed set of Y . Then by (2), we have

Cl⋆(f−1(σ1σ2-Int(K))) = Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(K)))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Int(K)))

= f−1(K).

(3) ⇒ (4): Let V be any σ1σ2-open set of Y . Then, σ1σ2-Cl(V ) is (σ1, σ2)r-closed in
Y . By (3), Cl⋆(f−1(V )) ⊆ Cl⋆(f−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(4) ⇒ (5): Let V be any σ1σ2-open set of Y . Since Y −σ1σ2-Cl(V ) is σ1σ2-open in Y ,
by (4) we have

X − Int⋆(f−1(σ1σ2-Cl(V ))) = Cl⋆(f−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − f−1(V )

and hence f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))).
(5) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (5),

x ∈ f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))). Put W = Int⋆(f−1(σ1σ2-Cl(V ))). Then, W is
⋆-open set of X containing x such that f(W ) ⊆ σ1σ2-Cl(V ). Thus, f is weakly τ⋆(σ1, σ2)-
continuous at x. This shows that f is weakly τ⋆(σ1, σ2)-continuous.

(1) ⇒ (6): Let V be any (σ1, σ2)p-open set of Y and x ̸∈ f−1(σ1σ2-Cl(V )). Then,
f(x) ̸∈ σ1σ2-Cl(V ) and there exists a σ1σ2-open set G of Y containing f(x) such that
G ∩ V = ∅. Since V is (σ1, σ2)p-open, we have

V ∩ σ1σ2-Cl(G) ⊆ σ1σ2-Int(σ1σ2-Cl(V )) ∩ σ1σ2-Cl(G)

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )) ∩G)



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 16 of 18

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ) ∩G))

⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ∩G)))

⊆ σ1σ2-Cl(V ∩G) = ∅.

Since f is weakly τ⋆(σ1, σ2)-continuous at x, there exists a ⋆-open set W of X containing
x such that f(W ) ⊆ σ1σ2-Cl(G). Thus, f(W ) ∩ V = ∅ and hence W ∩ f−1(V ) = ∅.
Therefore, x ̸∈ Cl⋆(f−1(V ). This shows that Cl⋆(f−1(V )) ⊆ f−1(σ1σ2-Cl(V )).

(6) ⇒ (7): Let V be any (σ1, σ2)p-open set of Y . Then, Y − σ1σ2-Cl(V ) is σ1σ2-open
and hence Y − σ1σ2-Cl(V ) is (σ1, σ2)p-open in Y . Then by (6), we have

X − Int⋆(f−1(σ1σ2-Cl(V ))) = Cl⋆(X − f−1(σ1σ2-Cl(V )))

= Cl⋆(f−1(Y − σ1σ2-Cl(V )))

⊆ f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

= f−1(Y − σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Int(σ1σ2-Cl(V )))

⊆ X − f−1(V )

and hence f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))).
(7) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since V is

(σ1, σ2)p-open in Y and by (7), x ∈ f−1(V ) ⊆ Int⋆(f−1(σ1σ2-Cl(V ))). Put

U = Int⋆(f−1(σ1σ2-Cl(V ))).

Then, U is a ⋆-open set of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). Thus, f is
weakly τ⋆(σ1, σ2)-continuous at x. This shows that f is weakly τ⋆(σ1, σ2)-continuous.

Theorem 17. For a function (X, τ,I ) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is weakly τ⋆(σ1, σ2)-continuous;

(2) f(Cl⋆(A)) ⊆ (σ1, σ2)θ-Cl(f(A)) for every subset A of X;

(3) Cl⋆(f−1(B)) ⊆ f−1((σ1, σ2)θ-Cl(B)) for every subset B of Y .

Proof. (1) ⇒ (2): Let A be any subset of X. Suppose that x ∈ Cl⋆(A) and G is
any σ1σ2-open set of Y containing f(x). Since f is weakly τ⋆(σ1, σ2)-continuous, there
exists a ⋆-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(G). Since x ∈ Cl⋆(A),
we have U ∩ A ̸= ∅. It follows that ∅ ̸= f(U) ∩ f(A) ⊆ σ1σ2-Cl(G) ∩ f(A). Thus,
f(x) ∈ (σ1, σ2)θ-Cl(f(A)) and hence f(Cl⋆(A)) ⊆ (σ1, σ2)θ-Cl(f(A)).

(2) ⇒ (3): Let B be any subset of Y . Then, we have

f(Cl⋆(f−1(B))) ⊆ (σ1, σ2)θ-Cl(f(f
−1(B))) ⊆ (σ1, σ2)θ-Cl(B)



N. Viriyapong, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 18 (3) (2025), 6568 17 of 18

and hence Cl⋆(f−1(B)) ⊆ f−1((σ1, σ2)θ-Cl(B)).
(3) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since

σ1σ2-Cl(V ) ∩ (Y − σ1σ2-Cl(V )) = ∅,

f(x) ̸∈ (σ1, σ2)θ-Cl(Y − σ1σ2-Cl(V )) and hence x ̸∈ f−1((σ1, σ2)θ-Cl(Y − σ1σ2-Cl(V ))).
By (3), x ̸∈ Cl⋆(f−1(Y − σ1σ2-Cl(V ))) and there exists a ⋆-open set U of X containing
x such that U ∩ f−1(Y − σ1σ2-Cl(V )) = ∅; hence f(U) ∩ (Y − σ1σ2-Cl(V )) = ∅. Thus,
f(U) ⊆ σ1σ2-Cl(V ) and so f is weakly τ⋆(σ1, σ2)-continuous at x. This shows that f is
weakly τ⋆(σ1, σ2)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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