EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1869-1876
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Properties of sθ̃-Open Sets in Generalized Topological
Spaces

Jeeranunt Khampakdee

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

Abstract. A study of θ̃-open sets in generalized topological spaces started in 2011 by Min [4]. In
this paper, we introduce the concepts of sθ̃-open sets, sθ̃-closed sets, sθ̃-continuous functions and
sθ̃-irresolute functions on generalized topological spaces. We also study some basic properties of
such sets and functions.
2020 Mathematics Subject Classifications: 54A05
Key Words and Phrases: sθ̃-open set, sθ̃-continuous function, sθ̃-irresolute function.

1. Introduction

In 2002, Császár [1] defined the concept of a generalized topological space which is
an extension of the idea from a topological space. In addition, the concepts of the interior
of sets and the closure of sets in topological spaces were introduced. If A is a subset of
X, then the symbols iµ(A) and cµ(A) are used to represent the interior of A and the
closure of A, respectively, in a generalized topological space (X,µ). In [1], the concept
of (µ, µ′)-continuous functions on generalized topological spaces was also introduced. In
2005, Császár [2] used the concepts of semi-open sets and semi-closed sets in topological
spaces to define such sets in generalized topological spaces, and proved that if A ⊆ X, then
iµ(iµ(A)) = iµ(A) and cµ(cµ(A)) = cµ(A). Moreover, if A ⊆ B ⊆ X, then iµ(A) ⊆ iµ(B)
and cµ(A) ⊆ cµ(B). In 2008, Császár [3] defined the family θ(µ) = θ ⊆ P (X) in a
generalized topological space (X,µ). A subset A of (X,µ) is an element of θ(µ) if and
only if there exists M ∈ µ such that x ∈ M and M ⊆ cµ(M) ⊆ A for all x ∈ A. The
elements of θ(µ) are called θ-open sets in (X,µ). The complements of θ-open sets are
called θ-closed sets. In 2011, the concept of the collection θ(µ) was developed into the
collection θ̃(µ) = θ̃ ⊆ P (X) by Min [4]. Additionally, he also concluded that θ ⊆ θ̃ ⊆ µ.
In 2011, Roy [5] studied some properties of (µ, µ′)-continuous functions on generalized
topological spaces. In this paper, we use the concept of the above researches to define new
types of sets in generalized topological spaces, along with studying some basic properties
of such sets.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5256

Email address: jeeranunt.k@msu.ac.th (J. Khampakdee)

https://www.ejpam.com 1869 © 2024 EJPAM All rights reserved.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1870

2. Preliminaries

Definition 1. [1] Let X be a non-empty set, P (X) denotes the power set of X, we call

a collection µ ⊆ P (X) a generalized topology on X if ∅ ∈ µ and
⋃
α∈J

Gα ∈ µ for each

Gα ∈ µ and α ∈ J . The elements in µ is called µ-open sets in X. The complement of
each µ-open set is called µ-closed in X. The pair (X,µ) is called a generalized topological
space. For A ⊆ X, the symbol iµ(A) represents the interior of A which is the union of
all µ-open sets contained in A, and the symbol cµ(A) represents the closure of A, which
means the intersection of all µ-closed sets containing A.

Lemma 1. [2] Let A be a subset of a generalized topological space (X,µ). Then:
1. iµ(A) is the largest µ-open set contained in A,
2. cµ(A) is the smallest µ-closed set containing A,
3. cµ(X −A) = X − iµ(A).

Definition 2. [2] Let (X,µ) be a subset of a generalized topological space. Then, a subset
A of X is called a µ-semi-open set if A ⊆ cµ(iµ(A)). The complement of µ-semi-open
sets is called µ-semi-closed sets. The intersection of all µ-semi-closed sets containing A is
denoted by cσ(A).

Definition 3. [3] Let A be a subset of a generalized topological space (X,µ). Then, A is
an element of a collection θ(µ) = θ ⊆ P (X) if and only if there exists M ∈ µ such that
x ∈M and M ⊆ cµ(M) ⊆ A for all x ∈ A. The elements of θ(µ) are called θ-open sets in
X. The complements of θ-open sets are called θ-closed sets.

Definition 4. [4] Let (X,µ) be a generalized topological space and θ̃(µ) = θ̃ ⊆ P (X).
A subset A of X is an element of θ̃ if and only if there exists M ∈ µ such that x ∈
M ⊆ cµ(M) ∩Mµ ⊆ A for all x ∈ A, where Mµ = ∪{M ⊆ X : M ∈ µ}. The elements
of θ̃ are called θ̃-open sets in X. The complements of θ̃-open sets are called θ̃-closed
sets. Additionally, in [4], the relationship between θ, θ̃ and µ was concluded as follows:
θ ⊆ θ̃ ⊆ µ .

Definition 5. [1] Let (X,µ) and (Y, µ′) be generalized topological spaces. Then a function
f from (X,µ) into (Y, µ′) is called (µ, µ′)-continuous if f−1(G) is µ-open in X for each
µ′-open set G in Y .

3. sθ̃-open sets

In this section, we introduce the concept of sθ̃-open sets. Furthermore, some properties
of sθ̃-open sets are studied.

Definition 6. Let (X,µ) be a generalized topological space and A ⊆ X. Define the col-
lection sθ̃(µ) = sθ̃ ⊆ P (X) by A ∈ sθ̃ if for each x ∈ A there exists M ∈ µ such that
x ∈ M ⊆ cσ(M) ∩Mµ ⊆ A, where Mµ = ∪{M ⊆ X : M ∈ µ}. The elements in sθ̃ are
called sθ̃-open sets and the complements are called sθ̃-closed sets in X.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1871

Theorem 1. Let (X,µ) be a generalized topological space. Then, sθ̃ is a generalized
topology on X.

Proof. By Definition 6, we can clearly see that ∅ ∈ sθ̃. Assume that Ai ∈ sθ̃ for all

i ∈ J . Let x ∈
⋃
i∈J

Ai, we get x ∈ Ai for some i ∈ J . By the assumption, there exists

M ∈ µ such that x ∈M ⊆ cσ(M) ∩Mµ ⊆ Ai ⊆
⋃
i∈J

Ai, where Mµ = ∪{M ⊆ X : M ∈ µ}.

Thus,
⋃
i∈J

Ai ∈ sθ̃. Therefore, sθ̃ is a generalized topology on X.

Theorem 2. Let (X,µ) be a generalized topological space. Then sθ̃ ⊆ µ.

Proof. Let A ⊆ X and A ∈ sθ̃. If A = ∅, then A ∈ µ. If A ̸= ∅, let x ∈ A.
Since A ∈ sθ̃, there exists Mx ∈ µ such that x ∈ Mx ⊆ cσ(Mx) ∩ Mµ ⊆ A for each

x ∈ A. Hence,
⋃
x∈A
{x} ⊆

⋃
x∈A

Mx ⊆ A. As A =
⋃
x∈A
{x}, A ⊆

⋃
x∈A

Mx ⊆ A. Consequently,

A =
⋃
x∈A

Mx ∈ µ. Therefore, sθ̃ ⊆ µ.

Corollary 1. Let A be a subset of a generalized topological space (X,µ). If A is an
sθ̃-closed set, then A is µ-closed.

Theorem 3. Let (X,µ) be a generalized topological space. Then, θ̃ ⊆ sθ̃.

Proof. Let A be an arbitrary element in θ̃. Assume that x ∈ A. Since A ∈ θ̃, there
exists M ∈ µ such that x ∈ M ⊆ cµ(M) ∩Mµ ⊆ A. As M ⊆ cσ(M) ⊆ cµ(M) for all
x ∈ A. Accordingly, A ∈ sθ̃. Therefore, θ̃ ⊆ sθ̃.

In a generalized topological space (X,µ), sθ̃-open sets may not be θ̃-open sets as
the following example.

Example 1. Let X = {a, b, c, d}, µ = {∅, {a}, {b}, {a, b}, {a, b, c}} and A = {b}. Then,
X, {b, c, d}, {a, c, d}, {c, d}, {d} are µ-closed and Mµ = {a, b, c}.
Moreover, only ∅, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, c, d}, {b, c, d}
and X are µ-semi-closed sets. Consider b ∈ A, then there exists {b} ∈ µ such that
b ∈ {b} ⊆ cσ({b})∩Mµ = {b}. Hence, A ∈ sθ̃. Consider each µ-open set M containing b.

If M = {b} ∈ µ, then b ∈ {b} ⊆ cµ({b}) ∩Mµ = {b, c} ̸⊆ {b}.
If M = {a, b} ∈ µ, then b ∈ {a, b} ⊆ cµ({a, b}) ∩Mµ = {a, b, c} ̸⊆ {b}.
If M = {a, b, c} ∈ µ, then b ∈ {a, b, c} ⊆ cµ({a, b, c}) ∩Mµ = {a, b, c} ̸⊆ {b}. From

the above three cases, A ̸∈ θ̃. Therefore, sθ̃ ̸⊆ θ̃.

By the generalized topological space (X,µ) in Example 1, we can A is sθ̃-open but
not θ̃-open. Even though µ is a topology on X, sθ̃-open sets may not be θ̃-open as in the
following example.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1872

Example 2. Let X = {a, b, c, d}, µ = {∅, {a}, {b}, {a, b}, {a, b, c}, X} and A = {b}.
Clearly, X, {b, c, d}, {a, c, d}, {c, d}, {d}, ∅ are µ-closed and Mµ = X.
Moreover, only ∅, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d},{c, d}, {a, c, d}, {b, c, d}, X are
µ-semi-closed sets. Consider b ∈ A, then there exists {b} ∈ µ such that b ∈ {b} ⊆
cσ({b}) ∩Mµ = {b}. Thus, A ∈ sθ̃. Since {b}, {a, b}, {a, b, c}, X are µ-open sets contain-
ing b. Consider each µ-open set M containing b.

If M = {b} ∈ µ, then b ∈ {b} ⊆ cµ({b}) ∩Mµ = {b, c, d} ̸⊆ {b}.
If M = {a, b} ∈ µ, then b ∈ {a, b} ⊆ cµ({a, b}) ∩Mµ = X ̸⊆ {b}.
If M = {a, b, c} ∈ µ, then b ∈ {a, b, c} ⊆ cµ({a, b, c}) ∩Mµ = X ̸⊆ {b}.
If M = X ∈ µ, then b ∈ X ⊆ cµ(X) ∩Mµ = X ̸⊆ {b}. From the above four cases,

A ̸∈ θ̃. Therefore, sθ̃ ̸⊆ θ̃ even though µ is a topology on X.

Corollary 2. Let (X,µ) be a generalized topological space. Then θ ⊆ θ̃ ⊆ sθ̃ ⊆ µ.

In generalized topological spaces, µ-open sets need not be sθ̃-open as can be seen from
the following example.

Example 3. By the generalized topological space (X,µ) in Example 2. Let A = {a, b, c},
then A ∈ µ. Consider c ∈ A. Since only {a, b, c} and X are µ-open sets containing c.
Consider each µ-open set M containing c.

If M = {a, b, c} ∈ µ, then c ∈ {a, b, c} ⊆ cµ({a, b, c}) ∩Mµ = X ̸⊆ {a, b, c}.
If M = X ∈ µ, then c ∈ X ⊆ cµ(X) ∩Mµ = X ̸⊆ {a, b, c}. From the above two

cases, A ̸∈ sθ̃.

In [4], Min defined the set γθ̃(A), where A is a subset of a generalized topological space
(X,µ) as follows: x ∈ γθ̃(A) if cµ(G)∩Mµ ∩A ̸= ∅ for each G ∈ µ such that x ∈ G, where
Mµ = ∪{M ⊆ X : M ∈ µ}. In this paper, the aforementioned concepts are used to define
the set γsθ̃(A) as follows.

Definition 7. Let A be a subset of a generalized topological space (X,µ). The set γsθ̃(A)
is a subset of X defined by x ∈ γsθ̃(A) if cσ(G) ∩Mµ ∩ A ̸= ∅ for each G ∈ µ such that
x ∈ G, where Mµ = ∪{M ⊆ X : M ∈ µ}.

Theorem 4. Let A and B be subsets of a generalized topological space (X,µ). If A ⊆
B ⊆ X, then γsθ̃(A) ⊆ γsθ̃(B).

Proof. Let A ⊆ B ⊆ X and x ∈ γsθ̃(A). Thus, cσ(G) ∩Mµ ∩A ̸= ∅ for all G ∈ µ such
that x ∈ G, where Mµ = ∪{M ⊆ X : M ∈ µ}. Let H be an arbitrary µ-open set such that
x ∈ H. By the assumption, cσ(H) ∩Mµ ∩ B ̸= ∅. Consequently, x ∈ γsθ̃(B). Therefore,
γsθ̃(A) ⊆ γsθ̃(B).

Theorem 5. Let (X,µ) be a generalized topological space and A ⊆ X. Then, A ⊆
γsθ̃(A) ⊆ γθ̃(A).

Proof. Assume that x ∈ A. If x /∈ Mµ, then x ∈ X −Mµ. It follows that x ∈ X and
x /∈ Mµ, where Mµ = ∪{M ⊆ X : M ∈ µ}. Thus, there is no G ∈ µ such that x ∈ G.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1873

By Definition 7, x ∈ γsθ̃(A). If x ∈ Mµ, then cσ(G) ∩Mµ ∩ A ̸= ∅ for each G ∈ µ such
that x ∈ G. Hence, x ∈ γsθ̃(A). Thus, A ⊆ γsθ̃(A). By Definition 7, we can see that
γsθ̃(A) ⊆ γθ̃(A), where A is a subset of a generalized topological space (X,µ). Therefore,
A ⊆ γsθ̃(A) ⊆ γθ̃(A).

We use some properties of the operation γsθ̃ to characterize sθ̃-closed sets as the fol-
lowing theorem.

Theorem 6. Let (X,µ) be a generalized topological space and A ⊆ X. Then A is sθ̃-closed
if and only if γsθ̃(A) = A.

Proof. (→) Let A be an sθ̃-closed. Then, X −A is sθ̃-open. Assume that x ∈ X −A.
Hence, there exists M ∈ µ such that x ∈M ⊆ cσ(M)∩Mµ ⊆ X−A, where Mµ = ∪{M ⊆
X : M ∈ µ}. Thus, cσ(M) ∩Mµ ∩ A = ∅, for some M ∈ µ and x ∈ M . It follows that
x /∈ γsθ̃(A). Consequently, x ∈ X − γsθ̃(A). Accordingly, X −A ⊆ X − γsθ̃(A). Therefore,
γsθ̃(A) ⊆ A. By Theorem 5, γsθ̃(A) = A.
(←) Suppose that γsθ̃(A) = A. Then, X − A = X − γsθ̃(A). Assume that x ∈ X − A. It
follows that x /∈ γsθ̃(A). Hence, there existsM ∈ µ such that x ∈M and cσ(M)∩Mµ∩A =

∅. Thus, x ∈M ⊆ cσ(M)∩Mµ ⊆ X −A. Accordingly, X −A is sθ̃-open. Therefore, A is
sθ̃-closed.

Corollary 3. γsθ̃(X) = X, where (X,µ) is a generalized topological space.

Proof. By Theorem 1, ∅ is sθ̃-open. It follows that X is sθ̃-closed. By Theorem 6,
γsθ̃(X) = X.

Definition 8. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, a function f
from (X,µ) into (Y, µ′) is called θ̃-continuous if f−1(G) is θ̃-open in X for each µ′-open
set G in Y .

Example 4. Let X = {a, b, c, d}, Y = {t, v, w}. µ = {∅, {a}, {b}, {a, b}, {a, b, c}} and σ =
{∅, {t}, {t, v}}. Let f be a function from (X,µ) into (Y, µ′) defined by f(a) = t, f(b) = t,
f(c) = t, f(d) = w. Then, only ∅ and {a, b, c} are θ̃-open sets in X. Consider ∅, {t} and
{t, v} are µ′-open sets in Y . Then f−1(∅) = ∅, f−1({t}) = {a, b, c} and f−1({t, v}) =
{a, b, c}. Thus, f−1(G) is θ̃-open in X for each µ′-open set G in Y . Therefore, f is a
θ̃-continuous function.

Theorem 7. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, f : (X,µ)→
(Y, µ′) is θ̃-continuous if and only if f−1(V ) is θ̃-closed in X for each µ′-closed set V in
Y .

Proof. (→) Assume that f is a θ̃-continuous function. Let V be a µ′-closed set in Y .
Then, Y −V is µ′-open in Y . Hence, f−1(Y −V ) = X−f−1(V ) is θ̃-open in X. Therefore,
f−1(V ) is θ̃-closed in X.
(←) Assume that f−1(V ) is θ̃-closed in X for each µ′-closed set V in Y . Let H be µ′-open
in Y . Then Y −H is µ′-closed in Y . By the assumption, f−1(Y −H) is θ̃-closed in X. It
follows that f−1(H) is θ̃-open in X. Hence, f is θ̃-continuous.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1874

Definition 9. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, a function f
from (X,µ) into (Y, µ′) is called sθ̃-continuous if f−1(G) is sθ̃-open in X for each µ′-open
set G in Y .

It is easy to see that all θ̃-continuous functions are sθ̃-continuous. But the converse
need not to be true as the following example.

Example 5. By the generalized topological spaces (X,µ) and (Y, µ′) in Example 4, if f is a
function from (X,µ) into (Y, µ′) defined by f(a) = t, f(b) = t, f(c) = v, f(d) = w. Then,
only ∅, {a}, {b}, {a, b} and {a, b, c} are sθ̃-open in X. It can be checked that f−1(∅) = ∅,
f−1({t}) = {a, b} and f−1({t, v}) = {a, b, c}. Thus, f−1(G) is sθ̃-open in X for each
sθ̃-open set G in Y . Consequently, f is sθ̃-continuous. Since there exists a µ′-open set
{t} in Y such that f−1({t}) = {a, b} is not θ̃-open in X. Therefore, f is sθ̃-continuous
but f is not θ̃-continuous.

Theorem 8. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, for a function
f from (X,µ) into (Y, µ′) the followings are equivalent:

a) f is sθ̃-continuous;
b) f−1(F ) is sθ̃-closed in X for each µ′-closed set F in Y ;
c) for all x ∈ X, there exists an sθ̃-open set H in X such that x ∈ H and f(H) ⊆ G,

for all G ∈ µ′ such that f(x) ∈ G

Proof.
a) → b) Let f be an sθ̃-continuous function. Then, f−1(G) is sθ̃-open in X for all

µ′-open set G in Y . Assume that F is µ′-closed in Y . We get Y − F is µ′-open in Y .
Hence f−1(Y −F ) is sθ̃-open in X. Thus, X−f−1(F ) is sθ̃-open in X. Therefore, f−1(F )
is sθ̃-closed in X.

b) → c) Assume that f−1(F ) is sθ̃-closed in X for each µ′-closed set F in Y . Let
x ∈ X and G ∈ µ′ such that f(x) ∈ G. Then, there exists a µ′-closed set F in Y such that
G = X−F . Since f(x) ∈ G, x ∈ f−1(G) = f−1(X−F ) = Y −f−1(F ). By the assumption,
f−1(F ) is sθ̃-closed in Y . It follows that f−1(G) is sθ̃-open in X and f(f−1(G)) ⊆ G.

c) → a) Assume that for all x ∈ X, there exists an sθ̃-open set H in X such that
x ∈ H and f(H) ⊆ G, for all G ∈ µ′ such that f(x) ∈ G. Let K be µ′-open in Y .

Case 1. f−1(K) = ∅. Since sθ̃ is a generalized topology on X, ∅ ∈ sθ̃. Hence,
f−1(K) ∈ sθ̃. Thus, f−1(K) is sθ̃-open in X.

Case 2. f−1(K) ̸= ∅. Let x ∈ f−1(K), then f(x) ∈ K. Hence, x ∈ X such that
f(x) ∈ K and K is µ′-open in Y . By the assumption, there exists an sθ̃-open set Hx in
X such that x ∈ Hx and f(Hx) ⊆ K. It follows that Hx ⊆ f−1(f(Hx)) ⊆ f−1(K). Thus,
x ∈ Hx ⊆ f−1(K). As x is an arbitrary element in f−1(K), {x} ⊆ Hx ⊆ f−1(K) for all

x ∈ f−1(K). Hence,
⋃

x∈f−1(K)

{x} ⊆
⋃

x∈f−1(K)

Hx ⊆ f−1(K). Thus f−1(K) =
⋃

x∈f−1(K)

Hx.

As sθ̃ is a generalized topology on X and Hx is sθ̃-open for each x ∈ f−1(K),
⋃

x∈f−1(K)

Hx

is sθ̃-open in X. Consequently, f−1(K) is sθ̃-open in X. Therefore, f is a sθ̃-continuous
function.



J. Khampakdee / Eur. J. Pure Appl. Math, 17 (3) (2024), 1869-1876 1875

Theorem 9. Let (X,µ), (Y, µ′) and (Z, µ′′) be generalized topological spaces. If f : X → Y
is sθ̃-continuous and g : Y → Z is (µ, µ′)-continuous, then g◦f : X → Z is sθ̃-continuous.

Proof. Assume that V ∈ µ′′. Since g is (µ, µ′)-continuous, g−1(V ) is µ′-open in Y . As
f is sθ̃-continuous, f−1(g−1(V )) is sθ̃-open in X. It follows that (g ◦ f)−1(V ) is sθ̃-open
in X for all V ∈ µ′′. Therefore, g ◦ f : X → Z is sθ̃-continuous.

Theorem 10. Let (X,µ) and (Y, µ′) be generalized topological spaces. If f : X → X is an
identity function and g : X → Y is sθ̃-continuous, then g ◦ f : X → Y is sθ̃-continuous.

Proof. Assume that V is µ′-open. Since g is sθ̃-continuous, g−1(V ) is sθ̃-open in X. As
f is an identity function, g−1(V ) = f−1(g−1(V )). Consequently, (g ◦ f)−1(V ) = g−1(V ) is
sθ̃-open in X for all V ∈ µ′. Thus, g ◦ f : X → Y is sθ̃-continuous.

Definition 10. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, a function
f from (X,µ) into (Y, µ′) is called sθ̃-irresolute if f−1(G) is sθ̃-open in X for each sθ̃-open
set G in Y .

Theorem 11. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, f : (X,µ)→
(Y, µ′) is sθ̃-irresolute if and only if f−1(F ) is sθ̃-closed in X for each sθ̃-closed set F in
Y .

Proof. (→) Let f be a sθ̃-irresolute function. Suppose that F is a sθ̃-closed set in Y .
Then, Y − F is sθ̃-open in Y . Consequently, X − f−1(F ) is sθ̃-open in X. It follows that
f−1(F ) is sθ̃-closed in X.
(←) Assume that f−1(F ) is sθ̃-closed in X for each sθ̃-closed set F in Y . Let K be sθ̃-open
in Y . Then, Y −K is sθ̃-closed in Y . By the assumption, f−1(Y −K) is sθ̃-closed in X.
It follows that f−1(K) is sθ̃-open in X. Hence, f is sθ̃-irresolute.

Theorem 12. Each sθ̃-irresolute function is sθ̃-continuous.

Proof. Assume that V is µ′-open in Y . By Theorem 3, V is sθ̃-open. Since f is
sθ̃-irresolute, f−1(V ) is sθ̃-open in X. Thus, f is sθ̃-continuous.

Theorem 13. Let (X,µ), (Y, µ′) and (Z, µ′′) be generalized topological spaces. Then,
g ◦ f : X → Z is sθ̃-continuous if f : X → Y is sθ̃-irresolute and g : X → Z is sθ̃-
continuous.

Proof. Assume that H is µ′′-open. Since g is sθ̃-continuous, g−1(H) is sθ̃-open in Y .
As f is sθ̃-irresolute, f−1(g−1(H)) = (g ◦ f)−1(H) is sθ̃-open in X for all H ∈ µ′′. Thus,
g ◦ f is sθ̃-continuous.

Theorem 14. Let (X,µ) and (Y, µ′) be generalized topological spaces. Then, g◦f : X → Y
is sθ̃-irresolute if f : X → X is an identity function and g : X → Y is sθ̃-irresolute.

Proof. Suppose that M is sθ̃-open in Y . As g is sθ̃-irresolute, g−1(M) is sθ̃-open in
X. Since f is an identity function, f−1(g−1(M)) = g−1(M) is sθ̃-open in X for each M is
sθ̃-open in Y . Thus, g ◦ f : X → Y is sθ̃-irresolute.



REFERENCES 1876

Acknowledgements

The author would like to thank the referees for their valuable suggestions, which en-
hanced the presentation of the paper.

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