EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3730-3742
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

δ(τ1, τ2)-Continuous Functions

Chatchadaporn Prachanpol1, Chawalit Boonpok1, Chokchai Viriyapong1,∗

1 Mathematics and Applied Mathematics Research Unit, Department of Mathematics,
Faculty of Science, Mahasarakham University, Maha Sarakham, 44150, Thailand

Abstract. This paper introduces a new class of functions called δ(τ1, τ2)-continuous functions.
Several characterizations of δ(τ1, τ2)-continuous functions are investigated. The relationships be-
tween δ(τ1, τ2)-continuity and the other types of δ(τ1, τ2)-continuity are also discussed.

2020 Mathematics Subject Classifications: 54C05, 54C08, 54E55
Key Words and Phrases: δ(τ1, τ2)-open set, δ(τ1, τ2)-continuous function

1. Introduction

The field of the mathematical science which goes under the name of topology is con-
cerned with all questions directly or indirectly related to continuity. By using various forms
of open sets many authors introduced and studied various types of continuity. In 1968,
Veličko [22] introduced a new class of open sets in topological spaces called δ-open sets
and investigated some properties of δ-closed sets and δ-open sets. The class of open sets
including the class of δ-open sets. In 1980, Noiri [17] introduced and studied the notion
of δ-continuous functions. Munshi and Bassan [16] defined and developed the concept of
super-continuity. The concept has been investigated further by Reilly and Vamanamurthy
[21] where super-continuity is characterized in terms of the semi-regularization topology.
Super-continuity is related to the concepts of δ-continuity and strong θ-continuity devel-
oped by Noiri [17]. In particular, super-continuity is strictly between strong θ-continuity
and δ-continuity and strictly between complete continuity [1] and δ-continuity. Raychaud-
huri and Mukherjee [20] introduced the concept of δ-preopen sets which is weaker than
that of preopen sets and used this concept to define the notion of δ-almost continuous func-
tions as a generalization of precontinuous functions due to Mashhour et al. [15]. Baker
[2] introduced and investigated the notion of weakly δ-continuous functions. The class of
weakly δ-continuous functions is a generalization of δ-continuous functions.

In 1997, Park et al. [19] introduced the notion of δ-semiopen sets by using δ-open
sets due to Valičko [22]. In 2005, Ekici and Navalagi [13] introduced and investigated

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5335

Email addresses: prachanpol.ch@gmail.com (C. Prachanpol),
chawalit.b@msu.ac.th (C. Boonpok), chokchai.v@msu.ac.th (C. Viriyapong)

https://www.ejpam.com 3730 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3731

the concept of δ-semicontinuous functions. The class of δ-semicontinuous functions is
a weaker form of the classes of perfectly continuous functions [18], strongly θ-continuous
functions [14] and super-continuous functions. Ekici [12] introduced the notion of almost δ-
semicontinuous functions which generalize R-maps [11] and δ-continuous functions. Yüksel
et al. [25] extended the concept of δ-open sets to ideal topological spaces and defined δ-
I-continuous functions. Moreover, some characterizations of θ-I -continuous functions,
⋆-continuous functions and θ(⋆)-precontinuous functions were presented in [3], [5] and [6],
respectively. In [9], the present authors introduced and studied the concept of (τ1, τ2)-
continuous functions. Futhermore, several characterizations of almost (τ1, τ2)-continuous
functions and weakly (τ1, τ2)-continuous functions were established in [8] and [7], respec-
tively. In this paper, we introduce the notions of δ(τ1, τ2)-continuous functions, almost
δ(τ1, τ2)-continuous functions, and weakly δ(τ1, τ2)-continuous functions. We also investi-
gate several characterizations of δ(τ1, τ2)-continuous functions, almost δ(τ1, τ2)-continuous
functions, and weakly δ(τ1, τ2)-continuous functions. Finally, the relationships among
δ(τ1, τ2)-continuous functions, almost δ(τ1, τ2)-continuous functions, and weakly δ(τ1, τ2)-
continuous functions are discussed.

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 ex-
plicitly 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), respec-
tively, for i = 1, 2. A subset A of a bitopological space (X, τ1, τ2) is called τ1τ2-closed
[10] 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 [10]
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 [10] of A and is denoted by τ1τ2-Int(A). A subset A
of a bitopological space (X, τ1, τ2) is called (τ1, τ2)r-open [23] (resp. (τ1, τ2)s-open [4],
(τ1, τ2)p-open [4], (τ1, τ2)β-open [4], α(τ1, τ2)-open) [24]) 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))),
A ⊆ τ1τ2-Int(τ1τ2-Cl(τ1τ2-Int(A)))). The complement of a (τ1, τ2)r-open (resp. (τ1, τ2)s-
open, (τ1, τ2)p-open, (τ1, τ2)β-open, α(τ1, τ2)-open) set is called (τ1, τ2)r-closed (resp.
(τ1, τ2)s-closed, (τ1, τ2)p-closed, (τ1, τ2)β-closed, α(τ1, τ2)-closed). Let A be a subset of
a bitopological space (X, τ1, τ2). A point x of X is called a δ(τ1, τ2)-cluster point of A
if V ∩ A ̸= ∅ for every (τ1, τ2)r-open set V containing x. The set of all δ(τ1, τ2)-cluster
points of A is called the δ(τ1, τ2)-closure of A and is denoted by δ(τ1, τ2)-Cl(A). A subset
A of a bitopological space (X, τ1, τ2) is called δ(τ1, τ2)-closed if A = δ(τ1, τ2)-Cl(A). The
complement of a δ(τ1, τ2)-closed set is called δ(τ1, τ2)-open (τ1τ2-δ-open [8]). The fam-
ily of all δ(τ1, τ2)-open (resp. δ(τ1, τ2)-closed) sets of a bitopological space (X, τ1, τ2) is
denoted by δ(τ1, τ2)O(X) (resp. δ(τ1, τ2)C(X)). The δ(τ1, τ2)-interior of A denoted by



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3732

δ(τ1, τ2)-Int(A) is defined as follows:

δ(τ1, τ2)-Int(A) = ∪{G ⊆ X | G ∈ δ(τ1, τ2)O(X) and G ⊆ A}.

Lemma 1. For a subset A of a bitopological space (X, τ1, τ2), x ∈ δ(τ1, τ2)-Cl(A) if and
only if V ∩A ̸= ∅ for every V ∈ δ(τ1, τ2)O(X) containing x.

Lemma 2. For a subset A of a bitopological space (X, τ1, τ2), the following properties
hold:

(1) δ(τ1, τ2)-Int(X −A) = X − δ(τ1, τ2)-Cl(A).

(2) δ(τ1, τ2)-Cl(X −A) = X − δ(τ1, τ2)-Int(A).

3. On δ(τ1, τ2)-continuous functions

In this section, we introduce the notion of δ(τ1, τ2)-continuous functions. We also
discuss several characterizations of δ(τ1, τ2)-continuous functions.

Definition 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called δ(τ1, τ2)-continuous at
x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a δ(τ1, τ2)-open set
U of X containing x such that f(U) ⊆ V . A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said
to be δ(τ1, τ2)-continuous if f is δ(τ1, τ2)-continuous at each point of X.

Example 1. Let X = {a, b, c} with topologies τ1 = {∅, {a}, {b}, {a, b}, {a, c}, X} and τ2 =
{∅, {a}, {b}, {a, b}, X}. Let Y = {1, 2, 3} with topologies σ1 = {∅, {1}, {3}, {1, 2}, {1, 3}, Y }
and σ2 = {∅, {1}, {3}, {1, 3}, Y }. Define a function f : (X, τ1, τ2) → (Y, σ1, σ2) as follows:
f(a) = f(c) = 2 and f(b) = 3. Then, f is δ(τ1, τ2)-continuous.

Theorem 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is δ(τ1, τ2)-continuous at x;

(2) x ∈ δ(τ1, τ2)-Int(f
−1(V )) for every σ1σ2-open set V of Y containing f(x);

(3) x ∈ f−1(σ1σ2-Cl(f(A))) for every A ⊆ X such that x ∈ δ(τ1, τ2)-Cl(A);

(4) x ∈ f−1(σ1σ2-Cl(B)) for every B ⊆ Y such that x ∈ δ(τ1, τ2)-Cl(f
−1(B));

(5) x ∈ δ(τ1, τ2)-Int(f
−1(B)) for every B ⊆ Y such that x ∈ f−1(σ1σ2-Int(B));

(6) x ∈ f−1(F ) for every σ1σ2-closed set F of Y such that x ∈ δ(τ1, τ2)-Cl(f
−1(F )).

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y containing f(x). By (1), There
exists a δ(τ1, τ2)-open set U of X containing x such that f(U) ⊆ V . Hence, x ∈ U ⊆
f−1(V ). Therefore, x ∈ δ(τ1, τ2)-Int(f

−1(V )).
(2) ⇒ (3): Let A ⊆ X such that x ∈ δ(τ1, τ2)-Cl(A) and V be any σ1σ2-open set of

Y containing f(x). By (2), x ∈ δ(τ1, τ2)-Int(f
−1(V )). Then, there exists a δ(τ1, τ2)-open



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3733

set U of X containing x such that U ⊆ f−1(V ). By Lemma 1, we have U ∩ A ̸= ∅ and
hence ∅ ≠ f(U ∩ A) ⊆ f(U) ∩ f(A) ⊆ V ∩ f(A). Thus, f(x) ∈ σ1σ2-Cl(f(A)) and so
x ∈ f−1(σ1σ2-Cl(f(A))).

(3) ⇒ (4): Let B be any subset of Y and x ∈ δ(τ1, τ2)-Cl(f
−1(B)). Then by (3), we

have x ∈ f−1(σ1σ2-Cl(f(f
−1(B)))) ⊆ f−1(σ1σ2-Cl(B)).

(4) ⇒ (5): Let B be any subset of Y and x ∈ f−1(σ1σ2-Int(B)). Suppose that
x /∈ δ(τ1, τ2)-Int(f

−1(B)). Then, x ∈ X − δ(τ1, τ2)-Int(f
−1(B)). Since

X − δ(τ1, τ2)-Int(f
−1(B)) = δ(τ1, τ2)-Cl(f

−1(Y −B))

and by (4), we obtain that x ∈ f−1(σ1σ2-Cl(Y −B)) = X − f−1(σ1σ2-Int(B)) and hence
x /∈ f−1(σ1σ2-Int(B)), which is a contradiction that x ∈ f−1(σ1σ2-Int(B)). Therefore,
x ∈ δ(τ1, τ2)-Int(f

−1(B)).
(5) ⇒ (6): Let F be any σ1σ2-closed set of Y and x ∈ δ(τ1, τ2)-Cl(f

−1(F )). Suppose
that x /∈ f−1(F ). Since Y − F is σ1σ2-open in Y , x ∈ X − f−1(F ) = f−1(Y − F ) =
f−1(σ1σ2-Int(Y − F )). By (5) and Lemma 2 (1), we have

x ∈ δ(τ1, τ2)-Int(f
−1(Y − F )) = X − δ(τ1, τ2)-Cl(f

−1(F ))

and hence x ̸∈ δ(τ1, τ2)-Cl(f
−1(F )). This is a contradiction. Therefore, x ∈ f−1(F ).

(6) ⇒ (2): Let V be any σ1σ2-open set of Y containing f(x). Then, x ∈ f−1(V ).
Suppose that x ̸∈ δ(τ1, τ2)-Int(f

−1(V )). By Lemma 2 (2), x ∈ X−δ(τ1, τ2)-Int(f
−1(V )) =

δ(τ1, τ2)-Cl(f
−1(Y − V )). Since Y − V is σ1σ2-closed in Y and by (6), we have

x ∈ f−1(Y − V ) = X − f−1(V ).

This implies that x /∈ f−1(V ), which is a contradiction. Thus, x ∈ δ(τ1, τ2)-Int(f
−1(V )).

(2) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (2),
we have x ∈ δ(τ1, τ2)-Int(f

−1(V )). Then, there exists a δ(τ1, τ2)-open set U such that
x ∈ U ⊆ f−1(V ). Thus, f(U) ⊆ V . This shows that f is δ(τ1, τ2)-continuous at x.

Theorem 2. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is δ(τ1, τ2)-continuous;

(2) f−1(V ) is δ(τ1, τ2)-open in X for every σ1σ2-open set V of Y ;

(3) f(δ(τ1, τ2)-Cl(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X;

(4) δ(τ1, τ2)-Cl(f
−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ δ(τ1, τ2)-Int(f
−1(B)) for every subset B of Y ;

(6) f−1(F ) is δ(τ1, τ2)-closed in X for every σ1σ2-closed set F of Y .



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3734

Proof. (1) ⇒ (2): Let V be any σ1σ2-open of Y and x ∈ f−1(V ). By (1), there exists
a δ(τ1, τ2)-open set U of X containing x such that f(U) ⊆ V . Thus, x ∈ U ⊆ f−1(V )
and hence x ∈ δ(τ1, τ2)-Int(f

−1(V )). This implies that f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(V )).

Therefore, f−1(V ) is δ(τ1, τ2)-open in X.
(2) ⇒ (3): Let A be any subset of X and let x ∈ δ(τ1, τ2)-Cl(A). Then, we have

f(x) ∈ f(δ(τ1, τ2)-Cl(A)). Let V be any σ1σ2-open set of Y containing f(x). By (2),
f−1(V ) is δ(τ1, τ2)-open in X. Therefore, x ∈ f−1(V ) = δ(τ1, τ2)-Int(f

−1(V )). Then,
there exists a δ(τ1, τ2)-open set U of X containing x such that U ⊆ f−1(V ). Since
x ∈ δ(τ1, τ2)-Cl(A), then U ∩ A ̸= ∅. Hence, ∅ ̸= f(U ∩ A) ⊆ f(U) ∩ f(A) ⊆ V ∩ f(A).
Thus, f(x) ∈ σ1σ2-Cl(f(A)) and so f(δ(τ1, τ2)-Cl(A)) ⊆ σ1σ2-Cl(f(A)).

(3) ⇒ (4): Let B be any subset of Y . Then by (3), f(δ(τ1, τ2)-Cl(f
−1(B))) ⊆

σ1σ2-Cl(f(f
−1(B))) ⊆ σ1σ2-Cl(B). Therefore, δ(τ1, τ2)-Cl(f

−1(B)) ⊆ f−1(σ1σ2-Cl(B)).
(4) ⇒ (5): Let B be any subset of Y . By (4) and Lemma 2 (2), we have

X − δ(τ1, τ2)-Int(f
−1(B)) = δ(τ1, τ2)-Cl(f

−1(Y −B))

⊆ f−1(σ1σ2-Cl(Y −B))

= X − f−1(σ1σ2-Int(B))

and hence f−1(σ1σ2-Int(B)) ⊆ δ(τ1, τ2)-Int(f
−1(B)).

(5) ⇒ (6): Let F be any σ1σ2-closed set of Y . Then, Y −F is σ1σ2-open in Y . By (5)
and Lemma 2 (1), we obtain that X − f−1(F ) = f−1(Y − F ) = f−1(σ1σ2-Int(Y − F )) ⊆
δ(τ1, τ2)-Int(f

−1(Y − F )) = X − δ(τ1, τ2)-Cl(f
−1(F )). Therefore, δ(τ1, τ2)-Cl(f

−1(F )) ⊆
f−1(F ). This shows that f−1(F ) is δ(τ1, τ2)-closed in X.

(6) ⇒ (2): Let V be any σ1σ2-open set of Y . Then, Y −V is σ1σ2-closed in Y . By (6),
we have X−f−1(V ) = f−1(Y −V ) is δ(τ1, τ2)-closed in X. Thus, f−1(V ) is δ(τ1, τ2)-open
in X.

(2) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By (2),
f−1(V ) is δ(τ1, τ2)-open in X. Hence, x ∈ f−1(V ) = δ(τ1, τ2)-Int(f

−1(V )). Then, there
exists a δ(τ1, τ2)-open set U such that x ∈ U ⊆ f−1(V ). Therefore, f(U) ⊆ V . Thus, f is
δ(τ1, τ2)-continuous at x. This shows that f is δ(τ1, τ2)-continuous.

4. On almost δ(τ1, τ2)-continuous functions

In this section, we introduce the notion of almost δ(τ1, τ2)-continuous functions and
investigate some characterizations of almost δ(τ1, τ2)-continuous functions. Moreover, the
relationships between δ(τ1, τ2)-continuous functions and almost δ(τ1, τ2)-continuous func-
tions are considered.

Definition 2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost δ(τ1, τ2)-
continuous at x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists
a δ(τ1, τ2)-open set U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). A
function f : (X, τ1, τ2) → (Y, σ1, σ2) is called almost δ(τ1, τ2)-continuous if f is almost
δ(τ1, τ2)-continuous at each point of X.



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3735

Remark 1. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following implication holds:

δ(τ1, τ2)-continuity ⇒ almost δ(τ1, τ2)-continuity.

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

Example 2. Let X = {1, 2, 3, 4} with topologies τ1 = {∅, {1}, {2}, {1, 2}, X} and

τ2 = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}, X}.

Let Y = {a, b, c} with topologies σ1 = {∅, {a}, {b}, {a, b}, Y } and

σ2 = {∅, {a}, {b}, {a, b}, {a, c}, Y }.

A function f : (X, τ1, τ2) → (Y, σ1, σ2) is defined as follows: f(1) = a, f(2) = b and
f(3) = f(4) = c. Then f is almost δ(τ1, τ2)-continuous, but f is not δ(τ1, τ2)-continuous.

Theorem 3. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost δ(τ1, τ2)-continuous at x;

(2) x ∈ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y con-

taining f(x);

(3) x ∈ δ(τ1, τ2)-Int(f
−1(V )) for every (σ1, σ2)r-open set V of Y containing f(x);

(4) for each x ∈ X and each (σ1, σ2)r-open set V of Y containing f(x), there exists a
δ(τ1, τ2)-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). Since f is
almost δ(τ1, τ2)-continuous at x ∈ X. There exists a δ(τ1, τ2)-open set U of X containing
x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Thus, x ∈ U ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V ))).
Therefore, x ∈ δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))).
(2) ⇒ (3): Let V be any (σ1, σ2)r-open set of Y containing f(x). Then, V is σ1σ2-open

in Y . By (2), x ∈ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) = δ(τ1, τ2)-Int(f

−1(V )).
(3) ⇒ (4): Let V be any (σ1, σ2)r-open set of Y containing f(x). Thus by (3), we

have x ∈ δ(τ1, τ2)-Int(f
−1(V )). Then, there exists a δ(τ1, τ2)-open set U of X such that

x ∈ U ⊆ f−1(V ). Therefore, f(U) ⊆ V .
(4) ⇒ (1): Let x ∈ X and V be any σ1σ2-open set of Y containing f(x). Since

σ1σ2-Int(σ1σ2-Cl(V )) is a (σ1, σ2)r-open set and by (4), there exists a δ(τ1, τ2)-open set
U of X containing x such that f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Consequently, f is almost
δ(τ1, τ2)-continuous at x.

Theorem 4. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3736

(1) f is almost δ(τ1, τ2)-continuous;

(2) f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y ;

(3) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(σ1σ2-Int(F )))) ⊆ f−1(F ) for every σ1σ2-closed set F of Y ;

(4) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(B))))) ⊆ f−1(σ1σ2-Cl(B)) for every sub-

set B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))) for every sub-

set B of Y ;

(6) f−1(V ) is δ(τ1, τ2)-open in X for every (σ1, σ2)r-open set V of Y ;

(7) f−1(F ) is δ(τ1, τ2)-closed in X for every (σ1, σ2)r-closed set F of Y .

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Since f is
almost δ(τ1, τ2)-continuous, there exists a δ(τ1, τ2)-open set U of X containing x such that
f(U) ⊆ σ1σ2-Int(σ1σ2-Cl(V )). Then, we have x ∈ U ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V ))) and
hence x ∈ δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))). This implies that

f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))).

(2) ⇒ (3): Let F be any σ1σ2-closed set of Y . Then, Y −F is σ1σ2-open in Y . Thus by
(2), we have X− f−1(F ) = f−1(Y −F ) ⊆ δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(Y −F )))) =
δ(τ1, τ2)-Int(f

−1(Y−σ1σ2-Cl(σ1σ2-Int(F )))) = X−δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(σ1σ2-Int(F ))))

and hence δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(σ1σ2-Int(F )))) ⊆ f−1(F ).

(3) ⇒ (4): Let B be any subset of Y . Then, σ1σ2-Cl(B) is σ1σ2-closed in Y and by
(3), δ(τ1, τ2)-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 − δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(Y −B)))))

= δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))).

Therefore, f−1(σ1σ2-Int(B)) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(B))))).

(5) ⇒ (6): Let V be any (σ1, σ2)r-open set of Y . Then, V is a σ1σ2-open set of
Y and V = σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(V ))). By (5), f−1(V ) = f−1(σ1σ2-Int(V )) ⊆
δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Int(V ))))) = δ(τ1, τ2)-Int(f
−1(V )) and so f−1(V )

is δ(τ1, τ2)-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). Then,

Y − V is (σ1, σ2)r-closed in Y . Thus by (7), we have X − f−1(V ) = f−1(Y − V ) =
δ(τ1, τ2)-Cl(f

−1(Y −V )) = X−δ(τ1, τ2)-Int(f
−1(V )) and hence x ∈ δ(τ1, τ2)-Int(f

−1(V )).
Then, there exists a δ(τ1, τ2)-open set U of X containing x such that U ⊆ f−1(V ). Thus,
f(U) ⊆ V . By Theorem 3 (4), f is almost δ(τ1, τ2)-continuous at x. This shows that f is
almost δ(τ1, τ2)-continuous.



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3737

Theorem 5. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is almost δ(τ1, τ2)-continuous;

(2) δ(τ1, τ2)-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open set V of Y ;

(3) δ(τ1, τ2)-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open set V of Y ;

(4) f−1(V ) ⊆ δ(τ1, τ2)-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)β-open set of Y . Then, σ1σ2-Cl(V ) is
(σ1, σ2)r-closed in Y . By Theorem 4 (7), f−1(σ1σ2-Cl(V )) is δ(τ1, τ2)-closed in X. Hence,
δ(τ1, τ2)-Cl(f

−1(V )) ⊆ δ(τ1, τ2)-Cl(f
−1(σ1σ2-Cl(V ))) = f−1(σ1σ2-Cl(V )).

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

By (3), we have δ(τ1, τ2)-Cl(f
−1(F )) ⊆ f−1(σ1σ2-Cl(F )) = f−1(F ) and hence f−1(F ) is

δ(τ1, τ2)-closed in X. By Theorem 4 (7), f is almost δ(τ1, τ2)-continuous.
(1) ⇒ (4): Let V be any (σ1, σ2)p-open set of Y . Then, σ1σ2-Int(σ1σ2-Cl(V )) is

(σ1, σ2)r-open in Y . By Theorem 4 (6), f−1(σ1σ2-Int(σ1σ2-Cl(V ))) is δ(τ1, τ2)-open in X.
Thus, f−1(σ1σ2-Int(σ1σ2-Cl(V ))) = δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))) and hence
f−1(V ) ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V ))) = δ(τ1, τ2)-Int(f

−1(σ1σ2-Int(σ1σ2-Cl(V )))).
(4) ⇒ (1) : Let V be any (σ1, σ2)r-open set of Y . Then, V = σ1σ2-Int(σ1σ2-Cl(V )) and

V is (σ1, σ2)p-open in Y . By (4), f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) =

δ(τ1, τ2)-Int(f
−1(V )). Therefore, f−1(V ) is δ(τ1, τ2)-open in X. By Theorem 4 (6), f is

almost δ(τ1, τ2)-continuous.

5. On weakly δ(τ1, τ2)-continuous functions

In this section, we introduce and investigate the concept of weakly δ(τ1, τ2)-continuous
functions. Furthermore, we discuss the relationships between almost δ(τ1, τ2)-continuous
functions and weakly δ(τ1, τ2)-continuous functions.

Definition 3. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly δ(τ1, τ2)-
continuous at x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists
a δ(τ1, τ2)-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is called weakly δ(τ1, τ2)-continuous if f is weakly δ(τ1, τ2)-
continuous at each point of X.

Remark 2. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following implication holds:

almost δ(τ1, τ2)-continuity ⇒ weak δ(τ1, τ2)-continuity.

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



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3738

Example 3. Let X = {a, b, c} with topologies τ1 = {∅, {a}, {b}, {a, b}, {a, c}, X} and
τ2 = {∅, {a}, {b}, {a, b}, X}. Let Y = {1, 2, 3} with topologies σ1 = {∅, {1}, {3}, {1, 3}, Y }
and σ2 = {∅, {1}, {3}, {1, 2}, {1, 3}, Y }. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is defined
as follows: f(a) = f(b) = 2 and f(c) = 1. Then f is weakly δ(τ1, τ2)-continuous, but f is
not almost δ(τ1, τ2)-continuous.

Theorem 6. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly δ(τ1, τ2)-continuous at
x ∈ X if and only if x ∈ δ(τ1, τ2)-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). Since f is
weakly δ(τ1, τ2)-continuous at x, so there exists a δ(τ1, τ2)-open set U of X containing x
such that f(U) ⊆ σ1σ2-Cl(V ). Therefore, x ∈ U ⊆ f−1(σ1σ2-Cl(V )). This implies that
x ∈ δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))).
Conversely let V be any σ1σ2-open set of Y containing f(x). By the hypothesis, we

obtain that x ∈ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(V ))). Then, there exists a δ(τ1, τ2)-open set U

of X containing x such that U ⊆ f−1(σ1σ2-Cl(V )). Thus, f(U) ⊆ σ1σ2-Cl(V ). This shows
that f is weakly δ(τ1, τ2)-continuous at x.

Theorem 7. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly δ(τ1, τ2)-continuous if and
only if f−1(V ) ⊆ δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y .

Proof. Let V be any σ1σ2-open set of Y and x ∈ f−1(V ). Since f is weakly δ(τ1, τ2)-
continuous, by Theorem 6, x ∈ δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))). Therefore, f−1(V ) ⊆
δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))).
Conversely, let x ∈ X and V be any σ1σ2-open set of Y containing f(x). By the

hypothesis, x ∈ f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(V ))). By Theorem 6, f is weakly

δ(τ1, τ2)-continuous at x. This shows that f is weakly δ(τ1, τ2)-continuous.

Theorem 8. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is weakly δ(τ1, τ2)-continuous;

(2) f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(V ))) for every σ1σ2-open set V of Y ;

(3) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(F ))) ⊆ f−1(F ) for every σ1σ2-closed F of Y ;

(4) δ(τ1, τ2)-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)) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))) for every subset B of

Y ;

(6) δ(τ1, τ2)-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every σ1σ2-open set V of Y ;

(7) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(F ))) ⊆ f−1(F ) for every (σ1, σ2)r-closed set F of Y ;



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3739

(8) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)β-open

set V of Y ;

(9) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)s-open

set V of Y ;

(10) δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open

set V of Y ;

(11) δ(τ1, τ2)-Cl(f
−1(V )) ⊆ f−1(σ1σ2-Cl(V )) for every (σ1, σ2)p-open set V of Y ;

(12) f−1(V ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(V ))) for every (σ1, σ2)p-open set V of Y .

Proof. (1) ⇒ (2): Let V be any σ1σ2-open set of Y . It follows from Theorem 7, we
obtain that f−1(V ) ⊆ δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))).
(2) ⇒ (3): Let F be any σ1σ2-closed set of Y . Then, Y − F is σ1σ2-open in Y . Thus

by (2), we have

X − f−1(F ) = f−1(Y − F ) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(Y − F )))

= δ(τ1, τ2)-Int(f
−1(Y − σ1σ2-Int(F )))

= δ(τ1, τ2)-Int(X − f−1(σ1σ2-Int(F )))

= X − δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(F )))

and hence δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(F ))) ⊆ f−1(F ).

(3) ⇒ (4): Let B be any subset of Y . Since σ1σ2-Cl(B) is σ1σ2-closed in Y and by
(3), δ(τ1, τ2)-Cl(f

−1(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 − δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(Y −B))))

= δ(τ1, τ2)-Int(X − f−1(σ1σ2-Int(σ1σ2-Cl(Y −B))))

= δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))).

Thus, f−1(σ1σ2-Int(B)) ⊆ δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(σ1σ2-Int(B)))).

(5) ⇒ (6): Let V be any σ1σ2-open set of Y . Suppose that x /∈ f−1(σ1σ2-Cl(V )).
Then, f(x) ̸∈ σ1σ2-Cl(V ). There exists a σ1σ2-open set U of Y containing f(x) such that
U ∩ V = ∅. Hence, σ1σ2-Cl(U) ∩ V = ∅. By (5), x ∈ f−1(U) = f−1(σ1σ2-Int(U)) ⊆
δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(σ1σ2-Int(U)))) = δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(U))). Then, there

exists a δ(τ1, τ2)-open set G of X such that x ∈ G ⊆ f−1(σ1σ2-Cl(U)). Therefore,

f−1(V ) ∩G ⊆ f−1(V ) ∩ f−1(σ1σ2-Cl(U)) = f−1(V ∩ σ1σ2-Cl(U)) = ∅.

Thus, x ̸∈ δ(τ1, τ2)-Cl(f
−1(V )) and hence δ(τ1, τ2)-Cl(f

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

Y . By (6), δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(F ))) ⊆ f−1(σ1σ2-Cl(σ1σ2-Int(F ))) = f−1(F ).



C. Prachanpol, C. Boonpok, C. Viriyapong / Eur. J. Pure Appl. Math, 17 (4) (2024), 3730-3742 3740

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

V ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V )))

and so σ1σ2-Cl(V ) is (σ1, σ2)r-closed. By (7), δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆

f−1(σ1σ2-Cl(V )).
(8) ⇒ (9): The proof is obvious since every (σ1, σ2)s-open set is (σ1, σ2)β-open.
(9)⇒ (10): Let V be any (σ1, σ2)p-open set of Y . Then, V ⊆ σ1σ2-Int(σ1σ2-Cl(V )) and

σ1σ2-Cl(V ) ⊆ σ1σ2-Cl(σ1σ2-Int(σ1σ2-Cl(V ))). Therefore, σ1σ2-Cl(V ) is (σ1, σ2)s-open in
Y . Thus by (9),

δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) = δ(τ1, τ2)-Cl(f

−1(σ1σ2-Int(σ1σ2-Cl(σ1σ2-Cl(V )))))

⊆ f−1(σ1σ2-Cl(σ1σ2-Cl(V )))

= f−1(σ1σ2-Cl(V )).

(10)⇒ (11): Let V be any (σ1, σ2)p-open set of Y . Then by (10), δ(τ1, τ2)-Cl(f
−1(V )) ⊆

δ(τ1, τ2)-Cl(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) ⊆ f−1(σ1σ2-Cl(V )).

(11) ⇒ (12): Let V be any (σ1, σ2)p-open set of Y . Thus by (11),

f−1(V ) ⊆ f−1(σ1σ2-Int(σ1σ2-Cl(V )))

= X − f−1(σ1σ2-Cl(Y − σ1σ2-Cl(V )))

⊆ X − δ(τ1, τ2)-Cl(f
−1(Y − σ1σ2-Cl(V )))

= δ(τ1, τ2)-Int(f
−1(σ1σ2-Cl(V ))).

(12) ⇒ (1): Let V be any σ1σ2-open set of Y . Then, V is (σ1, σ2)p-open in Y and
by (12), f−1(V ) ⊆ δ(τ1, τ2)-Int(f

−1(σ1σ2-Cl(V ))). It follows from Theorem 7 that f is
weakly δ(τ1, τ2)-continuous.

6. Conclusion

This paper deals with the concepts of δ(τ1, τ2)-continuous functions, almost δ(τ1, τ2)-
continuous functions, and weakly δ(τ1, τ2)-continuous functions. Moreover, some charac-
terizations and several properties concerning δ(τ1, τ2)-continuous functions, almost δ(τ1, τ2)-
continuous functions, and weakly δ(τ1, τ2)-continuous functions are obtained. For a func-
tion f : (X, τ1, τ2) → (Y, σ1, σ2), the following implications hold:

δ(τ1, τ2)-continuity ⇒ almost δ(τ1, τ2)-continuity ⇒ weak δ(τ1, τ2)-continuity.

Acknowledgements

This research project was financially supported by Mahasarakham University.



REFERENCES 3741

References

[1] S. P. Arya and R. Gupta. On strongly continuous mappings. Kyungpook Mathematical
Journal, 14:131–143, 1974.

[2] C. W. Baker. On super continuous functions. Bulletin of the Korean Mathematical
Society, 22(1):17–22, 1985.

[3] C. Boonpok. On characterizations of ⋆-hyperconnected ideal topological spaces. Jour-
nal of Mathematics, 2020:9387601, 2020.

[4] C. Boonpok. (τ1, τ2)δ-semicontinuous multifunctions. Heliyon, 6:e05367, 2020.

[5] C. Boonpok. On some spaces via topological ideals. Open Mathematics, 21:20230118,
2023.

[6] C. Boonpok. θ(⋆)-precontinuity. Mathematica, 65(1):31–42, 2023.

[7] C. Boonpok and C. Klanarong. On weakly (τ1, τ2)-continuous functions. European
Journal of Pure and Applied Mathematics, 17(1):416–425, 2024.

[8] C. Boonpok and P. Pue-on. Characterizations of almost (τ1, τ2)-continuous functions.
International Journal of Analysis and Applications, 22:33, 2024.

[9] C. Boonpok and N. Srisarakham. (τ1, τ2)-continuity for functions. Asia Pacific Jour-
nal of Mathematics, 11:21, 2024.

[10] C. Boonpok, C. Viriyapong, and M. Thongmoon. On upper and lower (τ1, τ2)-
precontinuous multifunctions. Journal of Mathematics and Computer Science,
18:282–293, 2018.

[11] D. Carnahan. Some properties of related to compactness in topological spaces. Ph. D.
Thesis, University of Arkansas, 1973.

[12] E. Ekici. On δ-semiopen sets and a generalization of functions. Boletim da Sociedade
Paranaense de Matemática, 23(1-2):73–84, 2005.

[13] E. Ekici and G. Navalagi. δ-semicontinuous functions. Mathematical Forum, 17:29–42,
2005.

[14] P. E. Long and L. L. Herrington. Strongly θ-continuous functions. Journal of the
Korean Mathematical Society, 18:21–28, 1981.

[15] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb. On precontinuous and
weak precontinuous mappings. Proceedings of the Mathematical and Physical Society
of Egypt, 53:47–53, 1982.

[16] B. M. Munshi and D. S. Bassan. Super-continuous mappings. Indian Journal of Pure
and Applied Mathematics, 13:229–239, 1982.



REFERENCES 3742

[17] T. Noiri. On δ-continuous functions. Journal of the Korean Mathematical Society,
16:161–166, 1980.

[18] T. Noiri. Supercontinuity and some strong forms of continuity. Indian Journal of
Pure and Applied Mathematics, 15:241–250, 1984.

[19] J. H. Park, B. Y. Lee, and M. J. Son. On δ-semiopen sets in topological spaces.
Journal of Indian Academy of Mathematics, 19(1):59–67, 1997.

[20] S. Raychaudhuri and N. Mukherjee. On δ-almost continuity and δ-preopen sets.
Bulletin of the Institute of Mathematics, Academia Sinica, 21:357–366, 1993.

[21] I. L. Reilly and M. K. Vamanamurthy. On super-continuous mappings. Indian Journal
of Pure and Applied Mathematics, 14:767–772, 1983.

[22] N. V. Veličko. H-closed topological spaces. American Mathematical Society Transla-
tions, 78(2):102–118, 1968.

[23] C. Viriyapong and C. Boonpok. (τ1, τ2)α-continuity for multifunctions. Journal of
Mathematics, 2020:6285763, 2020.

[24] N. Viriyapong, S. Sompong, and C. Boonpok. (τ1, τ2)-extremal disconnectedness in
bitopological spaces. International Journal of Mathematics and Computer Science,
19(3):855–860, 2024.

[25] S. Yüksel, A. Açikgöz, and T. Noiri. On δ-I-continuous functions. Turkish Joutnal of
Mathematics, 29:39–51, 2005.