EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 977-986
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On (µ1, µ2, µ3)-Weakly Generalized Closed Sets

Breix Michael G. Agua1,∗, Rolando N. Paluga1

1 Department of Mathematics, Caraga State University, Ampayon, Butuan City, Philippines

Abstract. This paper defines a new generalization of closed sets in a tri-generalized topological
space called (µ1, µ2, µ3)-weakly generalized closed set (or briefly (µ1, µ2, µ3)-wg closed set) which is
defined as follows: A subset A of X is (µ1, µ2, µ3)-weakly generalized closed set if clµ

1
(intµ

2
(A)) ⊆

U whenever A ⊆ U and U is µ3-open in X. At least fifteen defined closed sets found in literature
are considered special cases of (µ1, µ2, µ3)-weakly generalized closed set under some conditions.
Furthermore, some properties of (µ1, µ2, µ3)-weakly generalized closed sets are obtained.

2020 Mathematics Subject Classifications: 54A05, 54A10

Key Words and Phrases: Tri-generalized topological space, generalized topological space,
weakly generalized closed set

1. Introduction

Studies concerning generalized topologies have been in literature since its introduction
in 2002 by Csaszar as cited in [1]. Topological properties of generalized topologies have
also been explored by some researchers including those by Tabadkan and Taghavi in 2011
[2], and those by Khayyeri and Mohamadian also in 2011 [3]. Other authors named a
generalized topology as a supra topology and derived some important definitions and
properties such as those of Al-Shami in 2016 and 2018 [11,12], and El-Shafie, et al. in
2020 [13]. New developments of researches pertaining to generalized topologies have been
extended to bi-generalized topologies wherein two generalized topologies were considered
in the study. Two of which include the researches of Dungthaisong, et al. in 2011 [4]
and of Rara and Baculta in 2015 [5]. Several researches involving closed sets, generalized
closed sets and many more have been available in literature. In the paper of Mishra,
et al. [6], sixteen (16) definitions of closed sets were enumerated. Seven (7) definitions
of closed sets were also listed in the paper of Cao, et al.[7] in 1999. In this paper, the
researcher defines a new generalization of closed sets called (µ1, µ2, µ3)-weakly generalized
closed set containing fifteen literature-defined closed sets. Furthermore, the corresponding
(µ1, µ2, µ3)-weakly generalized open sets are characterized. Moreover, some properties of
(µ1, µ2, µ3)-weakly generalized closed sets are obtained.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3758

Email addresses: bgagua@carsu.edu.ph (B. Agua), rnpaluga@carsu.edu.ph (R. Paluga)

https://www.ejpam.com 977 c© 2020 EJPAM All rights reserved.



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 978

2. Preliminaries

Definition 1. [1] Let X be a nonempty set. A collection µ of subsets of X is a generalized
topology (or briefly GT) in X if it satisfies: i. ∅ ∈ µ, and ii. if {Mi : i ∈ I} ⊆ µ, then⋃
i∈IMi ∈ µ.

If µ is a GT in X, then (X,µ) is called a generalized topological space (or briefly GT
space), and the elements of µ are called µ-open sets in X. If µ1 and µ2 are GTs in X,
then (X,µ1, µ2) is called a bi-generalized topological space.If µ1 µ2 and µ3 are GTs in X,
then (X,µ1, µ2, µ3) is called a tri-generalized topological space.

Definition 2. [1,3] Let µ be a GT in X. A subset F of X is said to be a µ-closed set
if the complement of F (F c) is µ-open. The µ-closure of a subset A of X, denoted by
clµ(A), is the intersection of all µ-closed sets in X containing A while the µ-interior of a
subset A of X, denoted by intµ(A), is the union of all µ-open subsets of A in X.

The succeeding Theorems 1 to 5 are fundamental properties of any GT µ and can be
easily proven.

Theorem 1. Let X 6= ∅ and µ be a GT in X. If {Ai : i ∈ I} is a collection of µ-closed
sets, then

⋂
i∈I Ai is a µ-closed set.

Theorem 2. Let X 6= ∅ and µ be a GT in X. Suppose also that A and B are subsets of
X. Then,

i. If A ⊆ X, then intµ(A) ⊆ A.

ii. intµ(A) is the largest open subset of A.

iii. A is µ-open if and only if intµ(A) = A.

iv. If A ⊆ B, then intµ(A) ⊆ intµ(B).

Theorem 3. Let X 6= ∅ and µ be a GT in X. Suppose also that A and B are subsets of
X. Then,

i. If A ⊆ X, then A ⊆ clµ(A).

ii. clµ(A) is the smallest closed superset of A.

iii. A is µ-closed if and only if clµ(A) = A.

iv. If A ⊆ B, then clµ(A) ⊆ clµ(B).

Theorem 4. Let X 6= ∅ and µ1 ⊆ µ2 where µ1 and µ2 are GTs in X. Then A is µ2-open
(µ2-closed) whenever A is µ1-open (µ1-closed).

Theorem 5. Let X 6= ∅, A ⊆ X, and µ be a generalized topology. Then, [intµ(A)]c =
clµ(Ac).



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 979

Definition 3. [8] Let X be a non empty set, then the collection of all subsets of X is
called the discrete topology. We denote this collection as D.

Definition 4. Let Y be a subset of X. A set A is called a µ-open set in Y if A = Y
⋂
G

for some µ-open set G in X.

Definition 5. [1,5,6,10] Let X be a non empty set and µ be a generalized topology in X.
Then,

i. A subset A ofX is called generalized open (briefly g-open) set if F ⊆ int(A) whenever
F ⊆ A and F is closed. We denote the collection of g-open sets in X as G(X).

ii. A subset A of X is called semi-open set if A ⊆ cl(int(A)). We denote the collection
of semi-open sets in X as SO(X).

iii. A subset A of X is called α-open set if A ⊆ int(cl(int(A))). We denote the collection
of α-open sets in X as AO(X).

iv. A subset A of X is called semi-preopen set if A ⊆ cl(int(cl(A))). We denote the
collection of semi-preopen sets in X as SPO(X).

v. A subset A of X is called b-open set if A ⊆ cl(int(A))
⋃
int(cl(A)). We denote the

collection of b-open sets in X as BO(X).

The complements of the above-mentioned open sets are their respective closed sets.

Theorem 6. Let X 6= ∅. The following can be shown using Definitions 1 and 5, and
Theorems 2 and 3.

i. The collection of g-open sets in X is a generalized topology.

ii. The collection of semi-open sets in X is a generalized topology.

iii. The collection of α-open sets in X is a generalized topology.

iv. The collection of semi-preopen sets in X is a generalized topology.

v. The collection of pre-open sets in X is a generalized topology.

vi. The collection of b-open sets in X is a generalized topology.

Definition 6. [1,6,7,9] Let X 6= ∅ and µ be a topology on X. Then, a subset A of X is:

i. Generalized closed (briefly g-closed) set if cl(A) ⊆ U whenever A ⊆ U and U is open
in X.

ii. Strongly generalized closed (or briefly g*-closed) set if cl(A) ⊆ U whenever A ⊆ U
and U is g-open in X.

iii. Generalized preclosed (or briefly gp-closed) set if pcl(A) ⊆ U whenever A ⊆ U and
U is open in X.



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 980

iv. Semi-generalized closed (or briefly sg-closed) set if scl(A) ⊆ U whenever A ⊆ U and
U is semi-open in X.

v. Generalized semiclosed (or briefly gs-closed) set if scl(A) ⊆ U whenever A ⊆ U and
U is open in X.

vi. Generalized b-closed (or briefly gb-closed) set if bcl(A) ⊆ U whenever A ⊆ U and U
is open in X.

vii. Generalized α-b-closed (or briefly gαb-closed) set if scl(A) ⊆ U whenever A ⊆ U and
U is α-open in X.

viii. Semi generalized b-closed set (or briefly sbg-closed) set if bcl(A) ⊆ U whenever A ⊆ U
and U is semi-open in X.

ix. Weakly closed (or briefly w-closed) set if cl(A) ⊆ U whenever A ⊆ U and U is
semi-open in X.

x. Generalized semi-preclosed (or briefly gsp-closed) set if spcl(A) ⊆ U whenever A ⊆ U
and U is open in X.

xi. Generalized α closed (or briefly g-α-closed) set if α−cl(int(A)) ⊆ U whenever A ⊆ U
and U is α-open in X.

xii. α-generalized closed (or briefly αg-closed) set if α− cl(int(A)) ⊆ U whenever A ⊆ U
and U is open in X.

xiii. Weakly generalized closed (or briefly wg-closed) set if cl(int(A)) ⊆ U whenever
A ⊆ U and U is open in X.

xiv. Mildly generalized closed (or briefly mildly g-closed) set if cl(int(A)) ⊆ U whenever
A ⊆ U and U is g-open in X.

xv. Semi weakly generalized closed (or briefly swg-closed) set if cl(int(A)) ⊆ U whenever
A ⊆ U and U is semi-open in X.

3. Main Results

3.1. (µ1, µ2, µ3)-Weakly Generalized Closed Sets

Definition 7. Let A be a subset of a nonempty set X and µ1, µ2, and µ3 be general-
ized topologies in X. We say that A is (µ1, µ2, µ3)-weakly generalized closed (or briefly
(µ1, µ2, µ3)-wg closed) set if clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is µ3-open in X.
We call the complement of every (µ1, µ2, µ3)-weakly generalized closed set as (µ1, µ2, µ3)-
weakly generalized open (or briefly (µ1, µ2, µ3)-wg open) set in X.



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 981

3.2. Special Cases of (µ1, µ2, µ3)-Weakly Generalized Closed Sets

Let X be a non empty set and A ⊆ X. Then the following are special cases of
(µ1, µ2, µ3)-weakly generalized closed sets with the corresponding conditions.

i. Generalized closed (or briefly g-closed) set
If µ1 = µ3 is a topology for X and µ2 is the discrete topology, then a (µ1, µ2, µ3)-
weakly generalized closed set is just a g-closed set since the condition “clµ1 (intµ2 (A)) ⊆
U whenever A ⊆ U and U is µ3-open in X” becomes “cl(A) ⊆ U whenever A ⊆ U
and U is open in X”.

ii. Strongly generalized closed (or briefly g*-closed) set
If µ1 is a topology in X, µ2 is the discrete topology in X, and µ3 = G(X) where G(X)
is the collection of g-open sets in X, then a (µ1, µ2, µ3)-weakly generalized closed set
is just the strongly generalized closed set since the condition “clµ1 (intµ2 (A)) ⊆ U
whenever A ⊆ U and U is µ3-open in X” becomes “cl(A) ⊆ U whenever A ⊆ U and
U is g-open in X”.

iii. Generalized preclosed (or briefly gp-closed) set
If µ1 = PO(X) where PO(X) is the collection of pre-open sets sets inX, µ2 is the dis-
crete topology in X, and µ3 is a topology in X, then a (µ1, µ2, µ3)-weakly generalized
closed set is just the generalized preclosed set since the condition “clµ1 (intµ2 (A)) ⊆ U
whenever A ⊆ U and U is µ3-open in X” becomes “pcl(A) ⊆ U whenever A ⊆ U
and U is open in X”.

iv. Semi-generalized closed (or briefly sg-closed) set
If µ1 = µ3 = SO(X) where SO(X) is the collection of semi-open sets in X, and µ2
is a discrete topology in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just a
semi-generalized closed set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U
and U is µ3-open in X” becomes “scl(A) ⊆ U whenever A ⊆ U and U is semi-open
in X”.

v. Generalized b-closed (or briefly gs-closed) set
If µ1 = SO(X) where SO(X) is the collection of semi-open sets in X, µ2 is the dis-
crete topology in X, and µ3 is a topology in X, then a (µ1, µ2, µ3)-weakly generalized
closed set is just a generalized semi-closed set since the condition “clµ1 (intµ2 (A)) ⊆ U
whenever A ⊆ U and U is µ3-open in X” becomes “scl(A) ⊆ U whenever A ⊆ U
and U is open in X”.

vi. Generalized semi-closed (or briefly gb-closed) set
If µ1 = BO(X) where BO(X) is the collection of b-open sets sets in X, µ2 is the dis-
crete topology in X, and µ3 is a topology in X, then a (µ1, µ2, µ3)-weakly generalized
closed set is just the generalized b-closed set since the condition “clµ1 (intµ2 (A)) ⊆ U
whenever A ⊆ U and U is µ3-open in X” becomes “bcl(A) ⊆ U whenever A ⊆ U
and U is open in X”.



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 982

vii. Generalized αb-closed (or briefly gαb-closed) set
If µ1 = BO(X) where BO(X) is the collection of b-open sets sets in X, µ2 is the
discrete topology in X, and µ3 = AO(X) where AO(X) is the collection of α-open
sets in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just the generalized
b-closed set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is
µ3-open in X” becomes “scl(A) ⊆ U whenever A ⊆ U and U is α-open in X”.

viii. Semi generalized b-closed (or briefly sbg-closed) set
If µ1 = BO(X) where BO(X) is the collection of b-open sets sets in X, µ2 is the
discrete topology in X, and µ3 = SO(X) where SO(X) is the collection of semi-open
sets inX, then a (µ1, µ2, µ3)-weakly generalized closed set is just the semi-generalized
b-closed set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is µ3-
open in X” becomes “bcl(A) ⊆ U whenever A ⊆ U and U is semi-open in X”.

ix. Weakly closed (or briefly w-closed) set
If µ1 is a topology in X, µ2 is the discrete topology in X, and µ3 = SO(X) where
SO(X) is the collection of semi-open sets in X, then a (µ1, µ2, µ3)-weakly generalized
closed set is just the weakly closed set since the condition “clµ1 (intµ2 (A)) ⊆ U
whenever A ⊆ U and U is µ3-open in X” becomes “cl(A) ⊆ U whenever A ⊆ U and
U is semi-open in X”.

x. Generalized semi-preclosed (or briefly gsp-closed) set
If µ1 = SPO(X) where SPO(X) is the collection of semi-preopen sets in X, µ2 is
the discrete topology in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just
the generalized semi-preclosed set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever
A ⊆ U and U is µ3-open in X” becomes “spcl(A) ⊆ U whenever A ⊆ U and U is
open in X”.

xi. Generalized α closed (or briefly gα-closed) set
If µ1 = µ3 = AO(X) where AO(X) is the collection of α-open sets in X, and
µ2 is a topology in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just the
generalized gα-closed set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U
and U is µ3-open in X” becomes “αcl(int(A)) ⊆ U whenever A ⊆ U and U is α-open
in X”.

xii. α-generalized closed (or briefly αg-closed) set
If µ1 = AO(X) where AO(X) is the collection of α-open sets in X, µ2 is the dis-
crete topology in X, and µ3 is a topology in X, then a (µ1, µ2, µ3)-weakly gen-
eralized closed set is just the generalized α generalized closed set since the con-
dition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is µ3-open in X” becomes
“αcl(int(A)) ⊆ U whenever A ⊆ U and U is open in X”.

xiii. Weakly generalized closed (or briefly wg-closed) set
If µ1 = µ2 = µ3 is a topology in X, then a (µ1, µ2, µ3)-weakly generalized closed
set is just the weakly generalized closed set since the condition “clµ1 (intµ2 (A)) ⊆ U



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 983

whenever A ⊆ U and U is µ3-open in X” becomes “cl(int(A)) ⊆ U whenever A ⊆ U
and U is open in X”.

xiv. Mildly generalized closed (or briefly mildly g-closed) set
If µ1 = µ2 is a topology in X, and µ3 = G(X) where G(X) is the collection of g-open
sets in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just the weakly closed
set since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is µ3-open in
X” becomes “cl(int(A)) ⊆ U whenever A ⊆ U and U is g-open in X”.

xv. Semi-weakly generalized closed (or briefly swg-closed) set
If µ1 = µ2 in X, and µ3 = SO(X) where SO(X) is the collection of semi-open sets
in X, then a (µ1, µ2, µ3)-weakly generalized closed set is just the weakly closed set
since the condition “clµ1 (intµ2 (A)) ⊆ U whenever A ⊆ U and U is µ3-open in X”
becomes “cl(int(A)) ⊆ U whenever A ⊆ U and U is semi-open in X”.

Theorem 7. Let (X,µ1, µ2, µ3) be a trigeneralized space and A ⊆ X. Then, A is
(µ1, µ2, µ3)-weakly generalized open set if and only if F ⊆ intµ1 (clµ2 (A)) whenever F ⊆ A
and F is µ3-closed in X.

Proof. Let A be (µ1, µ2, µ3)-wg open set and F ⊆ A such that F is µ3-closed. Then
Ac is (µ1, µ2, µ3)-wg closed, F c is µ3-open, and Ac ⊆ F c. That is F ⊆ [clµ1(intµ2)]c =
intµ1(clµ2(A)). Hence, F ⊆ intµ1 (clµ2 (A)) whenever F ⊆ A and F is µ3-closed.

Now, let F ⊆ A and F be µ3-closed set in X such that F ⊆ intµ1 (clµ2 (A)). Taking

complementation, we have
[
intµ1 (clµ2 (A))

]c ⊆ F c whenever Ac ⊆ F c and F c is µ3-open
in X. But [

intµ1 (clµ2 (A))
]c

= clµ1
[
(clµ2 (A))c

]
= clµ1 (intµ2 (Ac))

So clµ1 (intµ2 (Ac)) ⊆ F c whenever Ac ⊆ F c and F c is µ3-open in X. This means that Ac

is (µ1, µ2, µ3)-wg closed set. Therefore A is (µ1, µ2, µ3)-wg open set.

Theorem 8. If A is µ1-closed then A is (µ1, µ2, µ3)-wg closed set.
Proof. Let A be µ1-closed and U be µ3-open such that A ⊆ U . By Theorem 2(i.),

intµ2 (A) ⊆ A and applying Theorem 3(iv.), clµ1 (intµ2 (A)) ⊆ clµ1(A). Since A is µ1 closed,
clµ1(A) = A. Thus, clµ1 (intµ2 (A)) ⊆ A ⊆ U . Therefore, A is (µ1, µ2, µ3)-wg closed set.

Theorem 9. If F ⊆ A and A is µ1-closed, then F is (µ1, µ2, µ3)-wg closed set.
Proof. Let A be µ1-closed and F ⊆ A. Suppose A ⊆ U and U is µ3-open. Since

F ⊆ A, intµ2(F ) ⊆ intµ2(A) using Theorem 2(iv.). Consequently, by Theorem 3(iv.),
clµ1 (intµ2 (F )) ⊆ clµ1 (intµ2 (A)). Moreover, by Theorem 8, clµ1 (intµ2 (A)) ⊆ U , thus
clµ1 (intµ2 (F )) ⊆ U . Therefore, F is (µ1, µ2, µ3)-wg closed set.

Theorem 10. If A is (µ1, µ2, µ3)-wg closed subset of X and A ⊆ B ⊆ clµ1 (intµ2 (A)),
then B is (µ1, µ2, µ3)-wg closed set.



B. Agua, R. Paluga / Eur. J. Pure Appl. Math, 13 (4) (2020), 977-986 984

Proof. Suppose thatA is (µ1, µ2, µ3)-wg closed subset ofX andA ⊆ B ⊆ clµ1 (intµ2 (A)).
Let U be µ3-open and B ⊆ U . Since A ⊆ B, then A ⊆ U . Also, since A is (µ1, µ2, µ3)-
wg closed set, clµ1 (intµ2 (A)) ⊆ U . Now, since B ⊆ clµ1 (intµ2 (A)) and using Theorem
2(i.), intµ2 (B) ⊆ B ⊆ clµ1 (intµ2 (A)). Consequently, by Theorem 3(iv.), clµ1(intµ2 (B)) ⊆
clµ1(B) ⊆ clµ1 (intµ2 (A)). In effect, clµ1 (intµ2 (B)) ⊆ U . Hence, B is (µ1, µ2, µ3)-wg closed
set.

Theorem 11. If A is (µ1, µ2, µ3)-wg closed set, then clµ1 (intµ2 (A)) − A contains no
nonempty µ3-closed set.

Proof. Let A be (µ1, µ2, µ3)-wg closed set and F be a nonempty µ3-closed set such
that F ⊆ clµ1 (intµ2 (A) − A. Then F ⊆ clµ1 (intµ2 (A))

⋂
Ac. This implies that F ⊆

clµ1 (intµ2 (A)) and F ⊆
⋂
Ac. Note that F c is µ3-open and A ⊆ F c. Now, since A is

(µ1, µ2, µ3)-wg closed, then clµ1 (intµ2 (A)) ⊆ F c. Thus, F ⊆ clµ1 (intµ2 (A)) ⊆ F c. This
means that F = F

⋂
F c = ∅. This is a contradiction. Hence if A is (µ1, µ2, µ3)-wg closed

set, then clµ1 (intµ2 (A))−A contains no nonempty µ3-closed set.

Remark 1. The converse of Theorem 11 is not necessarily true.

Example 1. Let X = {1, 2, 3}, µ1 = {∅, {1}, {1, 3}}, µ2 = {∅, {1}, {2}, {1, 2}}, and
µ3 = {∅, {2}, {3}, {2, 3}}. The µ1-closed sets are X, {2, 3} and {2}. Also, the µ2-open sets
are ∅, {1}, {2}, and {1, 2}. The µ3-open sets on the other hand are ∅, {2}, {3}, and {2, 3}
whose corresponding µ3-closed sets are X, {1, 3}, {1, 2}, and {1}. Observe that considering
all the possible subsets of X, only the sets: ∅ and {3} are not (µ1, µ2, µ3)-wg closed sets.
If A = ∅, intµ2(∅) = ∅. Consequently, clµ1(intµ2(∅)) = {2}. Hence, clµ1(intµ2(∅))\∅ = {2}
which is not a µ3-closed set. So “ clµ1(intµ2(A))\A contains no nonempty µ3-closed set”
is satisfied. Thus, when A = ∅ the statement “ If clµ1(intµ2(A))\A contains no nonempty
µ3-closed set, then A is (µ1, µ2, µ3)-wg closed sets.” is false.
Moreover, if A = {3}, intµ2({3}) = ∅. Consequently, clµ1(intµ2({3})) = {2}. Thus,
clµ1(intµ2({3}))\{3} = {2} which is not a µ3-closed set. That is, “clµ1(intµ2(A))\A con-
tains no nonempty µ3-closed” is satisfied. Thus, ifA = {3} the statement “clµ1(intµ2(A))\A
contains no nonempty µ3-closed set, then A is (µ1, µ2, µ3)-wg closed sets.” is false.

Remark 1. states that the converse of Theorem 11 is not necessarily true. This is
illustrated by Example 1. However, considering µ1 ⊆ µ3 where µ1 and µ3 are GTs in X,
then we can consider the statement “If clµ1 (intµ2 (A))−A contains no nonempty µ3-closed
set, then A is (µ1, µ2, µ3)-wg closed set.”

Theorem 12. Let µ1 ⊆ µ3. If clµ1 (intµ2 (A)) − A contains no nonempty µ3-closed set,
then A is (µ1, µ2, µ3)-wg closed set.

Proof. Let A ⊆ X. Suppose clµ1 (intµ2 (A))−A contains no nonempty µ3-closed set and
A is not a (µ1, µ2, µ3)-wg closed set. Then there exists µ3-open set U such that A ⊆ U and
clµ1 (intµ2 (A)) * U . Now, clµ1 (intµ2 (A)) * U implies that clµ1 (intµ2 (A))

⋂
U c 6= ∅. Let

F = clµ1 (intµ2 (A))
⋂
U c. By Theorem 3(ii.), clµ1 (intµ2 (A)) is µ1-closed. Since µ1 ⊆ µ3,



REFERENCES 985

applying Theorem 4, clµ1 (intµ2 (A)) is µ3-closed. Thus since U c is µ3-closed and using
Theorem 1, F = clµ1 (intµ2 (A))

⋂
U c is µ3-closed. Now, F 6= ∅ and

F = clµ1 (intµ2 (A))
⋂
U c ⊆ clµ1 (intµ2 (A))

⋂
Ac = clµ1 (intµ2 (A))−A.

This is a contradiction. Therefore, if clµ1 (intµ2 (A))− A contains no nonempty µ3-closed
set, then A is (µ1, µ2, µ3)-wg closed set.

Theorem 13. If A be µ1-closed and µ2-open, then A is (µ1, µ2, µ3)-wg closed set.
Proof. Let A ⊆ U such that A is both µ1-closed and µ2-open, and U is µ3-open

such that A ⊆ U . Since A is µ2-open and µ1-closed, then applying Theorem 2(iii.) and
Theorem 3(iii.), intµ2(A) = A and clµ1(A) = A. Thus, clµ1 (intµ2 (A)) = clµ1 (A) = A. In
effect, clµ1 (intµ2 (A)) ⊆ U . Hence, A is (µ1, µ2, µ3)-wg closed set.

Definition 8. Let A ⊆ Y ⊆ X.Then A is (µ1, µ2, µ3)-wg closed set in Y if clµ1 (intµ2 (A)) ⊆
U whenever A ⊆ U and U is µ3-open in Y .

Theorem 14. Let X 6= ∅ and A ⊆ Y ⊆ X. If A is (µ1, µ2, µ3)-wg closed set in X and Y
is µ1-closed set in X, then A is (µ1, µ2, µ3)-wg closed set in Y .

Proof. Let A ⊆ U such that U is µ3-open in Y . Then U = Y
⋂
G for some µ3-open set

G in X. Note that A ⊆ G. Since A is (µ1, µ2, µ3)-wg closed in X, then clµ1 (intµ2 (A)) ⊆ G.
Now, by Theorem 2(i.), intµ2 (A) ⊆ A. Since A ⊆ Y , intµ2 (A) ⊆ Y . By Theorem 3(iv.),
clµ1 (intµ2 (A)) ⊆ clµ1 (Y ). But Y is µ1-closed, so clµ1 (Y ) = Y . Thus, clµ1 (intµ2 (A)) ⊆
Y . Accordingly, clµ1 (intµ2 (A)) = clµ1 (intµ2 (A))

⋂
Y ⊆ G

⋂
Y = U . Therefore, A is

(µ1, µ2, µ3)-wg closed set in Y .

Acknowledgements

The authors would like to express their thanks to the referees who gave their suggestions
for the improvement of the paper.

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