EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 2, 2022, 354-374
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Generalized ωe∗-closed Sets

Pınar Şaşmaz1, Murad Özkoç2,∗

1 Muğla Sıtkı Koçman University, Graduate School of Applied and Natural Sciences,
Mathematics, 48000 Menteşe-Muğla, Turkey
2 Muğla Sıtkı Koçman University, Faculty of Science, Department of Mathematics, 48000
Menteşe-Muğla, Turkey

Abstract. The aim of this paper is to introduce and study a new type of generalized closed sets,
called generalized ωe∗-closed (briefly, gωe∗-closed) sets, via ωe∗-closure operator. We examine the
fundamental properties of the class of these sets. The notion of gωe∗-closed set is weaker than
the notions of gωβ-closed set and ωe∗-closed set in the literature. Also, we define and discuss the
notions of generalized ωe∗-continuous and generalized ωe∗-irresolute functions.
2020 Mathematics Subject Classifications: 54C05, 54C08
Key Words and Phrases: Generalized ωe∗-closed set, generalized ωe∗-open set, generalized
ωe∗-neighborhood, gωe∗-continuity, gωe∗-irresoluteness

1. Introduction

The notion of the generalized closed set is an important concept in the area of general
topology. It was first introduced by Levine [14] in 1970. Since then, many forms of
this notion such as gα-closed [15], gs-closed [7], gp-closed [16], gb-closed [18], gβ-closed
[21], ge-closed [8], πge-closed [9], gω-closed [5], and generalized ωβ-closed [4] have been
defined and studied by many mathematicians. Moreover, the authors have introduced
many new concepts via these new types of sets. They have also investigated some of
their fundamental properties and characterizations of these concepts. Furthermore, they
have not only discussed their fundamental properties but also put forth the relationships
between them and the notions in the literature.

In this study, we define a new concept called generalized ωe∗-closed sets via the ωe∗-
closure operator. We examine the relationships among this new concept and some other
concepts existing in the literature such as generalized β-closed, generalized e∗-closed, gen-
eralized ω-closed, and generalized ωβ-closed. In addition, by giving the notion of ωe∗-limit
point, we prove that the union of two generalized ωe∗-closed sets is a generalized ωe∗-closed
set under a special condition. Furthermore, the notions of generalized ωe∗-continuity and
generalized ωe∗-irresoluteness have been introduced and finally many basic properties of
such functions are obtained.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4340

Email addresses: pinarsasmaz@posta.mu.edu.tr (P. Şaşmaz), murad.ozkoc@mu.edu.tr (M. Özkoç)

https://www.ejpam.com 354 © 2022 EJPAM All rights reserved.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 355

2. Preliminaries

Throughout this present paper, X and Y represent topological spaces. For a subset A
of a space X, cl(A) and int(A) denote the closure of A and the interior of A, respectively.
The family of all closed (resp. open) sets of X is denoted C(X) (resp. O(X) or τ) and
the family of all closed (resp. open) sets of X containing a point x of X is denoted by
C(X,x) (resp. O(X,x)). The family of all neighborhood of a point x ∈ X is denoted by
N (x).

Definition 1. A subset A of a space X is called:
(a) regular open [20] if A = int(cl(A)). The complement of a regular open set is called

regular closed. A point x ∈ X is said to be the δ-cluster point [22] of A if int(cl(U))∩A ̸= ∅
for each open neighborhood U of x. The set of all δ-cluster points of A is called the δ-closure
of A and is denoted by δ-cl(A). If A = δ-cl(A), then A is called δ-closed [22], and the
complement of a δ-closed set is called δ-open. The set {x|(∃U ∈ O(X,x))(int(cl(U)) ⊆ A)}
is called the δ-interior of A and is denoted by δ-int(A).

(b) β-open [1] if A ⊆ cl(int(cl(A))). The complement of a β-open set is called β-closed.
The intersection of all β-closed sets containing A is called the β-closure of A and is denoted
by β-cl(A). The union of all β-open sets of X contained in A is called the β-interior of A
and is denoted by β-int(A).

(c) a-open [10] if A ⊆ int(cl(δ-int(A))). The complement of an a-open set is called
a-closed [10]. The intersection of all a-closed sets containing A is called the a-closure [10]
of A and is denoted by a-cl(A). The union of all a-open sets of X contained in A is called
the a-interior [10] of A and is denoted by a-int(A).

(d) e∗-open [11] if A ⊆ cl(int(δ-cl(A))). The complement of an e∗-open set is called
e∗-closed [11]. The intersection of all e∗-closed sets containing A is called the e∗-closure
[11] of A and is denoted by e∗-cl(A). The union of all e∗-open sets of X contained in A is
called the e∗-interior [11] of A and is denoted by e∗-int(A).

(e) ω-open [6] (resp. ωβ-open [2]) if for every x ∈ A there exists an open (resp. β-
open) set U containing x such that U \A is countable. The complement of an ω-open set
(resp. wβ-open set) is said to be ω-closed (resp. ωβ-closed).

The family of all regular open (resp. regular closed, β-open, β-closed, a-open, a-
closed, e∗-open, e∗-closed, ω-open, ω-closed, ωβ-open, ωβ-closed) subsets of X is denoted
by RO(X) (resp. RC(X), βO(X), βC(X), aO(X), aC(X), e∗O(X), e∗C(X), ωO(X),
ωC(X), ωβO(X), ωβC(X)). The family of all regular open (resp. regular closed, β-open,
β-closed, a-open, a-closed, e∗-open, e∗-closed, ω-open, ω-closed, ωβ-open, ωβ-closed) sets
of X containing a point x of X is denoted by RO(X,x) (resp. RC(X,x), βO(X,x),
βC(X,x), aO(X,x), aC(X,x), e∗O(X,x), e∗C(X,x), ωO(X,x), ωC(X,x), ωβO(X,x),
ωβC(X,x)).

Definition 2. Let A be a subset of a space X. A is said to be ωe∗-open [19] (resp. ωa-
open [19]) if for every x ∈ A, there exists an e∗-open (resp. a-open) set U containing x
such that U \A is countable. The complement of an ωe∗-open (resp. ωa-open) set is called



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 356

ωe∗-closed (resp. ωa-closed). The family of all ωe∗-open (resp. ωe∗-closed, ωa-open, ωa-
closed) sets of X will be denoted by ωe∗O(X) (resp. ωe∗C(X), ωaO(X), ωaC(X)). The
family of all ωe∗-open (resp. ωe∗-closed, ωa-open, ωa-closed) sets of X containing a point
x of X will be denoted by ωe∗O(X,x) (resp. ωe∗C(X,x), ωaO(X,x), ωaC(X,x)).

open → β-open → e∗-open
↓ ↓ ↓

ω-open → ωβ-open → ωe∗-open ← ωa-open

Definition 3. [19] Let A be a subset of a space X. The union of all ωe∗-open subsets of
X contained in A is called the ωe∗-interior of A and is denoted by ωe∗-int(A).

Theorem 1. [19] Let A be a subset of a space X. Then the following properties hold:
(a) ωe∗-int(A) ⊆ A,
(b) ωe∗-int(A) ∈ ωe∗O(X),
(c) x ∈ ωe∗-int(A) if and only if there exists U ∈ ωe∗O(X,x) such that U ⊆ A,
(d) A ⊆ B ⇒ ωe∗-int(A) ⊆ ωe∗-int(B),
(e) ωe∗-int(A) ∪ ωe∗-int(B) ⊆ ωe∗-int(A ∪B),
(f) ωe∗-int(A ∩B) ⊆ ωe∗-int(A) ∩ ωe∗-int(B),
(g) A ∈ ωe∗O(X) if and only if A = ωe∗-int(A),
(h) ωe∗-int(ωe∗-int(A)) = ωe∗-int(A).

Definition 4. [19] Let A be a subset of a space X. The intersection of all ωe∗-closed
subsets of X containing A is called the ωe∗-closure of A and is denoted by ωe∗-cl(A).

Theorem 2. [19] Let A and B be two subsets of a space X. Then the following properties
hold:
(a) A ⊆ ωe∗-cl(A),
(b) ωe∗-cl(A) ∈ ωe∗C(X),
(c) x ∈ ωe∗-cl(A) if and only if A ∩ U ̸= ∅ for every U ∈ ωe∗O(X,x),
(d) A ⊆ B ⇒ ωe∗-cl(A) ⊆ ωe∗-cl(B),
(e) ωe∗-cl(A) ∪ ωe∗-cl(B) ⊆ ωe∗-cl(A ∪B),
(f) ωe∗-cl(A ∩B) ⊆ ωe∗-cl(A) ∩ ωe∗-cl(B),
(g) A ∈ ωe∗C(X) if and only if A = ωe∗-cl(A),
(h) ωe∗-cl(ωe∗-cl(A)) = ωe∗-cl(A),
(i) ωe∗-cl(X \A) = X \ ωe∗-int(A).

Lemma 1. [19] Let X be a topological space. Then the following properties hold:
(a) The union of any family of we∗-open sets is we∗-open,
(b) The intersection of an wa-open set and an we∗-open set is we∗-open.

Definition 5. Let A be a subset of a space X. The intersection of all open sets in X
containing A is called the kernel [17] of A and is denoted by ker(A).

Lemma 2. [17] The followings hold for subsets A and B of a space X.
(a) x ∈ ker(A) if and only if A ∩ F ̸= ∅ for any F ∈ C(X,x),



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 357

(b) A ⊆ ker(A),
(c) If A is open in X, then A = ker(A),
(d) If A ⊆ B, then ker(A) ⊆ ker(B).

Definition 6. A function f : X → Y is called:
(a) e∗-continuous [11] if f−1[V ] ∈ e∗O(X) for each open set V of Y,
(b) β-continuous [1] if f−1[V ] ∈ ωβO(X) for each open set V of Y,
(c) ω-continuous [13] if f−1[V ] ∈ ωO(X) for each open set V of Y,
(d) ωβ-continuous [3] if for each x ∈ X and each open set V in Y containing f(x), there
exists an ωβ-open U in X containing x such that f [U ] ⊆ V,
(e) ωe∗-continuous [19] at a point x ∈ X if for every open set V in Y containing f(x),
there exists an ωe∗-open set U in X containing x such that f [U ] ⊆ V .

Definition 7. Let A be a subset of a space X. A is said to be generalized closed [14](briefly,
g-closed) (resp. generalized ω-closed [5](briefly, gω-closed), generalized β-closed [21](briefly,
gβ-closed), generalized e∗-closed [12](briefly, ge∗-closed), generalized ωβ-closed [4](briefly,
gωβ-closed)) if cl(A) ⊆ U (resp. ω-cl(A) ⊆ U , β-cl(A) ⊆ U , e∗-cl(A) ⊆ U , ωβ-cl(A) ⊆ U)
whenever U ∈ O(X) and A ⊆ U. The complement of a g-closed (resp. gω-closed [5],
gβ-closed [21], ge∗-closed [12], gωβ-closed [4]) set is called a generalized open (briefly, g-
open)(resp. generalized ω-open [5](briefly, gω-open), generalized β-open [21](briefly, gβ-
open), generalized e∗-open [12](briefly, ge∗-open), generalized ωβ-open [4](briefly, gωβ-
open)). The family of all g-closed (resp. gω-closed [5], gβ-closed [21], ge∗-closed [12],
gωβ-closed [4]) sets of X will be denoted by gC(X) (resp. gωC(X), gβC(X), ge∗C(X),
gωβC(X)). The family of all g-open (resp. gω-open [5], gβ-open [21], ge∗-open[12],
gωβ-open [4]) sets of X will be denoted by gO(X) (resp. gωO(X), gβO(X), ge∗O(X),
gωβO(X)).

3. Generalized ωe∗-closed Sets

Definition 8. A subset A of a space X is called generalized ωe∗-closed set (briefly,
gωe∗-closed set) if ωe∗-cl(A) ⊆ U whenever U ∈ O(X) and A ⊆ U. We denote the family
of all generalized ωe∗-closed subsets of a space X by gωe∗C(X).

Proposition 1. Let X be a topological space. Then the followings hold:
(a) If X is a countable space, then gωe∗C(X) = 2X ,
(b) If ωe∗O(X) = ωe∗C(X), then gωe∗C(X) = 2X .

Proof. (a) Let A ∈ 2X and A ⊆ U ∈ O(X).

|X| ≤ ℵ0 ⇒ ωe∗C(X) = 2X

A ∈ 2X

}
⇒ A ∈ ωe∗C(X)⇒ ωe∗-cl(A) = A

A ⊆ U ∈ O(X)

}
⇒

⇒ ωe∗-cl(A) ⊆ U
This means that A ∈ gωe∗C(X). Then we have 2X ⊆ gωe∗C(X). On the other hand, we
have always gωe∗C(X) ⊆ 2X . Therefore gωe∗C(X) = 2X .



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 358

(b) Let A ∈ 2X and A ⊆ U ∈ O(X).
A ⊆ U ∈ O(X)

O(X) ⊆ ωe∗O(X) = ωe∗C(X)

}
⇒ ωe∗-cl(A) ⊆ ωe∗-cl(U) = U

This is means that A ∈ gωe∗C(X). Then we have 2X ⊆ gωe∗C(X). On the other hand,
we have always gωe∗C(X) ⊆ 2X . Therefore gωe∗C(X) = 2X .

Remark 1. The following diagram follows immediately from the definitions in which none
of the implications is reversible. Also, examples for the other implications are shown in
the related papers.

ω-closed

gω-closed

gωβ-closed

gωe∗-closed

ωe∗-closed

closed

g-closed

gβ-closed

ge∗-closed

e∗-closed

Figure 1: Relationships between some types of closed sets

Example 1. Let X = {a, b, c} with the topology τ = {∅, X, {a}, {b}, {a, b}} and A = {a, b}.
Then A is gωe∗-closed since X is countable. But A is not ge∗-closed since A ⊆ {a, b} ∈
O(X) but e∗-cl(A) = X ⊈ {a, b}.

QUESTION: Is there an example of gωe∗-closed set which is not ωe∗-closed?

Theorem 3. Let A be a subset of a space X. If A is gωe∗-closed, then ωe∗-cl(A) \A does
not contain any non-empty closed sets.

Proof. Suppose that F ∈ C(X) \ {∅} and F ⊆ ωe∗-cl(A) \A.
(F ∈ C(X) \ {∅})(F ⊆ ωe∗-cl(A) \A)⇒ A ⊆ X \ F ∈ O(X)

A ∈ gωe∗C(X)

}
⇒

⇒ ωe∗-cl(A) ⊆ X \ F ⇒ F ⊆ X \ ωe∗-cl(A)
F ⊆ ωe∗-cl(A) \A

}
⇒ F = ∅

This contradicts with F ̸= ∅.

Theorem 4. Let A be a gωe∗-closed subset of a space X. Then A is ωe∗-closed if and
only if ωe∗-cl(A) \A is closed.

Proof. (⇒) : It is obvious.
(⇐) : Let ωe∗-cl(A) \A ∈ C(X).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 359

A ∈ gωe∗C(X)
Theorem 3⇒ (∀F ∈ C(X))[F ̸= ∅ ⇒ F ⊈ ωe∗-cl(A) \A]

ωe∗-cl(A) \A ∈ C(X)

}
⇒

⇒ ωe∗-cl(A) \A = ∅ ⇒ ωe∗-cl(A) ⊆ A
A ⊆ X ⇒ A ⊆ ωe∗-cl(A)

}
⇒ ωe∗-cl(A) = A⇒ A ∈ ωe∗C(X).

Definition 9. A space X is called an ωe∗-locally indiscrete space if every open set is
ωe∗-closed set.

Proposition 2. Let X be a topological space. Then the following are equivalent.
(a) X is ωe∗-locally indiscrete;
(b) Every subset of X is gωe∗-closed.

Proof. (a)⇒ (b) : Let A ⊆ U ∈ O(X).
A ⊆ U ∈ O(X)

Hypothesis

}
⇒ A ⊆ U ∈ ωe∗C(X)⇒ ωe∗-cl(A) ⊆ ωe∗-cl(U) = U.

(b)⇒ (a) : Let U ∈ O(X).

U ∈ O(X)
Hypothesis

}
⇒ U ∈ gωe∗C(X)

U ∈ O(X)

}
⇒ ωe∗-cl(U) ⊆ U ⇒ U ∈ ωe∗C(X).

Theorem 5. Let A be a subset of a space X. If A is both gωe∗-closed and open, then
ωe∗-cl(A) \A = ∅.

Proof. Let A ∈ O(X) ∩ gωe∗C(X).
A ∈ O(X) ∩ gωe∗C(X)⇒ (A ∈ O(X))(A ∈ gωe∗C(X))
⇒ (A ∈ O(X))(∀U ∈ O(X))(A ⊆ U ⇒ ωe∗-cl(A) ⊆ U)
⇒ ωe∗-cl(A) ⊆ A
⇒ ωe∗-cl(A) \A = ∅.

Theorem 6. Let A and B be subsets of a space X. If A is gωe∗-closed and B is any set
such that A ⊆ B ⊆ ωe∗-cl(A), then B is gωe∗-closed.

Proof. Let B ⊆ U ∈ O(X).
B ⊆ U ∈ O(X)

Hypothesis

}
⇒ (A ⊆ B ⊆ U)(A ⊆ B ⊆ ωe∗-cl(A) ⊆ U)

⇒ ωe∗-cl(A) ⊆ ωe∗-cl(B) ⊆ ωe∗-cl(ωe∗-cl(A)) = ωe∗-cl(A) ⊆ U.

Definition 10. Let A be a subset of a space X. A point x ∈ X is said to be an ωe∗-limit
point of A if for each ωe∗-open set U containing x, we have U ∩ (A \ {x}) ̸= ∅. The set of
all ωe∗-limit points of A is called the ωe∗-derived set of A and is denoted by Dωe∗(A).

Lemma 3. Let A be a subset of a space X. If D(A) = Dωe∗(A), then cl(A) = ωe∗-cl(A).

Proof. It is clear.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 360

Proposition 3. Let A and B be two subsets of a space X. Then the following properties
hold:
(a) Dωe∗(A) ⊆ ωe∗-cl(A),
(b) A ⊆ B ⇒ Dωe∗(A) ⊆ Dωe∗(B),
(c) Dωe∗(A) ∪Dωe∗(B) ⊆ Dωe∗-cl(A ∪B),
(d) Dωe∗-cl(A ∩B) ⊆ Dωe∗-cl(A) ∩Dωe∗-cl(B),
(e) Dωe∗(A) ⊆ D(A),
(f) A ∈ ωe∗C(X) if and only if Dωe∗(A) ⊆ A,
(g) A ∪Dωe∗(A) ∈ ωe∗C(X),
(h) ωe∗-cl(A) = A ∪Dωe∗(A).

Proof. The proofs of above results are standard. Hence, they are omitted.

Corollary 1. Let A be a subset of a space X. If D(A) ⊆ Dωe∗(A), then for any subsets
F and B of X, we have ωe∗-cl(F ∪B) = ωe∗-cl(F ) ∪ ωe∗-cl(B).

Proof. It is obvious.

Proposition 4. Let A and B be subsets of a space X. If A and B are gωe∗-closed sets
such that D(A) ⊆ Dωe∗(A) and D(B) ⊆ Dωe∗(B), then A ∪B is gωe∗-closed.

Proof. Let A ∪B ⊆ U ∈ O(X).
A ∪B ⊆ U ∈ O(X)⇒ (A ⊆ U ∈ O(X))(B ⊆ U ∈ O(X))

A,B ∈ gωe∗C(X)

}
⇒

⇒ (ωe∗-cl(A) ⊆ U)(ωe∗-cl(B) ⊆ U)
(D(A) ⊆ Dωe∗(A))(D(B) ⊆ Dωe∗(B))

}
⇒

⇒ cl(A ∪B) = cl(A) ∪ cl(B) = ωe∗-cl(A) ∪ ωe∗-cl(B) = ωe∗-cl(A ∪B) ⊆ U.

Proposition 5. Let A and B be subsets of a space X. Then the following properties hold:
(a) If A is open and gωe∗-closed and B is ωe∗-closed, then A ∩B is gωe∗-closed,
(b) If A is gωe∗-closed and B is closed, then A ∩B is gωe∗-closed.

Proof. (a) Let A ∈ O(X) ∩ gωe∗C(X).

A ∈ O(X) ∩ gωe∗C(X)
Theorem 5⇒ ωe∗-cl(A) \A = ∅ ⇒ ωe∗-cl(A) = A

⇒ A ∈ ωe∗C(X)
B ∈ ωe∗C(X)

}
⇒ A ∩B ∈ ωe∗C(X) ⊆ gωe∗C(X).

(b) Let A ∩B ⊆ U ∈ O(X).
A ∩B ⊆ U ∈ O(X)

B ∈ C(X)⇒ X \B ∈ O(X)

}
⇒

⇒ A ⊆ (A ∩B) ∪ (X \B) ⊆ U ∪ (X \B) ∈ O(X)
A ∈ gωe∗C(X)

}
⇒



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 361

⇒ ωe∗-cl(A) ⊆ U ∪ (X \B)⇒ ωe∗-cl(A ∩B) ⊆ ωe∗-cl(A) ∩ ωe∗-cl(B)
⊆ ωe∗-cl(A) ∩ cl(B)
= ωe∗-cl(A) ∩B
⊆ (U ∪ (X \B)) ∩B
= U ∩B
⊆ U.

Theorem 7. Let A be a subset of a space X. A is gωe∗-closed if and only if cl({x})∩A ̸= ∅
for every x ∈ ωe∗-cl(A).

Proof. (⇒) : Suppose that x ∈ ωe∗-cl(A) and cl({x}) ∩A = ∅.
cl({x}) ∩A = ∅ ⇒ A ⊆ X \ cl({x}) ∈ O(X)

A ∈ gωe∗C(X)

}
⇒ ωe∗-cl(A) ⊆ X \ cl({x})

⇒ x /∈ ωe∗-cl(A)
This contradicts with x ∈ ωe∗-cl(A).

(⇐) : Let A ⊆ U ∈ O(X) and x ∈ ωe∗-cl(A).
x ∈ ωe∗-cl(A)

Hypothesis

}
⇒ A ∩ cl({x}) ̸= ∅ ⇒ (∃y ∈ X)(y ∈ A ∩ cl({x}))

⇒ (y ∈ A)(y ∈ cl({x}))
A ⊆ U ∈ O(X)

}
⇒ (y ∈ A ⊆ U ∈ O(X))(y ∈ cl({x}))

⇒ (U ∈ O(X, y))(y ∈ cl({x}))⇒ U ∩ {x} ≠ ∅ ⇒ x ∈ U.

Theorem 8. Let X be a space. For an element x ∈ X, either {x} is closed or X \ {x} is
gωe∗-closed.

Proof. Suppose that {x} /∈ C(X).
{x} /∈ C(X)⇒ X \ {x} /∈ O(X)

X \ {x} ⊆ X ∈ O(X)

}
⇒ ωe∗-cl(X \ {x}) ⊆ ωe∗-cl(X) = X.

Definition 11. A space X is said to be an ωe∗-T 1
2
space if for every generalized ωe∗-closed

set is ωe∗-closed.

Example 2. Any set with indiscrete topology is an example for an ωe∗-T 1
2
space.

Theorem 9. Let X be a space. X is an ωe∗-T 1
2
space if and only if every singleton is

either closed or ωe∗-open.

Proof. (⇒) : Suppose that {x} /∈ C(X).

{x} /∈ C(X)
Theorem 8⇒ X \ {x} ∈ gωe∗C(X)

X is ωe∗-T 1
2
space

}
⇒ X \ {x} ∈ ωe∗C(X)

⇒ {x} ∈ ωe∗O(X).

(⇐) : Let A ∈ gωe∗C(X) and x ∈ ωe∗-cl(A).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 362

1st case: Let {x} ∈ C(X) and suppose that x /∈ A.
({x} ∈ C(X))(x /∈ A)

x ∈ ωe∗-cl(A)

}
⇒ x ∈ ωe∗-cl(A) \A⇒ {x} ⊆ ωe∗-cl(A) \A

This result contradicts with Theorem 3. Hence, x ∈ A. This means that ωe∗-cl(A) ⊆ A.
Then, we have A ∈ ωe∗-closed.

2nd case: Let {x} ∈ ωe∗O(X).
x ∈ ωe∗-cl(A)⇒ (∀U ∈ ωe∗O(X,x))(U ∩A ̸= ∅)

{x} ∈ ωe∗O(X)

}
⇒

⇒ ({x} ∈ ωe∗O(X,x))({x} ∩A ̸= ∅)⇒ x ∈ A
This means that A is ωe∗-closed.

Definition 12. A space X is said to be an e∗-anti-locally countable if each U ∈ e∗O(X) \
{∅} is uncountable.

Theorem 10. Let X be a space. If X is e∗-anti-locally countable and ωe∗-T 1
2
space, then

X is T1 space.

Proof. Let x ∈ X and suppose that {x} /∈ C(X).

{x} /∈ C(X)
Theorem 8⇒ X \ {x} ∈ gωe∗C(X)

X is ωe∗-T 1
2

}
⇒ X \ {x} ∈ ωe∗C(X)

⇒ x ∈ {x} ∈ ωe∗O(X)⇒ (∃U ∈ e∗O(X,x))(|U \ {x}| ≤ ℵ0)
This contradicts the fact that X is e∗-anti-locally countable. Then, {x} ∈ C(X) for all
x ∈ X. Namely, X is T1 space.

Proposition 6. Let A be a subset of a space X. A is gωe∗-closed set if and only if
ωe∗-cl(A) ⊆ ker(A).

Proof. (⇒) : Let A ∈ gωe∗C(X).

A ∈ gωe∗C(X)⇒ (∀U ∈ O(X))(A ⊆ U ⇒ ωe∗-cl(A) ⊆ U)
ker(A) := ∩{U |(A ⊆ U)(U ∈ O(X))}

}
⇒ ωe∗-cl(A) ⊆ ker(A).

(⇐) : Let A ⊆ U ∈ O(X).

A ⊆ U ⇒ ker(A) ⊆ ker(U)
Hypothesis

}
⇒ ωe∗-cl(A) ⊆ ker(A) ⊆ ker(U)

U ∈ O(X)⇒ U = ker(U)

}
⇒

⇒ ωe∗-cl(A) ⊆ U.

4. Generalized ωe∗-open Sets and Generalized ωe∗-neighborhoods

Definition 13. A subset A of a space X is called generalized ωe∗-open if its complement
is generalized ωe∗-closed. We denote the family of all generalized ωe∗-open subsets of a
space X by gωe∗O(X).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 363

Corollary 2. Let A be a subset of a space X. A is gωe∗-open set if and only if F ⊆
ωe∗-int(A), where F is closed set and F ⊆ A.

Proof. (⇒) : Let A ∈ gωe∗O(X) and F ∈ C(X) such that F ⊆ A.
A ∈ gωe∗O(X)⇒ X \A ∈ gωe∗C(X)

A ⊇ F ∈ C(X)⇒ X \A ⊆ X \ F ∈ O(X)

}
⇒

⇒ ωe∗-cl(X \A) = X \ ωe∗-int(A) ⊆ X \ F ⇒ F ⊆ ωe∗-int(A).

(⇐) : Let X \A ⊆ U ∈ O(X).
X \A ⊆ U ∈ O(X)⇒ (X \ U ∈ C(X))(X \ U ⊆ A)

Hypothesis

}
⇒ X \ U ⊆ ωe∗-int(A)

⇒ ωe∗-cl(X \A) = X \ ωe∗-int(A) ⊆ U.

Proposition 7. Let A and B be subsets of a space X. If ωe∗-int(A) ⊆ B ⊆ A and A is
gωe∗-open, then B is gωe∗-open.

Proof. It is clear from Theorem 6.

Proposition 8. Let A be a subset of a space X. If A is gωe∗-closed, then ωe∗-cl(A) \ A
is gωe∗-open.

Proof. It is clear from Theorem 4.

Remark 2. Let A be a subset of a space X. Then ωe∗-int(ωe∗-cl(A) \A) = ∅.

Proposition 9. Let A and B be two subsets of a space X. If A ⊆ B ⊆ X and ωe∗-cl(A)\A
is gωe∗-open, then ωe∗-cl(A) \B is gωe∗-open.

Proof. Let F ∈ C(X) such that F ⊆ ωe∗-cl(A) \B.
A ⊆ B ⇒ ωe∗-cl(A) \A ⊆ ωe∗-cl(A) \B)

(F ∈ C(X))(F ⊆ ωe∗-cl(A) \B)

}
⇒

⇒ (F ∈ C(X))(F ⊆ ωe∗-cl(A) \A ⊆ ωe∗-cl(A) \B)
ωe∗-cl(A) \A ∈ gωe∗O(X)

}
Corollary 2⇒

⇒ F ⊆ ωe∗-int(ωe∗-cl(A) \A) ⊆ ωe∗-int(ωe∗-cl(A) \B).

Proposition 10. Let A be a subset of a space X. If A is gωe∗-open, then U = X whenever
U is open in X and ωe∗-int(A) ∪ (X \A) ⊆ U.

Proof. Let ωe∗-int(A) ∪ (X \A) ⊆ U ∈ O(X).
ωe∗-int(A) ∪ (X \A) ⊆ U ∈ O(X)⇒ (X \ U ∈ C(X))(X \ U ⊆ X \ [ωe∗-int(A) ∪ (X \A)]

⇒ (X \ U ∈ C(X))(X \ U ⊆ X \ [ωe∗-int(A) ∪ (X \A)] = ωe∗-cl(X \A) \ (X \A)
A ∈ gωe∗O(X)⇒ X \A ∈ gωe∗C(X)

}
⇒

Theorem 3⇒ X \ U = ∅ ⇒ X ⊆ U
U ⊆ X

}
⇒ U = X.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 364

Theorem 11. Let A and B be two subsets of a space X. Then the following properties
hold:
(a) If A is gωe∗-open and B is ωa-open, then A ∩B is gωe∗-open,
(b) If B is gωe∗-open and ωe∗-int(B) ⊆ A, then A ∩B is gωe∗-open.

Proof. (a) Let F ∈ C(X) such that F ⊆ A ∩B.
(F ∈ C(X))(F ⊆ A ∩B)⇒ (F ∈ C(X))(F ⊆ A ∩B ⊆ A)

A ∈ gωe∗O(X)

}
Corollary 2⇒

⇒ F ⊆ ωe∗-int(A)
B ∈ ωaO(X)

}
⇒ F = F ∩B ⊆ ωe∗-int(A) ∩B = ωe∗-int(A ∩B).

(b) Let B ∈ gωe∗O(X) and ωe∗-int(B) ⊆ A.
ωe∗-int(B) ⊆ A⇒ B ∩ ωe∗-int(B) ⊆ A ∩B ⊆ B

B ∈ gωe∗O(X)

}
Proposition 7⇒ A ∩B ∈ gωe∗O(X).

Definition 14. Let X be a space and x ∈ X. A subset N of X is called a gωe∗-
neighborhood of x if there exists a gωe∗-open set U such that x ∈ U ⊆ N. The set of
all gωe∗-neighborhoods of x is called the gωe∗-neighborhood system at x, and is denoted by
Ngωe∗(x).

Definition 15. Let X be a space and A ⊆ X. A subset N of X is called a gωe∗-
neighborhood of A if there exists a gωe∗-open set U such that A ⊆ U ⊆ N.

Corollary 3. Let X be a space and x ∈ X. Every neighborhood N of x is a gωe∗-
neighborhood of x.

Remark 3. A gωe∗-neighborhood N of x in a space X need not be a neighborhood of x
as shown by the following example.

Example 3. Let X = {a, b, c, d} with a topology τ = {∅, X, {a}, {b}, {a, b}, {a, b, c}} and
A = {a, c}. Since X is countable, gωe∗O(X) = 2X . Then, A is a gωe∗-neighborhood of the
point c, since {c} is gωe∗-open set such that c ∈ {c} ⊆ {a, c}. However, the set {a, c} is not
a neighborhood of the point c, since there exists no open set U such that c ∈ U ⊆ {a, c}.

Theorem 12. Let N be a subset of a space X and x ∈ X. If N is gωe∗-open, then N is
a gωe∗-neighborhood of x.

Proof. It is clear.

Theorem 13. Let N and F be two subsets of a space X and x ∈ X. If F is gωe∗-closed
and x ∈ X \ F , then there exists a gωe∗-neighborhood N of x such that N ∩ F = ∅.

Proof. It is clear.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 365

Theorem 14. Let N be a subset of a space X and x ∈ X. Then the following properties
hold:
(a) For all x ∈ X, Ngωe∗(x) ̸= ∅,
(b) If N ∈ Ngωe∗(x), then x ∈ N,
(c) If N ∈ Ngωe∗(x) and N ⊆M ⊆ X, then M ∈ Ngωe∗(x),
(d) If N ∈ Ngωe∗(x), then there exists M ∈ Ngωe∗(x) such that M ⊆ N and N ∈ Ngωe∗(y)
for every y ∈M.

Proof. Straightforward.

Definition 16. Let A be a subset of a space X. The intersection of all generalized ωe∗-
closed (resp. generalized closed [14]) subsets of X containing A is called the general-
ized ωe∗-closure (resp. generalized closure [14]) of A and is denoted by gωe∗-cl(A) (resp.
g-cl(A)).

The proofs of the following results are standard, hence they are omitted.

Theorem 15. Let A and B be subsets of a space X and x ∈ X. Then the following
properties hold:
(a) x ∈ gωe∗-cl(A) iff V ∩A ̸= ∅ for every gωe∗-open set V containing x,
(b) gωe∗-cl(∅) = ∅ and gωe∗-cl(X) = X,
(c) If A ⊆ B, then gωe∗-cl(A) ⊆ gωe∗-cl(B),
(d) A ⊆ gωe∗-cl(A) ⊆ ωe∗-cl(A) ⊆ cl(A),
(e) A ⊆ gωe∗-cl(A) ⊆ g-cl(A) ⊆ cl(A),
(f) gωe∗-cl(A) ∪ gωe∗-cl(B) ⊆ gωe∗-cl(A ∪B),
(g) gωe∗-cl(A ∩B) ⊆ gωe∗-cl(A) ∩ gωe∗-cl(B),
(h) A ∈ gωe∗C(X) if and only if A = gωe∗-cl(A),
(i) gωe∗-cl(A) = gωe∗-cl(gωe∗-cl(A)),
(j) gωe∗-cl(A) ∈ gωe∗C(X).

Definition 17. Let X be a topological space.
(a) [14] τ∗ = {U ⊆ X|cl∗(X \ U) = X \ U},
(b) τ∗ωe∗ = {V ⊆ X|gωe∗-cl(X \ V ) = X \ V }.

Proposition 11. For a subset A of X, the following properties hold:
(a) τ ⊆ ωe∗O(X) ⊆ τ∗ωe∗ ,
(b) τ ⊆ gO(X) ⊆ τ∗ ⊆ τ∗ωe∗ .

Theorem 16. Let X be a topological space. If the family gωe∗O(X) is a topology on X,
then the family τ∗ωe∗ is a topology on X.

Proof. It is obvious that ∅, X ∈ τ∗ωe∗ . Let A,B ∈ τ∗ωe∗ .
A,B ∈ τ∗ωe∗ ⇒ (gωe∗-cl(X \A) = X \A)(gωe∗-cl(X \B) = X \B)

gωe∗O(X) is a topology on X

}
⇒

⇒ gωe∗-cl(X \A) ∪ gωe∗-cl(X \B) = (X \A) ∪ (X \B)
⇒ gωe∗-cl((X \A) ∪ (X \B)) = gωe∗-cl(X \ (A ∩B)) = X \ (A ∩B)



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 366

⇒ A ∩B ∈ τ∗ωe∗ .

Now, let A ⊆ τ∗ωe∗ .
A ∈ A ⊆ τ∗ωe∗ ⇒ gωe∗-cl(X \A) = X \A⇒ X \A ∈ gωe∗C(X)

gωe∗O(X) is a topology on X

}
⇒

⇒ X \ (∪A) = ∩A∈A(X \A) ∈ gωe∗C(X)
⇒ gωe∗-cl(∩A∈A(X \A)) = ∩A∈A(X \A) = X \ (∪A)
⇒ gωe∗-cl(X \ (∪A)) = X \ (∪A)
⇒ ∪A ∈ τ∗ωe∗ .

Theorem 17. Let X be a topological space. Then the following properties hold:
(a) A space X is ωe∗-T 1

2
if and only if τ∗ωe∗ = ωe∗O(X),

(b) Every gωe∗-closed is closed if and only if τ∗ωe∗ = τ.

Proof. (a) (⇒) : Let A ∈ τ∗ωe∗ .

A ∈ τ∗ωe∗ ⇒ X \A = gωe∗-cl(X \A)⇒ X \A ∈ gωe∗C(X)
X is ωe∗-T 1

2

}
⇒

⇒ X \A ∈ ωe∗C(X)⇒ A ∈ ωe∗O(X).

(⇐) : Let A ∈ gωe∗C(X).
A ∈ gωe∗C(X)⇒ A = gωe∗-cl(A)⇒ X \A ∈ τ∗ωe∗

Hypothesis

}
⇒ X \A ∈ ωe∗O(X)

⇒ A ∈ ωe∗C(X).

(b) (⇒) : Let A ∈ τ∗ωe∗ .
A ∈ τ∗ωe∗ ⇒ X \A = gωe∗-cl(X \A)⇒ X \A ∈ gωe∗C(X)

Hypothesis

}
⇒

⇒ X \A ∈ C(X)⇒ A ∈ τ.

(⇐) : Let A ∈ gωe∗C(X).
A ∈ gωe∗C(X)⇒ A = gωe∗-cl(A)⇒ X \A ∈ τ∗ωe∗

Hypothesis

}
⇒ X \A ∈ τ

⇒ A ∈ C(X).

5. gωe∗-continuity, gωe∗-irresoluteness and gωe∗-closedness

Definition 18. A function f : X → Y is said to be gωe∗-continuous (resp. gωβ-
continuous [4]) if f−1[V ] is gωe∗-closed (resp. gωβ-closed [4]) in X for every closed
set V of Y.

Corollary 4. Let f : X → Y be a function. f is gωe∗-continuous if and only if the inverse
image of every open set in Y is gωe∗-open in X.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 367

Remark 4. Every continuous function is gωe∗-continuous but the converse need not to
be true as shown by the following example.

Example 4. Consider the real numbers R with usual topology and let Y = {1, 2} with the
topology τ = {∅, Y, {1}}. Define the function f : R→ Y by

f(x) =

{
1 , x ∈ Q
2 , x ∈ R \Q .

Then the function f is gωe∗-continuous but not continuous since f−1[{2}] = R \Q is not
closed in R.

Remark 5. Let f : X → Y be a function. Then the following properties hold:
(a) If τ∗ωe∗ = τ in X, then the notion of continuity and the notion of gωe∗-continuity
coincide.
(b) Every gωe∗-continuous function defined on ωe∗-T 1

2
space is ωe∗-continuous.

Remark 6. The following diagram follows immediately from the definitions in which none
of the implications is reversible.

continuous → ωβ-continuous → gωβ-continuous
↓ ↓ ↓

e∗-continuous → ωe∗-continuous → gωe∗-continuous

Theorem 18. Let f : X → Y be a function. If f is gωe∗-continuous, then f [gωe∗-cl(A)] ⊆
cl(f [A]) for every subset A of X.

Proof. Let A ⊆ X.
A ⊆ X ⇒ cl(f [A]) ∈ C(Y )

f is gωe∗-continuous

}
⇒ f−1[cl(f [A])] ∈ gωe∗C(X)

⇒ gωe∗-cl(f−1[cl(f [A])]) = f−1[cl(f [A])]
A ⊆ f−1[f [A]] ⊆ f−1[cl(f [A])]⇒ gωe∗-cl(A) ⊆ gωe∗-cl(f−1[cl(f [A])])

}
⇒

⇒ gωe∗-cl(A) ⊆ f−1[cl(f [A])]
⇒ f [gωe∗-cl(A)] ⊆ cl(f [A]).

Theorem 19. Let f : X → Y be a function. If for each point x ∈ X and each open set
V containing f(x) there exists a gωe∗-open set U containing x such that f [U ] ⊆ V , then
f [gωe∗-cl(A)] ⊆ cl(f [A]) for every subset A of X.

Proof. Let y ∈ f [gωe∗-cl(A)].
y ∈ f [gωe∗-cl(A)]⇒ (∃x ∈ gωe∗-cl(A))(f(x) = y)

⇒ (∀U ∈ gωe∗O(X,x))(U ∩A ̸= ∅)(f(x) = y)
Hypothesis

}
⇒

⇒ (∀V ∈ O(Y, f(x)))(U ∈ gωe∗O(X,x))(∅ ≠ f [U ∩A] ⊆ f [U ] ∩ f [A] ⊆ V ∩ f [A])
⇒ y = f(x) ∈ cl(f [A]).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 368

Theorem 20. Let f : X → Y be a function. Then the following statements are equivalent:
(a) f [gωe∗-cl(A)] ⊆ cl(f [A]) for every subset A of X;
(b) If τ∗ωe∗ is a topology on X, then f : (X, τ∗ωe∗)→ (Y, σ) is continuous.

Proof. (a)⇒ (b) : Let A ∈ C(Y ).
A ∈ C(Y )⇒ f−1[A] ⊆ X

Hypothesis

}
⇒ f [gωe∗-cl(f−1[A])] ⊆ cl(f [f−1[A]]) ⊆ cl(A) = A

⇒ gωe∗-cl(f−1[A]) ⊆ f−1[A]
f−1[A] ⊆ gωe∗-cl(f−1[A])

}
⇒ f−1[A] = gωe∗-cl(f−1[A])

⇒ X \ f−1[A] ∈ τ∗ωe∗ ⇒ f−1[A] ∈ C(X, τ∗ωe∗).

(b)⇒ (a) : Let A ⊆ X.
A ⊆ X ⇒ cl(f [A]) ∈ C(Y )

Hypothesis

}
⇒ X \ f−1[cl(f [A])] ∈ τ∗ωe∗

⇒ f−1[cl(f [A])] ∈ C(X, τ∗ωe∗)
⇒ gωe∗-cl(A) ⊆ gωe∗-cl(f−1[cl(f [A])]) = f−1[cl(f [A])]
⇒ f [gωe∗-cl(A)] ⊆ cl(f [A]).

Definition 19. A function f : X → Y is said to be pre-ωe∗-closed if f [F ] is ωe∗-closed
in Y for every ωe∗-closed set F of X.

Definition 20. A function f : X → Y is said to be pre-gωe∗-closed if f [U ] is gωe∗-closed
in Y for every gωe∗-closed set U of X.

Theorem 21. Let f : X → Y be a function. If f is continuous and pre-ωe∗-closed, then
f is pre-gωe∗-open.

Proof. Let A ∈ gωe∗C(X) and f [A] ⊆ U ∈ O(Y ).
(A ∈ gωe∗C(X))(f [A] ⊆ U ∈ O(Y ))

f is continuous

}
⇒

⇒ (A ⊆ f−1[U ] ∈ O(X))(ωe∗-cl(A) ⊆ f−1[U ])⇒ f [ωe∗-cl(A)] ⊆ U
f is pre-ωe∗-closed function

}
⇒

⇒ ωe∗-cl(f [A]) ⊆ ωe∗-cl(f [ωe∗-cl(A)]) = f [ωe∗-cl(A)] ⊆ U.

Definition 21. A function f : X → Y is said to be gωe∗-irresolute if f−1[V ] is gωe∗-closed
in X for every gωe∗-closed set V of Y.

Corollary 5. Let f : X → Y be a function. f is gωe∗-irresolute if f−1[V ] is gωe∗-open
in X for every gωe∗-open set V of Y.

Proposition 12. Let f : (X, τ) → (Y, σ) be a function. If f is gωe∗-continuous and
σ∗
ωe∗ = σ holds, then f is gωe∗-irresolute.

The proof follows from Remark 5.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 369

Theorem 22. Let f : X → Y be a function. If f is an ωe∗-irresolute open bijection, then
f is gωe∗-irresolute.

Proof. Let F ∈ gωe∗C(Y ) and f−1[F ] ⊆ U ∈ O(X).

f−1[F ] ⊆ U ∈ O(X)
f is open bijection

}
⇒ F ⊆ f [U ] ∈ O(Y )

F ∈ gωe∗C(Y )

}
⇒ ωe∗-cl(F ) ⊆ f [U ]

⇒ f−1[ωe∗-cl(F )] ⊆ U
f is ωe∗-irresolute

}
⇒

⇒ ωe∗-cl(f−1[F ]) ⊆ ωe∗-cl(f−1[ωe∗-cl(F )]) = f−1[ωe∗-cl(F )] ⊆ U.

Theorem 23. Let f : X → Y and g : Y → Z be any two functions. Then the following
properties hold:
(a) If g is continuous and f is gωe∗-continuous, then g ◦ f is gωe∗-continuous,
(b) If g is gωe∗-irresolute and f is gωe∗-irresolute, then g ◦ f is gωe∗-irresolute,
(c) If g is gωe∗-continuous and f is gωe∗-irresolute, then g ◦ f is gωe∗-continuous,
(d) If g is gωe∗-continuous and f is ωe∗-irresolute and Y is ωe∗-T 1

2
space, then g ◦ f is

ωe∗-continuous,
(e) If g and f are gωe∗-continuous and σ∗

ωe∗ = σ, then g ◦ f is ωe∗-continuous.

Proof. Straightforward.

Theorem 24. Let f : X → Y be a function. Then the following properties hold:
(a) If f is gωe∗-irresolute and X is ωe∗-T 1

2
space, then f is ωe∗-irresolute,

(b) If f is gωe∗-continuous and X is ωe∗-T 1
2
space, then f is ωe∗-continuous.

Proof. Straightforward.

Theorem 25. Let f : X → Y be a pre-ωe∗-closed and gωe∗-irresolute surjection. If X is
ωe∗-T 1

2
space, then Y is ωe∗-T 1

2
space.

Proof. Let F ∈ gωe∗C(Y ).

F ∈ gωe∗C(Y )
f is gωe∗-irresolute

}
⇒ f−1[F ] ∈ gωe∗C(X)

X is ωe∗-T 1
2

}
⇒ f−1[F ] ∈ ωe∗C(X)

⇒ f−1[F ] ∈ ωe∗C(X)
f is pre-ωe∗-closed surjection

}
⇒ f [f−1[F ]] = F ∈ ωe∗C(Y ).

Definition 22. A function f : X → Y is said to be g∗ωe∗-continuous if f−1[V ] is gωe∗-
closed in X for every ωe∗-closed set V of Y.



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 370

Remark 7. Recall that every gωe∗-irresolute function is g∗ωe∗-continuous function and
every g∗ωe∗-continuous function is gωe∗-continuous function.

Proposition 13. Let f : X → Y be a function. If f is an open bijection and g∗ωe∗-
continuous, then f is gωe∗-irresolute.

Proof. Let A ∈ gωe∗C(Y ) and f−1[A] ⊆ U ∈ O(X).

f−1[A] ⊆ U ∈ O(X)
f is open bijection

}
⇒ f [f−1[A]] = A ⊆ f [U ] ∈ O(Y )

A ∈ gωe∗C(Y )

}
⇒ ωe∗-cl(A) ⊆ f [U ]

⇒ f−1[ωe∗-cl(A)] ⊆ U
f is g∗ωe∗-continuous

}
⇒ f−1[ωe∗-cl(A)] ∈ gωe∗C(X)

⇒ ωe∗-cl(f−1[A]) ⊆ ωe∗-cl(f−1[ωe∗-cl(A)]) ⊆ U.

Proposition 14. Let f : X → Y be a pre-ωe∗-closed and g∗ωe∗-continuous bijection open
function. If X is ωe∗-T 1

2
space, then Y is ωe∗-T 1

2
space.

Proof. Let A ∈ gωe∗C(Y ).

A ∈ gωe∗C(Y )
f is g∗ωe∗-continuous open bijection

}
Proposition 13⇒ f−1[A] ∈ gωe∗C(X)

X is ωe∗-T 1
2
space

}
⇒

⇒ f−1[A] ∈ ωe∗C(X)
f is pre-ωe∗-closed bijection

}
⇒ f [f−1[A]] = A ∈ ωe∗C(Y ).

Definition 23. A function f : X → Y is said to be gωe∗-closed if f [F ] is gωe∗-closed in
Y for every closed set F of X.

Remark 8. Every closed function is gωe∗-closed function but not conversely.

Example 5. Let X = {1, 2} with the topologies τ = {X, ∅, {1}} and σ = {X, ∅, {2}}. Let
f : (X, τ) → (X,σ) be the identity function. Then f is gωe∗-closed but not closed since
f [{2}] = {2} is not closed in X.

Theorem 26. Let f : X → Y be a function. Then, f is gωe∗-closed if and only if for
each subset S of Y and for each open set U containing f−1[S], there exists a gωe∗-open
set V of Y such that S ⊆ V and f−1[V ] ⊆ U.

Proof. (⇒) : Let S ⊆ Y and f−1[S] ⊆ U ∈ O(X).
f−1[S] ⊆ U ∈ O(X)⇒ (X \ U ∈ C(X))(X \ U ⊆ X \ f−1[S])

f is gωe∗-closed

}
⇒

⇒ (f [X \ U ] ∈ gωe∗C(Y ))(f [X \ U ] ⊆ f [X \ f−1[S]] = f [f−1[Y \ S]] ⊆ Y \ S)
⇒ (Y \ f [X \ U ] ∈ gωe∗O(Y ))(S ⊆ Y \ f [X \ U ])

V := Y \ f [X \ U ]

}
⇒

⇒ (V ∈ gωe∗O(Y ))(S ⊆ V )(f−1[V ] ⊆ U).



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 371

(⇐) : Let F ∈ C(X).
F ∈ C(X)⇒ f−1[Y \ f [F ]] ⊆ X \ F ∈ O(X)

Hypothesis

}
⇒

⇒ (∃V ∈ gωe∗O(Y ))(Y \ f [F ] ⊆ V )(f−1[V ] ⊆ X \ F )
⇒ (∃V ∈ gωe∗O(Y ))(Y \ V ⊆ f [F ] ⊆ f [X \ f−1[V ]] ⊆ Y \ V )
⇒ (Y \ V ∈ gωe∗C(Y ))(Y \ V = f [F ])
⇒ f [F ] ∈ gωe∗C(Y ).

Theorem 27. Let f : X → Y be a function. If f is gωe∗-closed, then gωe∗-cl(f [A]) ⊆
f [cl(A)] for every subset A of X.

Proof. Let A ⊆ X.
A ⊆ X ⇒ cl(A) ∈ C(X)

f is gωe∗-closed

}
⇒ f [cl(A)] ∈ gωe∗C(Y )

⇒ gωe∗-cl(f [A]) ⊆ gωe∗-cl(f [cl(A)]) = f [cl(A)].

Theorem 28. Let f : X → Y be a function. If f is continuous, gωe∗-closed and A is a
g-closed subset of X, then f [A] is gωe∗-closed.

Proof. Let f [A] ⊆ U ∈ O(Y ).

f [A] ⊆ U ∈ O(Y )
f is continuous

}
⇒ A ⊆ f−1[f [A]] ⊆ f−1[U ] ∈ O(X)

A ∈ gC(X)

}
⇒ cl(A) ⊆ f−1[U ]

⇒ (cl(A) ∈ C(X))(f [A] ⊆ f [cl(A)] ⊆ f [f−1[U ]] ⊆ U)
f is gωe∗-closed

}
⇒

⇒ (f [cl(A)] ∈ gωe∗C(Y ))(f [A] ⊆ f [cl(A)] ⊆ U)
⇒ ωe∗-cl(f [A]) ⊆ ωe∗-cl(f [cl(A)]) ⊆ U.

Theorem 29. Let f : X → Y be an open bijection. If f is g∗ωe∗-continuous, then f is
gωe∗-irresolute.

Proof. Let V ∈ gωe∗C(Y ) and f−1[V ] ⊆ U ∈ O(X).
(V ∈ gωe∗C(Y ))(f−1[V ] ⊆ U ∈ O(X))

f is open bijection

}
⇒

⇒ (f [f−1[V ]] = V ⊆ f [U ] ∈ O(Y ))(ωe∗-cl(V ) ⊆ f [U ])
f is g∗ωe∗-continuous

}
⇒

⇒ (f−1[ωe∗-cl(V )] ∈ gωe∗C(X))(f−1[ωe∗-cl(V )] ⊆ U)
⇒ ωe∗-cl(f−1[V ]) ⊆ ωe∗-cl(f−1[ωe∗-cl(V )]) ⊆ U.

Theorem 30. Let f : X → Y be a function. If f is a continuous pre-ωe∗-closed bijection,
then the inverse function of f is gωe∗-irresolute.

Proof. Let A ∈ gωe∗C(X) and (f−1)−1[A] = f [A] ⊆ U ∈ O(Y ).

f [A] ⊆ U ∈ O(Y )
f is continuous

}
⇒ A ⊆ f−1[U ] ∈ O(X)

A ∈ gωe∗C(X)

}
⇒



P. Şaşmaz, M. Özkoç / Eur. J. Pure Appl. Math, 15 (2) (2022), 354-374 372

⇒ ωe∗-cl(A) ⊆ f−1[U ]
f is a pre-ωe∗-closed bijection

}
⇒

⇒ (f [ωe∗-cl(A)] ∈ ωe∗C(Y ))(f [A] ⊆ f [ωe∗-cl(A)] ⊆ f [f−1[U ]] = U)
⇒ ωe∗-cl(f [A]) ⊆ ωe∗-cl(f [ωe∗-cl(A)]) = f [ωe∗-cl(A)] ⊆ U.

Theorem 31. Let f : X → Y and g : Y → Z be two functions. If f is a continuous
surjection and g ◦ f is gωe∗-closed, then g is gωe∗-closed.

Proof. Let V ∈ C(Y ).

V ∈ C(Y )
f is continuous

}
⇒ f−1[V ] ∈ C(X)

f is surjective

}
⇒

⇒ g[f [f−1[V ]]] = (g ◦ f)[f−1[V ]] = g[V ]
g ◦ f is gωe∗-closed

}
⇒ g[V ] ∈ gωe∗C(X).

Theorem 32. Let f : X → Y be a function. If f is gωe∗-closed continuous and X is
normal, then Y is ωe∗-normal.

Proof. Let A,B ∈ C(Y ) and A ∩B = ∅.
(A,B ∈ C(Y ))(A ∩B = ∅)

f is continuous

}
⇒

⇒ (f−1[A], f−1[B] ∈ C(X))(f−1[A ∩B] = f−1[A] ∩ f−1[B] = f−1[∅] = ∅)
X is normal

}
⇒

⇒ (∃U ∈ O(X, f−1[A]))(∃V ∈ O(X, f−1[B]))(U ∩ V = ∅)
f is gωe∗-closed

}
Theorem 26⇒

⇒ (∃G,H ∈ gωe∗O(Y ))(A ⊆ G)(B ⊆ H)(f−1[G] ⊆ U)(f−1[H] ⊆ V )(U ∩ V = ∅)
⇒ (A ⊆ ωe∗-int(G))(B ⊆ ωe∗-int(H))(A ∩B ⊆ G ∩H)(f−1[G] ∩ f−1[H] = ∅)

(U ′ := ωe∗-int(G))(V ′ := ωe∗-int(H))

}
⇒

⇒ (U ′ ∈ ωe∗O(Y,A))(V ′ ∈ ωe∗O(Y,B))(U ′ ∩ V ′ = ∅).

Theorem 33. Let f : X → Y be a function. If f is a gωe∗-closed continuous surjection,
ωe∗-open and X is regular, then Y is ωe∗-regular.

Proof. Let y ∈ Y and U ∈ O(Y, y).

(y ∈ Y )(U ∈ O(Y, y))
f is continuous

}
⇒ (∃x ∈ X)(y = f(x))(f−1[U ] ∈ O(X,x))

X is regular

}
⇒

⇒ (∃V ∈ O(X,x))(V ⊆ cl(V ) ⊆ f−1[U ])
f is gωe∗-closed surjection

}
⇒

⇒ (∃V ∈ O(X,x))(f [cl(V )] ∈ gωe∗C(X))(y ∈ f [V ] ⊆ f [cl(V )] ⊆ U)
f is ωe∗-open

}
⇒

⇒ (f [V ] ∈ ωe∗O(Y, y))(ωe∗-cl(f [V ]) ⊆ ωe∗-cl(f [cl(V )]) ⊆ U).



REFERENCES 373

Conclusion

Many forms of generalized closed sets which are first defined by Levine [14] have been
studied by many authors in recent years. This paper is concerned with the notion of
generalized ωe∗-closed sets which are defined by utilizing the concept of ωe∗-open set.
We have seen that this concept is weaker than many generalized closed set forms in the
literature as will be seen in Figure 1. In addition, we gave some examples related to the
concept but we could not find an example generalized ωe∗-closed set which is not ωe∗-
closed. We believe that this study will help researchers to upgrade and support further
studies related to compactness and connectedness etc. Also, the objects considered in
the article may find an application in the area of both pure and applied sciences such as
computational topology and digital topology.

Acknowledgements

We would like to thank the anonymous reviewers for their careful reading of our
manuscript and their insightful comments and suggestions.

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