EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 300-309
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Almost strong θ(Λ, p)-continuity for functions

Chawalit Boonpok1, Jeeranunt Khampakdee1,∗

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

Abstract. Our main purpose is to introduce the concept of almost strongly θ(Λ, p)-continuous
functions. Moreover, some characterizations of almost strongly θ(Λ, p)-continuous functions are
considered.

2020 Mathematics Subject Classifications: 54A05, 54C08

Key Words and Phrases: θ(Λ, p)-open set, almost strongly θ(Λ, p)-continuous function

1. Introduction

The notion of θ-continuous functions was introduced by Fomin [10]. Noiri [20] studied
some properties of θ-continuous functions. Arya and Bhamini [1] introduced the notion
of θ-semi-continuous functions. Noiri [22] investigated several characterizations of θ-semi-
continuous functions. Moreover, Jafari and Noiri [15] obtained some properties of θ-semi-
continuous functions. Di Maio and Noiri [18] introduced the concept of quasi-irresolute
functions. It is shown in [8] that a function is quasi-irresolute if and only if it is θ-irresolute.
Noiri [24] introduced and investigated the notion of θ-preirresolute functions. The notion
of weakly β-irresolute functions has been defined and studied in [25]. These four classes of
functions have properties similar to the class of θ-continuous functions. In 1980, Noiri [21]
introduced the notion of strongly θ-continuous functions. Long et al. [17] studied some
properties of strongly θ-continuous functions. In 1998, Jafari and Noiri [12] introduced and
studied the concept of strongly θ-semi-continuous functions. Moreover, Jafari and Noiri
[14] studied the notion of strongly sober θ-continuous functions. Noiri [23] introduced the
concept of θ-precontinuous functions. In 2002, Noiri and Popa [27] introduced and investi-
gated the notion of strongly θ-β-continuous functions. In 2005, Noiri and Popa [29] defined
a new notion of strongly θ-M -continuous functions as functions from a set satisfying some
minimal conditions into a set satisfying some minimal conditions. Noiri and Kang [26]
introduced and studied the notion of almost strongly θ-continuous functions. Jafari and
Noiri [16] investigated some properties of almost strongly θ-continuous functions. Beceren

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4975

Email addresses: chawalit.b@msu.ac.th (C. Boonpok), jeeranunt.k@msu.ac.th (J. Khampakdee)

https://www.ejpam.com 300 © 2024 EJPAM All rights reserved.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 301

et al. [2] introduced and studied the concept of almost strongly θ-semi-continuous func-
tions. Furthermore, Jafari and Noiri [13] investigated several characterizations of almost
strongly θ-semi-continuous functions. Dube and Chauhan [9] introduced the notion of
strongly closure semi-continuous functions which are equivalent to almost strongly θ-semi-
continuous functions. These classes of functions have properties similar to the class of
θ-continuous functions. Noiri and Popa [28] introduced and studied the notion of almost
strongly θ-m-continuous functions as functions from a set satisfying some minimal condi-
tions into a topological space. In [7], the present authors introduced and investigated the
concept of almost (Λ, s)-continuous functions. The notions of (Λ, sp)-open sets, s(Λ, sp)-
open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets
were studied in [4]. Viriyapong and Boonpok [31] investigated some characterizations of
(Λ, sp)-continuous functions. Furthermore, several characterizations of pairwise almost
M -continuous functions were established in [3]. In this paper, we introduce the concept
of almost strongly θ(Λ, p)-continuous functions. In particular, several characterizations of
almost strongly θ(Λ, p)-continuous functions are discussed.

2. Preliminaries

Throughout the present paper, spaces (X, τ) and (Y, σ) (or simply X and Y ) always
mean topological spaces on which no separation axioms are assumed unless explicitly
stated. For a subset A of a topological space (X, τ), Cl(A) and Int(A), represent the
closure and the interior of A, respectively. A subset A of a topological space (X, τ) is said
to be preopen [19] if A ⊆ Int(Cl(A)). The complement of a preopen set is called preclosed.
The family of all preopen sets of a topological space (X, τ) is denoted by PO(X, τ). A
subset Λp(A) [11] is defined as follows: Λp(A) = ∩{U | A ⊆ U,U ∈ PO(X, τ)}. A subset A
of a topological space (X, τ) is called a Λp-set [6] (pre-Λ-set [11]) if A = Λp(A). A subset
A of a topological space (X, τ) is called (Λ, p)-closed [6] if A = T ∩C, where T is a Λp-set
and C is a preclosed set. The complement of a (Λ, p)-closed set is called (Λ, p)-open. The
family of all (Λ, p)-open (resp. (Λ, p)-closed) sets in a topological space (X, τ) is denoted
by ΛpO(X, τ) (resp. ΛpC(X, τ)). Let A be a subset of a topological space (X, τ). A point
x ∈ X is called a (Λ, p)-cluster point [6] of A if A∩U ̸= ∅ for every (Λ, p)-open set U of X
containing x. The set of all (Λ, p)-cluster points of A is called the (Λ, p)-closure [6] of A
and is denoted by A(Λ,p). The union of all (Λ, p)-open sets of X contained in A is called
the (Λ, p)-interior [6] of A and is denoted by A(Λ,p). The θ(Λ, p)-closure [6] of A, Aθ(Λ,p),
is defined as follows:

Aθ(Λ,p) = {x ∈ X | A ∩ U (Λ,p) ̸= ∅ for each (Λ, p)-open set U containing x}.

A subset A of a topological space (X, τ) is called θ(Λ, p)-closed [6] if A = Aθ(Λ,p). The
complement of a θ(Λ, p)-closed set is said to be θ(Λ, p)-open. Let A be a subset of a
topological space (X, τ). A point x ∈ X is called a θ(Λ, p)-interior point [30] of A if
x ∈ U ⊆ U (Λ,p) ⊆ A for some U ∈ ΛpO(X, τ). The set of all θ(Λ, p)-interior points of A is
called the θ(Λ, p)-interior [30] of A and is denoted by Aθ(Λ,p).



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 302

Lemma 1. [30] For subsets A and B of a topological space (X, τ), the following properties
hold:

(1) X −Aθ(Λ,p) = [X −A]θ(Λ,p) and X −Aθ(Λ,p) = [X −A]θ(Λ,p).

(2) A is θ(Λ, p)-open if and only if A = Aθ(Λ,p).

(3) A ⊆ A(Λ,p) ⊆ Aθ(Λ,p) and Aθ(Λ,p) ⊆ A(Λ,p) ⊆ A.

(4) If A ⊆ B, then Aθ(Λ,p) ⊆ Bθ(Λ,p) and Aθ(Λ,p) ⊆ Bθ(Λ,p).

(5) If A is (Λ, p)-open, then A(Λ,p) = Aθ(Λ,p).

A subset A of a topological space (X, τ) is said to be s(Λ, p)-open [6] (resp. p(Λ, p)-
open [6], β(Λ, p)-open [6], α(Λ, p)-open [32], r(Λ, p)-open [6]) if A ⊆ [A(Λ,p)]

(Λ,p) (resp.

A ⊆ [A(Λ,p)](Λ,p), A ⊆ [[A(Λ,p)](Λ,p)]
(Λ,p), A ⊆ [[A(Λ,p)]

(Λ,p)](Λ,p), A = [A(Λ,p)](Λ,p)). The
family of all s(Λ, p)-open (resp. p(Λ, p)-open, β(Λ, p)-open, α(Λ, p)-open, r(Λ, p)-open)
sets in a topological space (X, τ) is denoted by s(Λ, p)O(X, τ) (resp. p(Λ, p)O(X, τ),
β(Λ, p)O(X, τ), α(Λ, p)O(X, τ), r(Λ, p)O(X, τ)). The union of all s(Λ, p)-open (resp.
p(Λ, p)-open, α(Λ, p)-open) sets of X contained in A is called the s(Λ, p)-interior (resp.
p(Λ, p)-interior, α(Λ, p)-interior) of A and is denoted by As(Λ,p) (resp. Ap(Λ,p), Aα(Λ,p)).
The complement of a s(Λ, p)-open (resp. p(Λ, p)-open, β(Λ, p)-open, α(Λ, p)-open, r(Λ, p)-
open) set is called s(Λ, p)-closed (resp. p(Λ, p)-closed, β(Λ, p)-closed, α(Λ, p)-closed, r(Λ, p)-
closed). The family of all s(Λ, p)-closed (resp. p(Λ, p)-closed, β(Λ, p)-closed, α(Λ, p)-
closed, r(Λ, p)-closed) sets in a topological space (X, τ) is denoted by s(Λ, p)C(X, τ)
(resp. p(Λ, p)C(X, τ), β(Λ, p)C(X, τ), α(Λ, p)C(X, τ), r(Λ, p)C(X, τ)). The intersection
of all s(Λ, p)-closed (resp. p(Λ, p)-closed, α(Λ, p)-closed) sets of X containing A is called
the s(Λ, p)-closure (resp. p(Λ, p)-closure, α(Λ, p)-closure) of A and is denoted by As(Λ,p)

(resp. Ap(Λ,p), Aα(Λ,p)). Let A be a subset of a topological space (X, τ). A point x of X is
called a δ(Λ, p)-cluster point [5] of A if A ∩ [V (Λ,p)](Λ,p) ̸= ∅ for every (Λ, p)-open set V of
X containing x. The set of all δ(Λ, p)-cluster points of A is called the δ(Λ, p)-closure [5]
of A and is denoted by Aδ(Λ,p). If A = Aδ(Λ,p), then A is said to be δ(Λ, p)-closed [5]. The
complement of a δ(Λ, p)-closed set is said to be δ(Λ, p)-open. The union of all δ(Λ, p)-open
sets of X contained in A is called the δ(Λ, p)-interior [5] of A and is denoted by Aδ(Λ,p).

3. On almost strongly θ(Λ, p)-continuous functions

We begin this section by introducing the concept of almost strongly θ(Λ, p)-continuous
functions.

Definition 1. A function f : (X, τ) → (Y, σ) is said to be almost strongly θ(Λ, p)-
continuous functions at x ∈ X if for each (Λ, p)-open set V of Y containing f(x), there
exists a (Λ, p)-open set U of X containing x such that f(U (Λ,p)) ⊆ V s(Λ,p). A function
f : (X, τ) → (Y, σ) is said to be almost strongly θ(Λ, p)-continuous if f has the property
at each point x ∈ X.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 303

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

(1) f is almost strongly θ(Λ, p)-continuous;

(2) f−1(V ) is θ(Λ, p)-open in X for every r(Λ, p)-open set V of Y ;

(3) f−1(K) is θ(Λ, p)-closed in X for every r(Λ, p)-closed set K of Y ;

(4) for each x ∈ X and each r(Λ, p)-open set V of Y containing f(x),
there exists a (Λ, p)-open set U of X containing x such that f(U (Λ,p)) ⊆ V ;

(5) f−1(V ) is θ(Λ, p)-open in X for every δ(Λ, p)-open set V of Y ;

(6) f−1(K) is θ(Λ, p)-closed in X for every δ(Λ, p)-closed set K of Y ;

(7) f(Aθ(Λ,p)) ⊆ [f(A)]δ(Λ,p) for every subset A of X;

(8) [f−1(B)]θ(Λ,p) ⊆ f−1(Bδ(Λ,p)) for every subset B of Y ;

(9) f−1(Bδ(Λ,p)) ⊆ [f−1(B)]θ(Λ,p) for every subset B of Y ;

(10) f−1(V ) ⊆ [f−1(V s(Λ,p))]θ(Λ,p) for every (Λ, p)-open set V of Y .

Proof. (1) ⇒ (2): Let V be any r(Λ, p)-open set of Y and x ∈ f−1(V ). Since f
is almost strongly θ(Λ, p)-continuous, there exists a (Λ, p)-open set U of X containing x
such that f(U (Λ,p)) ⊆ V s(Λ,p) = V . Thus, x ∈ U ⊆ U (Λ,p) ⊆ f−1(V ) which implies that
x ∈ [f−1(V )]θ(Λ,p). This shows that f−1(V ) ⊆ [f−1(V )]θ(Λ,p). By Lemma 1, f−1(V ) =
[f−1(V )]θ(Λ,p) and hence f−1(V ) is θ(Λ, p)-open.

(2) ⇒ (3): Let K be any r(Λ, p)-closed set of Y . By (2), we have

f−1(K) = X − f−1(Y −K)

= X − [f−1(Y −K)]θ(Λ,p)

= X − [X − f−1(K)]θ(Λ,p)

= [f−1(K)]θ(Λ,p).

Thus, f−1(K) is θ(Λ, p)-closed in X.
(3) ⇒ (4): Let x ∈ X and V be any r(Λ, p)-open set of Y containing f(x). By (3),

X − f−1(V ) = f−1(Y − V ) = [f−1(Y − V )]θ(Λ,p) = X − [f−1(V )]θ(Λ,p). This implies that
f−1(V ) = [f−1(V )]θ(Λ,p). Then, there exists a (Λ, p)-open set U of X containing x such

that U (Λ,p) ⊆ f−1(V ); hence f(U (Λ,p)) ⊆ V .
(4) ⇒ (5): Let V be any δ(Λ, p)-open set of Y and x ∈ f−1(V ). There exists a

r(Λ, p)-open set G of Y such that f(x) ∈ G ⊆ V . By (4), there exists a (Λ, p)-open
set U of X containing x such that f(U (Λ,p)) ⊆ G. Thus, x ∈ U ⊆ U (Λ,p) ⊆ f−1(V )
which implies that x ∈ [f−1(V )]θ(Λ,p). Therefore, f−1(V ) ⊆ [f−1(V )]θ(Λ,p) and hence
f−1(V ) = [f−1(V )]θ(Λ,p), by Lemma 1, f−1(V ) is θ(Λ, p)-open.



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 304

(5) ⇒ (6): Let K be any δ(Λ, p)-closed set of Y . By (5), we have

f−1(K) = X − f−1(Y −K)

= X − [f−1(Y −K)]θ(Λ,p)

= [f−1(K)]θ(Λ,p).

Thus, f−1(K) = [f−1(K)]θ(Λ,p) and hence f−1(K) is θ(Λ, p)-closed.
(6) ⇒ (7): Let A be any subset of X. Since [f(A)]δ(Λ,p) is δ(Λ, p)-closed in Y , by

(6) we have f−1([f(A)]δ(Λ,p)) = [f−1([f(A)]δ(Λ,p))]θ(Λ,p). Let x ̸∈ f−1([f(A)]δ(Λ,p)). Then,
there exists a (Λ, p)-open set U of X containing x such that U (Λ,p)∩f−1([f(A)]δ(Λ,p)) = ∅.
This implies that U (Λ,p) ∩A = ∅. Thus, x ̸∈ Aθ(Λ,p) and hence f(Aθ(Λ,p)) ⊆ [f(A)]δ(Λ,p).

(7) ⇒ (8): Let B be any subset of Y . Then, by (7) we have f([f−1(B)]θ(Λ,p)) ⊆ Bδ(Λ,p)

and hence [f−1(B)]θ(Λ,p) ⊆ f−1(Bδ(Λ,p)).
(8) ⇒ (9): Let B be any subset of Y . Let x ∈ f−1(Bδ(Λ,p)). Then, f(x) ∈ Bδ(Λ,p)

and f(x) ̸∈ Y − Bδ(Λ,p) = [Y − B]δ(Λ,p). Therefore, x ̸∈ f−1([Y − B]δ(Λ,p)). By (8),

x ̸∈ [f−1(Y −B)]θ(Λ,p). There exists a (Λ, p)-open set U of X containing x such that

x ∈ U ⊆ U (Λ,p) ⊆ f−1(B).

Thus, x ∈ [f−1(B)]θ(Λ,p) and hence f−1(Bδ(Λ,p)) ⊆ [f−1(B)]θ(Λ,p).
(9) ⇒ (10): Let V be any (Λ, p)-open set of Y . Then, we have

V ⊆ [V (Λ,p)](Λ,p) ⊆ [V s(Λ,p)]δ(Λ,p)

and by (9), f−1(V ) ⊆ f−1([V s(Λ,p)]δ(Λ,p)) ⊆ [f−1(V s(Λ,p))]θ(Λ,p).
(10) ⇒ (1): Let x ∈ X and V be any (Λ, p)-open set of Y containing f(x). Then,

x ∈ f−1(V ) ⊆ [f−1(V s(Λ,p))]θ(Λ,p). Then, there exists a (Λ, p)-open set U of X containing

x such that x ∈ U ⊆ U (Λ,p) ⊆ f−1(V s(Λ,p)) which implies that f(U (Λ,p)) ⊆ V s(Λ,p). Thus,
f is almost strongly θ(Λ, p)-continuous.

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

(1) f is almost strongly θ(Λ, p)-continuous;

(2) [f−1([K(Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(K) for every (Λ, p)-closed set K of Y ;

(3) [f−1([[B(Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(B(Λ,p)) for every subset B of Y ;

(4) f−1(B(Λ,p)) ⊆ [f−1([[B(Λ,p)]
(Λ,p)](Λ,p))]θ(Λ,p) for every subset B of Y .

Proof. (1) ⇒ (2): Let K be any (Λ, p)-closed set of Y . Then, Y −K is (Λ, p)-open in
Y . Thus, by Theorem 1 and Lemma 1, we have

X − f−1(K) = f−1(Y −K)

⊆ [f−1([[Y −K](Λ,p)](Λ,p))]θ(Λ,p)



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 305

= [X − f−1([K(Λ,p)]
(Λ,p))]θ(Λ,p)

= X − [f−1([K(Λ,p)]
(Λ,p))]θ(Λ,p)

and hence [f−1([K(Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(K).

(2) ⇒ (3): Let B be any subset of Y . Then, B(Λ,p) is (Λ, p)-closed in Y and by (2),
[f−1([[B(Λ,p)](Λ,p)]

(Λ,p))]θ(Λ,p) ⊆ f−1(B(Λ,p)).
(3) ⇒ (4): Let B be any subset of Y . Then, we have

f−1(B(Λ,p)) = X − f−1([Y −B](Λ,p))

⊆ X − [f−1([[[Y −B](Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p)

= [f−1([[B(Λ,p)]
(Λ,p)](Λ,p))]θ(Λ,p)

and hence f−1(B(Λ,p)) ⊆ [f−1([[B(Λ,p)]
(Λ,p)](Λ,p))]θ(Λ,p).

(4) ⇒ (1): Let V be any r(Λ, p)-open set of Y . By (4), f−1(V ) ⊆ [f−1(V )]θ(Λ,p) and
hence f−1(V ) = [f−1(V )]θ(Λ,p). Thus, f−1(V ) is θ(Λ, p)-open and by Theorem 1, f is
almost strongly θ(Λ, p)-continuous.

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

(1) f is almost strongly θ(Λ, p)-continuous;

(2) [f−1(V )]θ(Λ,p) ⊆ f−1(V (Λ,p)) for every β(Λ, p)-open set V of Y ;

(3) [f−1(V )]θ(Λ,p) ⊆ f−1(V (Λ,p)) for every s(Λ, p)-open set V of Y ;

(4) f−1(V ) ⊆ [f−1([V (Λ,p)](Λ,p))]θ(Λ,p) for every p(Λ, p)-open set V of Y .

Proof. (1) ⇒ (2): Let V be any β(Λ, p)-open set of Y . Then, V (Λ,p) is r(Λ, p)-closed.
Since f is almost strongly θ(Λ, p)-continuous, by Theorem 2 we have

[f−1(V )]θ(Λ,p) ⊆ [f−1([[V (Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(V (Λ,p))

and hence [f−1(V )]θ(Λ,p) ⊆ f−1(V (Λ,p)).
(2) ⇒ (3): This is obvious since s(Λ, p)O(X, τ) ⊆ β(Λ, p)O(X, τ).
(3) ⇒ (4): Let V be any p(Λ, p)-open set of Y . Then, Y − V is p(Λ, p)-closed in Y

and hence [[Y − V ](Λ,p)]
(Λ,p) ⊆ Y − V . Since [[Y − V ](Λ,p)]

(Λ,p) is r(Λ, p)-closed, we have

[[Y − V ](Λ,p)]
(Λ,p) is s(Λ, p)-open in Y . Then by (3),

[f−1([[Y − V ](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1([[Y − V ](Λ,p)]

(Λ,p)) ⊆ f−1(Y − V ).

Thus,

f−1(V ) ⊆ X − [f−1([[Y − V ](Λ,p)]
(Λ,p))]θ(Λ,p)

= X − [X − f−1([V (Λ,p)](Λ,p))]
θ(Λ,p)



C. Boonpok, J. Khampakdee / Eur. J. Pure Appl. Math, 17 (1) (2024), 300-309 306

= [f−1([V (Λ,p)](Λ,p))]θ(Λ,p).

(4) ⇒ (1): Let V be any r(Λ, p)-open set of Y . Then, V is p(Λ, p)-open and

f−1(V ) ⊆ [f−1([V (Λ,p)](Λ,p))]θ(Λ,p) = [f−1(V )]θ(Λ,p).

Thus, f−1(V ) = [f−1(V )]θ(Λ,p) and by Lemma 1, f−1(V ) is θ(Λ, p)-open in X. It follows
from Theorem 1 that f is almost strongly θ(Λ, p)-continuous.

Lemma 2. For a topological space (X, τ), the following properties hold:

(1) V α(Λ,p) = V (Λ,p) for every V ∈ β(Λ, p)O(X, τ);

(2) V p(Λ,p) = V (Λ,p) for every V ∈ s(Λ, p)O(X, τ);

(3) V s(Λ,p) = [V (Λ,p)](Λ,p) for every V ∈ p(Λ, p)O(X, τ).

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

(1) f is almost strongly θ(Λ, p)-continuous;

(2) [f−1(V )]θ(Λ,p) ⊆ f−1(V α(Λ,p)) for every β(Λ, p)-open set V of Y ;

(3) [f−1(V )]θ(Λ,p) ⊆ f−1(V p(Λ,p)) for every s(Λ, p)-open set V of Y ;

(4) f−1(V ) ⊆ [f−1(V s(Λ,p))]θ(Λ,p) for every p(Λ, p)-open set V of Y .

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

(1) f is almost strongly θ(Λ, p)-continuous;

(2) [f−1([[Bδ(Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(Bδ(Λ,p)) for every subset B of Y ;

(3) [f−1([[B(Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(Bδ(Λ,p)) for every subset B of Y ;

(4) [f−1([[V (Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(V (Λ,p)) for every (Λ, p)-open set V of Y ;

(5) [f−1([[V (Λ,p)](Λ,p)]
(Λ,p))]θ(Λ,p) ⊆ f−1(V (Λ,p)) for every p(Λ, p)-open set V of Y .

Proof. (1) ⇒ (2): Let B be any subset of Y . Then, Bδ(Λ,p) is (Λ, p)-closed in Y . By
Theorem 2, [f−1([[Bδ(Λ,p)](Λ,p)]

(Λ,p))]θ(Λ,p) ⊆ f−1(Bδ(Λ,p)).

(2) ⇒ (3): This is obvious since B(Λ,p) ⊆ Bδ(Λ,p) for every subset B of Y .
(3) ⇒ (4): This is obvious since V (Λ,p) = V δ(Λ,p) for every (Λ, p)-open set V of Y .
(4) ⇒ (5): Let V be any p(Λ, p)-open set of Y . Then, we have V ⊆ [V (Λ,p)](Λ,p) and

V (Λ,p) = [[V (Λ,p)](Λ,p)]
(Λ,p). Thus, by (4), [f−1([[V (Λ,p)](Λ,p)]

(Λ,p))]θ(Λ,p) ⊆ f−1(V (Λ,p)).
(5) ⇒ (1): Let K be any r(Λ, p)-closed set of Y . Then, we have K(Λ,p) is p(Λ, p)-open

in Y and by (5),

[f−1(K)]θ(Λ,p) = [f−1([K(Λ,p)]
(Λ,p))]θ(Λ,p)



REFERENCES 307

= [f−1([[[K(Λ,p)]
(Λ,p)](Λ,p)]

(Λ,p))]θ(Λ,p)

⊆ f−1([K(Λ,p)]
(Λ,p))

= f−1(K).

Thus, [f−1(K)]θ(Λ,p) = f−1(K) and by Lemma 1, f−1(K) is θ(Λ, p)-closed in X. By
Theorem 1, f is almost strongly θ(Λ, p)-continuous.

Acknowledgements

This research project was financially supported by Mahasarakham University.

References

[1] S. P. Arya and M. P. Bhamini. Some weaker forms of semi-continuous functions.
Ganita, 33:124–134, 1982.

[2] Y. Beceren, S. Yuksel, and E. Hatir. On almost strongly θ-semi-continuous functions.
Bulletin of the Calcutta Mathematical Society, 87:329–334, 1995.

[3] C. Boonpok. M -continuous functions on biminimal structure spaces. Far East Journal
of Mathematical Sciences, 43(1):41–58, 2010.

[4] C. Boonpok and J. Khampakdee. (Λ, sp)-open sets in topological spaces. European
Journal of Pure and Applied Mathematics, 15(2):572–588, 2022.

[5] C. Boonpok and M. Thongmoon. δp(Λ, p)-open sets in topological spaces. European
Journal of Pure and Applied Mathematics, 16(3):1533–1542, 2023.

[6] C. Boonpok and C. Viriyapong. On (Λ, p)-closed sets and the related notions in
topological spaces. European Journal of Pure and Applied Mathematics, 15(2):415–
436, 2022.

[7] C. Boonpok and C. Viriyapong. On some forms of closed sets and related topics.
European Journal of Pure and Applied Mathematics, 16(1):336–362, 2023.

[8] F. Cammaroto and T. Noiri. Almost irresolute functions. Indian Journal of Pure and
Applied Mathematics, 20:472–482, 1989.

[9] K. K. Dube and S. S. Chauhan. Strongly closure semi-continuous mappings. The
Journal of the Indian Academy of Mathematics, 19:139–147, 1997.

[10] S. Fomin. Extensions of topological spaces. Doklady Akademii Nauk SSSR, 32:114–
116, 1941.

[11] M. Ganster, S. Jafari, and T. Noiri. On pre-Λ-sets and pre-V -sets. Acta Mathematica
Hungarica, 95:337–343, 2002.



REFERENCES 308

[12] S. Jafari and T. Noiri. Strongly θ-semi-continuous functions. Indian Journal of Pure
and Applied Mathematics, 29:1195–1201, 1998.

[13] S. Jafari and T. Noiri. On almost strongly θ-semi-continuous functions. Acta Math-
ematica Hungarica, 85:167–173, 1999.

[14] S. Jafari and T. Noiri. Strongly sober θ-continuous functions. Journal of Pure Math-
ematics, 16:9–17, 1999.

[15] S. Jafari and T. Noiri. Properties of θ-semi-continuous functions. Journal of Institute
of Mathematics and Computer Sciences. Mathematics Series, 13:123–128, 2000.

[16] S. Jafari and T. Noiri. Some properties of almost strongly θ-continuous functions.
Memoirs of the Faculty of Science Kochi University Series A Mathematics, 25:71–76,
2004.

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

[18] G. Di Maio and T. Noiri. Weak and strong forms of irresolute functions. Rendiconti
del Circolo Matematico di Palermo (2) Supplemento, 18:255–273, 1988.

[19] 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.

[20] T. Noiri. Properties of θ-continuous functions. Atti della Accademia Nazionale dei
Lincei, Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, Series (8),
58:887–891, 1975.

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

[22] T. Noiri. On θ-semi-continuous functions. Indian Journal of Pure and Applied Math-
ematics, 21:410–415, 1990.

[23] T. Noiri. Strongly θ-precontinuous functions. Acta Mathematica Hungarica, 90:307–
316, 2001.

[24] T. Noiri. On θ-preirresolute functions. Acta Mathematica Hungarica, 95:287–298,
2002.

[25] T. Noiri. Weak and strong forms of β-irresolute functions. Acta Mathematica Hun-
garica, 99:305–318, 2003.

[26] T. Noiri and S. M. Kang. On almost strongly θ-continuous functions. Indian Journal
of Pure and Applied Mathematics, 15:1–8, 1984.



REFERENCES 309

[27] T. Noiri and V. Popa. Strongly θ-β-precontinuous functions. Journal of Pure Math-
ematics, 19:31–39, 2002.

[28] T. Noiri and V. Popa. On almost strongly θ-m-continuous functions. Istanbul
Üniversitesi Fen Fakültesi Matematik, Fizik Astronomi Dergisi, 1:69–92, 2004-2005.

[29] T. Noiri and V. Popa. A unified theory for strongly θ-continuity for functions. Acta
Mathematica Hungarica, 106(3):167–186, 2005.

[30] M. Thongmoon and C. Boonpok. θ(Λ, p)-continuous functions. International Journal
of Mathematics and Computer Science, 19(2):475–479, 2024.

[31] C. Viriyapong and C. Boonpok. (Λ, sp)-continuous functions. WSEAS Transactions
on Mathematics, 21:380–385, 2022.

[32] N. Viriyapong and C. Boonpok. On (Λ, p)-extremally disconnected spaces. Interna-
tional Journal of Mathematics and Computer Science, 18(2):289–293, 2023.