EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2753-2762
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Characterizations of Faintly (τ1, τ2)-Continuous
Functions

Napassanan Srisarakham1, 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 deals with the concept of faintly (τ1, τ2)-continuous functions. Furthermore,
some characterizations of faintly (τ1, τ2)-continuous functions are investigated. The relationships
between faint (τ1, τ2)-continuity and other forms of (τ1, τ2)-continuity are considered.

2020 Mathematics Subject Classifications: 54C08, 54E55

Key Words and Phrases: (τ1, τ2)θ-open set, (τ1, τ2)θ-closed set, faintly (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. Semi-open sets [25],
preopen sets [27], α-open sets [29], β-open sets [22] and θ-open sets [38] play an impor-
tant role in researches of generalizations of continuity. Using these sets several authors
introduced and investigated various types of generalizations of continuity in topological
spaces. Viriyapong and Boonpok [40] studied some characterizations of (Λ, sp)-continuous
functions by utilizing the notions of (Λ, sp)-open sets and (Λ, sp)-closed sets due to Boon-
pok and Khampakdee [8]. Dungthaisong et al. [21] introduced and studied the concept
of g(m,n)-continuous functions. Duangphui et al. [20] introduced and investigated the no-

tion of (µ, µ′)(m,n)-continuous functions. Furthermore, several characterizations of almost
(Λ, p)-continuous functions, strongly θ(Λ, p)-continuous functions, almost strongly θ(Λ, p)-
continuous functions, θ(Λ, p)-continuous functions, weakly (Λ, b)-continuous functions,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5395

Email addresses: napassanan.sri@msu.ac.th (N. Srisarakham),
areeyuth.s@psu.ac.th (A. Sama-Ae), chawalit.b@msu.ac.th (C. Boonpok)

https://www.ejpam.com 2753 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 2753-2762 2754

(Λ, p(⋆))-continuous functions, θ(⋆)-precontinuous functions, ⋆-continuous functions, θ-
I -continuous functions, almost (g,m)-continuous functions, pairwise M -continuous func-
tions, almost quasi (τ1, τ2)-continuous functions and weakly quasi (τ1, τ2)-continuous func-
tions were presented in [35], [37], [9], [33], [12], [5], [7], [6], [3], [1], [2], [24] and [17], respec-
tively. Long and Herrington [26] introduced the notion of faintly continuous functions.
Moreover, some characterizations of faintly continuous functions were investigated in [28]
and [30], respectively. Three weak forms of faint continuity were introduced by Noiri and
Popa [31]. Nasef and Noiri [28] introduced and studied three strong forms of faint conti-
nuity under the names of strongly faint semi-continuity, strongly faint precontinuity and
strongly faint β-continuity. Jafari and Noiri [23] introduced and investigated the concept
of faintly α-continuous functions. Chananan et al. [15] introduced a new class of functions,
called faintly (m,µ)-continuous functions and established the relationships between faint
(m,µ)-continuity and other related generalized forms of (m,µ)-continuity. Noiri and Popa
[32] introduced the notion of faintly m-continuous functions as functions from a set X
satisfying some minimal conditions into a topological space and investigated several char-
acterizations of faintly m-continuous functions. Pue-on et al. [34] introduced the concept
of faintly (τ1, τ2)-continuous functions. In this paper, we investigate some characteriza-
tions of faintly (τ1, τ2)-continuous functions. We also discuss the relationships between
faintly (τ1, τ2)-continuous functions and other forms of (τ1, τ2)-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 [14] if
A = τ1-Cl(τ2-Cl(A)). The complement of a τ1τ2-closed set is called τ1τ2-open. A subset A
of a bitopological space (X, τ1, τ2) is said to be τ1τ2-clopen [14] if A is both τ1τ2-open and
τ1τ2-closed. Let A be a subset of a bitopological space (X, τ1, τ2). The intersection of all
τ1τ2-closed sets of X containing A is called the τ1τ2-closure [14] 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
[14] of A and is denoted by τ1τ2-Int(A).

Lemma 1. [14] 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. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 2753-2762 2755

(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 [39] (resp.
(τ1, τ2)s-open [4], (τ1, τ2)p-open [4], (τ1, τ2)β-open [4]) 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 called (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 [41] 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. Let A be a subset of a bitopological space (X, τ1, τ2). A point x ∈ X is
called a (τ1, τ2)θ-cluster point [39] of A if τ1τ2-Cl(U)∩A ̸= ∅ for every τ1τ2-open set U of X
containing x. The set of all (τ1, τ2)θ-cluster points of A is called the (τ1, τ2)θ-closure [39]
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 [39] 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 [39] of A and is denoted by (τ1, τ2)θ-Int(A).

3. Characterizations of faintly (τ1, τ2)-continuous functions

In this section, we investigate several characterizations of faintly (τ1, τ2)-continuous
functions.

Definition 1. [34] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called faintly (τ1, τ2)-continuous
at a point x ∈ X if for each (σ1, σ2)θ-open set V of Y containing f(x), there exists a τ1τ2-
open set U of X containing x such that f(U) ⊆ V . A function f : (X, τ1, τ2) → (Y, σ1, σ2)
is called faintly (τ1, τ2)-continuous if f has this property at every point of X.

Theorem 1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is faintly (τ1, τ2)-continuous at x ∈ X
if and only if for each (σ1, σ2)θ-open set V of Y containing f(x), x ∈ τ1τ2-Int(f

−1(V )).

Proof. Let x ∈ X and V be any (σ1, σ2)θ-open set of Y containing f(x). Then, there
exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V . Thus x ∈ U ⊆ f−1(V )
and hence x ∈ τ1τ2-Int(f

−1(V )).
Conversely, let V be any (σ1, σ2)θ-open set of Y containing f(x). By the hypothesis,

x ∈ τ1τ2-Int(f
−1(V )). Then, there exists a τ1τ2-open set U of X containing x such that

U ⊆ f−1(V ); hence f(U) ⊆ V . This shows that f is faintly (τ1, τ2)-continuous at x ∈ X.

Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-T2 [19] if for any pair
of distinct points x, y in X, there exist disjoint τ1τ2-open sets U and V of X containing x
and y, respectively.

Definition 2. A bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-T2 if for each distinct
points x, y ∈ X, there there exist (τ1, τ2)θ-open sets U and V of X containing x and y,
respectively, such that U ∩ V = ∅.

Theorem 2. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a faintly (τ1, τ2)-continuous injection and
(Y, σ1, σ2) is (σ1, σ2)θ-T2, then (X, τ1, τ2) is (τ1, τ2)-T2.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 2753-2762 2756

Proof. Let x, y be any distinct points of X. Then f(x) ̸= f(y). Since (Y, σ1, σ2)
is (σ1, σ2)θ-T2, there exist (σ1, σ2)θ-open sets U and V of Y containing f(x) and f(y),
respectively, such that U ∩ V = ∅. Since f is faintly (τ1, τ2)-continuous, there exist τ1τ2-
open sets G and W of X containing x and y, respectively, such that f(G) ⊆ U and
f(W ) ⊆ V . This implies that G ∩W = ∅. Thus, (X, τ1, τ2) is (τ1, τ2)-T2.

Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-compact [14] if every
cover of X by τ1τ2-open sets of X has a finite subcover. A subset K of X is said to be
τ1τ2-compact relative to (X, τ1, τ2) if every cover of K by τ1τ2-open sets of X has a finite
subcover.

Definition 3. A subset K of a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-compact
relative to (X, τ1, τ2) if every cover of K by (τ1, τ2)θ-open sets of X has a finite subcover. A
bitopological space (X, τ1, τ2) is said to be (τ1, τ2)θ-compact if the set X is (τ1, τ2)θ-compact
relative to (X, τ1, τ2).

Theorem 3. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a faintly (τ1, τ2)-continuous function and K
is τ1τ2-compact relative to (X, τ1, τ2), then f(K) is (σ1, σ2)θ-compact relative to (Y, σ1, σ2).

Proof. Let {Vγ : γ ∈ Γ} be any cover of f(K) by (σ1, σ2)θ-open sets of Y . For each
x ∈ K, there exists γ(x) ∈ Γ such that f(x) ∈ Vγ(x). Since f is faintly (τ1, τ2)-continuous,
there exist a τ1τ2-open set U(x) of X containing x such that f(U(x)) ⊆ Vγ(x). The family
{U(x) : x ∈ K} is a cover of K by τ1τ2-open sets of X. Since K is τ1τ2-compact relative
to (X, τ1, τ2), there exists a finite number of points, say, x1, x2, x3, ..., xn in K such that
K ⊆ ∪{U(xk) : xk ∈ K, 1 ≤ k ≤ n}. Thus,

f(K) ⊆ ∪{f(U(xk)) : xk ∈ K, 1 ≤ k ≤ n}
⊆ ∪{Vγ(xk) : xk ∈ K, 1 ≤ k ≤ n}.

This shows that f(K) is (σ1, σ2)θ-compact relative to (Y, σ1, σ2).

Recall that a bitopological space (X, τ1, τ2) is said to be τ1τ2-connected [14] if X cannot
be written as the union of two disjoint nonempty τ1τ2-open sets.

Lemma 2. [34] For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are
equivalent:

(1) f is faintly (τ1, τ2)-continuous;

(2) f−1(V ) is τ1τ2-open in X for each (σ1, σ2)θ-open set V of Y ;

(3) f−1(K) is τ1τ2-closed in X for each (σ1, σ2)θ-closed set K of Y ;

(4) for each x ∈ X and for each (σ1, σ2)θ-open set V of Y containing f(x), there exists
a τ1τ2-open set U of X containing x such that f(U) ⊆ V .

Theorem 4. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a faintly (τ1, τ2)-continuous surjection and
(X, τ1, τ2) is τ1τ2-connected, then (Y, σ1, σ2) is σ1σ2-connected.



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 2753-2762 2757

Proof. Assume that (Y, σ1, σ2) is not σ1σ2-connected. Then, there exist nonempty
σ1σ2-open sets V andW such that V ∩W = ∅ and V ∪W = Y . Thus, f−1(V )∩f−1(W ) = ∅
and f−1(V )∪f−1(W ) = X. Since f is surjective, f−1(V ) and f−1(W ) are nonempty. Since
V and W are σ1σ2-open and σ1σ2-closed, we have V and W are (σ1, σ2)θ-open sets of Y .
Since f is faintly (τ1, τ2)-continuous, by Lemma 2, f−1(V ) and f−1(W ) are τ1τ2-open in
X. Thus, (X, τ1, τ2) is not τ1τ2-connected. This is a contradiction and hence (Y, σ1, σ2) is
σ1σ2-connected.

The τ1τ2-frontier [13] of a subset A of a bitopological space (X, τ1, τ2), denoted by
τ1τ2-fr(A), is defined by

τ1τ2-fr(A) = τ1τ2-Cl(A) ∩ τ1τ2-Cl(X −A) = τ1τ2-Cl(A)− τ1τ2-Int(A).

Theorem 5. The set of all points x ∈ X at which a function f : (X, τ1, τ2) → (Y, σ1, σ2) is
not faintly (τ1, τ2)-continuous is identical with the union of the τ1τ2-frontier of the inverse
images of (σ1, σ2)θ-open sets of Y containing f(x).

Proof. Suppose that f is not faintly (τ1, τ2)-continuous at x ∈ X. Then, there exists a
(σ1, σ2)θ-open set V of Y containing f(x) such that f(U) is not contained in V for every
τ1τ2-open set U of X containing x. Then, U ∩ (X − f−1(V )) ̸= ∅ for every τ1τ2-open
set U of X containing x. Thus, x ∈ τ1τ2-Cl(X − f−1(V )). On the other hand, we have
x ∈ f−1(V ) ⊆ τ1τ2-Cl(f

−1(V )) and hence x ∈ τ1τ2-fr(A).
Conversely, suppose that f is faintly (τ1, τ2)-continuous at x ∈ X. Let V be any

(σ1, σ2)θ-open set of Y containing f(x). Then by Theorem 1, x ∈ τ1τ2-Int(f
−1(V )). Thus,

x ̸∈ τ1τ2-fr(f
−1(V )) for each (σ1, σ2)θ-open set V of Y containing f(x). This completes

the proof.

Definition 4. [36] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be slightly (τ1, τ2)-
continuous if for each x ∈ X and each σ1σ2-clopen set V of Y containing f(x), there
exists a τ1τ2-open set U of X containing x such that f(U) ⊆ V .

Theorem 6. If f : (X, τ1, τ2) → (Y, σ1, σ2) is faintly (τ1, τ2)-continuous, then f is slightly
(τ1, τ2)-continuous.

Proof. Let x ∈ X and V be any σ1σ2-clopen set of Y containing f(x). Then, V is
(σ1, σ2)θ-open in Y . Since f is faintly (τ1, τ2)-continuous, there exists a τ1τ2-open set U
of X containing x such that f(U) ⊆ V . This shows that f is slightly (τ1, τ2)-continuous.

4. On faint (τ1, τ2)-continuity and other forms of (τ1, τ2)-continuity

In this paper, we investigate the relationships between faintly (τ1, τ2)-continuous func-
tions and other forms of (τ1, τ2)-continuous functions.

Definition 5. [10] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-
continuous at a point x ∈ X if for each τ1τ2-open set V of Y containing f(x), there



N. Srisarakham, A. Sama-Ae, C. Boonpok / Eur. J. Pure Appl. Math, 17 (4) (2024), 2753-2762 2758

exists a τ1τ2-open set U of X containing x such that f(U) ⊆ σ1σ2-Cl(V ). A function
f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be weakly (τ1, τ2)-continuous if f has this property
at each point of X.

Theorem 7. If f : (X, τ1, τ2) → (Y, σ1, σ2) is weakly (τ1, τ2)-continuous, then f is faintly
(τ1, τ2)-continuous.

Proof. Let x ∈ X and V be any (σ1, σ2)θ-open set of Y containing f(x). There
exists a σ1σ2-open set W of Y such that f(x) ∈ W ⊆ σ1σ2-Cl(W ) ⊆ V . Since f is
weakly (τ1, τ2)-continuous, there exists a τ1τ2-open set U of X containing x such that
f(U) ⊆ σ1σ2-Cl(W ) ⊆ V . Thus, f is faintly (τ1, τ2)-continuous.

Definition 6. [13] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is called (τ1, τ2)-continuous at
a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists a τ1τ2-open
set U of X containing x such that f(U) ⊆ V . A function f : (X, τ1, τ2) → (Y, σ1, σ2) is
called (τ1, τ2)-continuous if f has this property at each point of X.

Recall that a bitopological space (X, τ1, τ2) is said to be (τ1, τ2)-regular [16] if for each
τ1τ2-closed set F and each point x ∈ X − F , there exist disjoint τ1τ2-open sets U and V
such that x ∈ U and F ⊆ V .

Lemma 3. [13] For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiv-
alent:

(1) f is (τ1, τ2)-continuous;

(2) f−1(V ) is τ1τ2-open in X for every σ1σ2-open set V of Y ;

(3) f(τ1τ2-Cl(A)) ⊆ σ1σ2-Cl(f(A)) for every subset A of X;

(4) τ1τ2-Cl(f
−1(B)) ⊆ f−1(σ1σ2-Cl(B)) for every subset B of Y ;

(5) f−1(σ1σ2-Int(B)) ⊆ τ1τ2-Int(f
−1(B)) for every subset B of Y ;

(6) f−1(K) is τ1τ2-closed in X for every σ1σ2-closed set K of Y .

Theorem 8. If f : (X, τ1, τ2) → (Y, σ1, σ2) is faintly (τ1, τ2)-continuous and (Y, σ1, σ2) is
a (σ1, σ2)-regular space, then f is (τ1, τ2)-continuous.

Proof. Let V be any σ1σ2-open set of Y . Since (Y, σ1, σ2) is a (σ1, σ2)-regular space, V
is (σ1, σ2)θ-open in Y . Since f is faintly (τ1, τ2)-continuous, by Lemma 2 we have f−1(V )
is τ1τ2-open in X and hence by Lemma 3, f is (τ1, τ2)-continuous.

Definition 7. [11] A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be almost (τ1, τ2)-
continuous at a point x ∈ X if for each σ1σ2-open set V of Y containing f(x), there exists
a τ1τ2-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 said to be almost (τ1, τ2)-continuous if f has this property
at each point of X.



REFERENCES 2759

Lemma 4. [11] For a function (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equiv-
alent:

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

(2) x ∈ τ1τ2-Int(f
−1(σ1σ2-Int(σ1σ2-Cl(V )))) for every σ1σ2-open set V of Y containing

f(x);

(3) x ∈ τ1τ2-Int(f
−1(V )) for every (σ1, σ2)r-open set V of Y containing f(x);

(4) for each (σ1, σ2)r-open set V of Y containing f(x), there exists a τ1τ2-open set U of
X containing x such that f(U) ⊆ V .

Recall that a bitopological space (X, τ1, τ2) is said to be almost (τ1, τ2)-regular [18] if
for each (τ1, τ2)r-closed set F and each x ̸∈ F , there exist disjoint τ1τ2-open sets U and V
such that x ∈ U and F ⊆ V .

Lemma 5. Let (X, τ1, τ2) be an almost (τ1, τ2)-regular space. Then, every (τ1, τ2)r-open
set is (τ1, τ2)θ-open.

Theorem 9. If f : (X, τ1, τ2) → (Y, σ1, σ2) is faintly (τ1, τ2)-continuous and (Y, σ1, σ2) is
almost (σ1, σ2)-regular, then f is almost (τ1, τ2)-continuous.

Proof. Let x ∈ X and V be any (σ1, σ2)r-open set of Y containing f(x). Then by
Lemma 5, V is (σ1, σ2)θ-open in Y . Since f is faintly (τ1, τ2)-continuous, there exists a
τ1τ2-open set U of X containing x such that f(U) ⊆ V . It follows from Lemma 4 that f
is almost (τ1, τ2)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.

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