EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 199-206
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Weakly Covering Spaces in Coc-open Sets

Fuad A. Abushaheen1,∗, Fadi Alrimawi2

1 Basic Science Department, Middle East University, Amman, Jordan
2 Department of Basic Sciences, Al-Ahliyya Amman University, Amman, Jordan

Abstract. In this paper, we define several types of weakly covering spaces concerning coc-open
sets, we study some implications between these weakly covering spaces. Also, we give several
product theorems under coc-almost open function.

2020 Mathematics Subject Classifications: 54B05, 54B10, 54D20
Key Words and Phrases: coc-weakly-compact, coc-almost-compact, coc-nearly-compact, coc-
R-compact

1. Introduction and Preliminaries

It is known that covering spaces play very important role in topology. After the con-
cept of compactness defined, many authors studied the relations between this notion and
other topological and analytical concepts. Moreover the topologists gave many different
generalizations of compactness stands on the types of covers, open sets and subcovers.
The discussion about these covering spaces is still a rich area to study in topology.

Also, the idea of giving weaker and stronger forms of open sets is also a hot topic in
research. For example semi-open set [9], pre-open [10], no where dense [12] and many
more.

In [8], a regular-open set is a set that equals to interior of its closure or equivalently
, a subset A of a topological space X is regular open if int(A) = A where the closure of
A and the interior of A are denoted by A and int(A), respectively. The complement of a
regular open said is said to be regular closed, or A = int(A). It is clearly that regularly
open is an open set. For further studied in regular open spaces in covering spaces and
separation axioms, see [1–5].

In 2012, Al-Ghour and Samarah [7] defined a new type of open sets called coc-compact,
a subset A of a topological space X is called coc-open set if A is a union of sets of the
form V −C, where V is open set and C is a compact subset of X. Also, they proved that
the family of all coc-open sets of a topological space (X, τ) forms a topology on X finer
than τ . For more results about coc-open subsets, see [11].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4248

Email addresses: fshaheen@meu.edu.jo (F.A. Abushaheen), f.rimawi@ammanu.edu.jo (F. Alrimawi)

http://www.ejpam.com 199 © 2022 EJPAM All rights reserved.



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 200

In this paper, no separation axioms to be assumed. A space X means nonempty
sets with topology. ω0 denoted the cardinal number of N, for a subset A of X, A

coc
and

intcoc(A) will denote the closure of A and the interior of A in τk, respectively. For notation
not mention here, we refer the reader to [6].

The following definitions are important for our paper.

Definition 1. [7] A subset A of a topological space X is called co-compact open set (nota-
tion: coc-open) if for every x ∈ A, there exists an open set U ⊆ X and a compact subset K
of X such that x ∈ U −K ⊆ A. The complement of a coc-open subset is called coc-closed.
The family of all coc-open subsets of a topological space (X, τ) will be denoted by τk.

Definition 2. [11] A topological space X is called coc-T2-space if and only if for all
x, y ∈ X with x ̸= y ∈ X, there exist U, V ∈ τk such that x ∈ U, y ∈ V with U ∩ V = ϕ.

Definition 3. A subset A of a topological space X is coc-regular open if intcoc(A
coc

) = A,
the complement of a coc-regular open is said to be coc-regular closed, or A = intcoc(A)

coc
.

Definition 4. [7] A family U ⊆ X is coc-open cover of X if U covers X and U is a
subfamily of τk.

Definition 5. A coc-open cover U = {Uα|α ∈ ∆} of X is called coc-regular cover, if
for each α ∈ ∆ there exists a coc-regular closed set Fα such that Fα ⊆ Uα and X =⋃
{intcoc(Fα)|α ∈ ∆}.

Definition 6. [7] A space (X, τ) is coc-compact if every coc-open cover has a finite sub-
cover.

Definition 7. A space (X, τ) is coc-almost-compact if every coc-open cover of X has a
finite collection such that the coc-closure of the union is X.

Definition 8. A space (X, τ) is coc-weakly-compact if every coc-regular cover of X has a
finite collection such that the coc-closure of the union is X.

Definition 9. A space (X, τ) is coc-nearly-compact if every coc-open cover of X by coc-
regular open sets has a finite subcover.

Clearly the following implications are true for a topological space X:
coc-compact ⇒ coc-nearly compact ⇒ coc-almost compact ⇒ coc-weakly compact.

The following definition gives a coc-covering space between coc-almost-compact and
coc-weakly-compact spaces.

Definition 10. A space X is called coc-R-compact if every coc-regular cover of X has a
finite subcover.

It is obvious that coc-R-compactness implies coc-weak-compactness.

Theorem 1. A coc-almost-compact space X is coc-R-compact-space.

Proof. Let U =
{
Uα|α ∈ ∆

}
be a coc-regular cover of X with Fα ⊆ Uα and X =⋃{

intcoc(Fα)|α ∈ ∆
}
, but X is coc-almost-compact, so there exists a finite subset ∆0 of

∆ such that X =
⋃{

intcoc(Fα)
coc|α ∈ ∆0

}
, so X =

⋃{
Uα|α ∈ ∆0

}
=

⋃{
Uα|α ∈ ∆0

}coc
,

hence X is coc-R-compact-space.



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 201

2. Main Results

In the beginning of this section, we explore some results concerning subspaces via
coc-weakly covering spaces.

Definition 11. A subset S of a space X is said to be coc-weakly-compact subset if S is
coc-weakly-compact subspace of X.

Definition 12. A subset S of a space X is said to be coc-weakly-compact relative to X,
if for each cover

{
Vα|α ∈ ∆

}
of S by coc-open sets of X satisfying the following condition

(∗∗), there exists a set ∆0 ⊆ ∆ such that S ⊆
⋃{

Vα
coc|α ∈ ∆0 ⊆ ∆, |∆0| < ω0

}
, where

the condition (∗∗) is :
”for each α ∈ ∆, there exists a nonempty coc-regular closed set Fα such that Fα ⊆ Vα and
S ⊆

⋃{
intcoc(Fα)|α ∈ ∆

}
” (∗∗).

Theorem 2. If A is a coc-weakly-compact subspace of a space X, then A is coc-weakly-
compact relative to X.

Proof. Let
{
Vα|α ∈ ∆

}
be a cover of A by coc-open sets of X satisfying the condition

(∗∗), for each α ∈ ∆ there exists a non empty coc-regular closed set Fα such that Fα ⊆ Vα

and A ⊆
⋃{

intcoc(Fα)|α ∈ ∆
}
. For each α ∈ ∆, intcoc(Fα) ∩A and Vα ∩A are coc-open

subsets in A and Fα ∩ A is coc-closed in A, so the family
{
Vα ∩ A|α ∈ ∆

}
is coc-open

cover of A, therefore for each α ∈ ∆, we have

intcoc(Fα) ∩A ⊆ intcoc(Fα) ∩A
coc(A) ⊆ Fα ∩A ⊆ Vα ∩A ⊆ A,

moreover
A =

⋃{
intcoc(Fα) ∩A|α ∈ ∆

}
and

intcoc(Fα) ∩A ⊆ intcoc(A)

(
intcoc(Fα ∩A

)coc(A)
.

Now, the set intcoc(Fα ∩A
)coc(A)

is coc-regular closed in A and
{
Vα ∩ A|α ∈ ∆

}
is a

coc-regular cover of A, so there exists a finite subset ∆0 ⊆ ∆ such that

A =
⋃{

Vα ∩A
coc(A)|α ∈ ∆

}
, but Vα ∩A

coc(A) ⊆ Vα
coc(A)

for each α ∈ ∆0, therefore

A ⊆
⋃{

Vα
coc|α ∈ ∆0

}
, hence the result.

Corollary 1. If A is coc-compact (coc-almost-compact, coc-nearly-compact) subspace of
X, then A is coc-weakly-compact relative to X.

Theorem 3. If every coc-regular closed subset of a space X is coc-weakly compact relative
to X, then X is coc-weakly compact.

Proof. Let
{
Vα|α ∈ ∆

}
be a coc-regular cover of X, for each α ∈ ∆, there exists

nonempty coc-regular set such that Fα ⊆ Vα and X =
⋃{

intcoc(Fα)|α ∈ ∆
}
. For α0 ∈ ∆

let K = X − intcoc(Fα0), then K is a coc-regular closed and K ⊆
⋃{

intcoc(Fα)|α ∈



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 202

∆− {α0}
}
, so V∗ =

{
Vα|α ∈ ∆− {α0}

}
is a cover of K by coc-open sets of X satisfying

condition (**) and hence for some finite set ∆0 ⊆ ∆ we have K ⊆
⋃{

Vα
coc|α ∈ ∆0

}
, thus

X = K ∪ intcoc(Fα0) ⊆
⋃{

Vα
coc|α ∈ ∆0 ∪ {α0}

}
,

so X is coc-weakly compact.

Corollary 2. If every proper coc-regular closed subset of a space X is coc-weakly compact,
then X is coc-weakly compact.

Proof. Clearly.

Corollary 3. If every proper coc-regular closed subset of a space X is coc-compact (coc-
almost-compact, coc-nearly-compact) subspace, then X is coc-weakly compact.

Theorem 4. If a subset A of a coc-weakly-compact space X is coc-clopen, then A is
coc-weakly-compact relative to X.

Proof. Let UA =
{
Uα|α ∈ ∆

}
be a cover of A by coc-open sets of X satisfying

condition (**), hence the family
{
Uα|α ∈ ∆

}
∪
{
X −A

}
is a coc-regular cover of X, but

X is coc-weakly-compact, so there exists a finite set ∆0 ⊆ ∆ such that

X ∪
( ⋃
α∈∆0

Vα
coc) ∪ (

X −A
coc)

=
( ⋃
α∈∆0

Vα
coc) ∪ (

X −A
)
,

therefore A ⊆
⋃

α∈∆0
Vα

coc
, therefore A is coc-weakly-compact relative to X.

The next definition makes our coc-weakly covering spaces equivalent.

Definition 13. A space X is said to be coc-almost regular if for each coc-regular closed
F and x ∈ X − F , there exist disjoint coc-open sets U, V such that F ⊆ U and x ∈ V .

Theorem 5. If a space X is coc-weakly-compact and coc-almost-regular, then X is coc-
nearly-compact.

Proof. Let U =
{
Uα|α ∈ ∆

}
be coc-regular open cover of X, so for each x ∈ X, there

exists α(x) ∈ ∆ such that x ∈ Uα(x), but X is coc-almost regular space, so there exist
coc-regular open sets Vα(x) and Wα(x) such that

x ∈ Vα(x) ⊆ Vα(x)
coc ⊆ Wα(x) ⊆ Wα(x)

coc ⊆ Uα(x),

and clearly the family
{
Wα(x)|α(x) ∈ ∆

}
is coc-regular cover of X, so there exists a finite

elements x1, x2, ..., xn ∈ X such that

X =

n⋃
i=1

Wα(xi)
coc ⊆

n⋃
i=1

Uα(xi),

therefore X is coc-nearly-compact space.



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 203

Corollary 4. A coc-almost-regular space is coc-weakly-compact if and only if it is coc-
nearly-compact.

Proof. Obvious.

Corollary 5. A coc-T2-space is coc-nearly-compact if and only if it is coc-weakly compact
and coc-almost regular.

Proof. (⇐) Clearly.
(⇒) Enough to show that X is coc-almost regular. Let A be a coc-regular closed set and
X − A, assume y ∈ A. Since X is coc-T2-space, there exists a coc-open set Vy contains
y with x /∈ Vy

coc
. The family V =

{
Vy|y ∈ A

}
∪
{
X − A

}
is coc-regular open cover of

X, so there exists a finite subcover V∗ of X, let V =
⋃{

Vα ∈ V∗|Vα ∩ A ̸= ϕ
}
, then V

is coc-open subset of X and A ⊆ V , therefore for each Vα ∈ V∗ we have Vα ⊆ Vy and
x /∈ V

coc
, hence X is coc-almost-regular.

Corollary 6. The following are equivalent for a coc-almost-regular space:

(i) X is coc-nearly-compact,

(ii) X is coc-almost-compact,

(iii) X is coc-R-compact,

(iv) X is coc-weakly-compact.

Proof. The prove comes from Theorems 5 and 1.

Corollary 7. Let (X, τk) be a T3-space and X is coc-weakly-compact, then X is coc-
compact space.

Proof. The result comes from Theorem 5 and the fact that every T3-space in (X, τk)
is coc-almost regular .

Definition 14. A function f : X → Y is said to be coc-almost open function if f(U) ⊆
intcoc

(
f(U)

coc)
for every coc-open set U of X.

Lemma 1. A function f : X → Y is coc-almost open if and only if f−1
(
V

coc) ⊆ f−1
(
V
)coc

for each coc-open subset V of Y .

Proof. (⇒): Let V be a coc-open subset of Y with x ∈ f−1
(
V

coc)
and U is a coc-open

subset ofX contains x, then f(x) ∈ f(U)∩V coc ⊆ intcoc
(
U

coc)∩V coc
, so V ∩intcoc

(
U

coc) ̸=
ϕ and U ∩ f−1(V ) ̸= ϕ, therefore x ∈ f−1

(
V
)coc

.
(⇐): If f is not coc-almost open function, so there is a coc-open set U with f(U) ⊈
intcocf

(
U

coc)
. Let V = Y−f(U)

coc
, then V ∩f(U) = ϕ, but V ⊆ Y−f(U), so V

coc∩f(U) ̸=
ϕ, so U ∩f−1

(
V

coc) ̸= ϕ and U ∩f−1
(
V
)coc

̸= ϕ, hence U ∩f−1(V ) ̸= ϕ which contradicts
the fact that f(U) ∩ V = ϕ, hence the result.



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 204

Definition 15. A function f : X → Y is called coc-perfect function if it is surjective,
coc-closed and f−1(y) coc-compact.

Theorem 6. Let f : X → Y be a coc-almost-open coc-perfect function. If K is a coc-
weakly-compact relative to Y , then f−1

(
K
)
is coc-weakly-compact relative to X.

Proof. Let K be coc-weakly-compact relative to Y and {Uα|α ∈ ∆} any cover of
f−1(K) by coc-open sets of X satisfies condition (∗∗). For each α ∈ ∆ there exists a
coc-regular closed set Fα of X such that Fα ⊆ Uα and
f−1(K) ⊆

⋃{
intcoc

(
Fα

)
|α ∈ ∆

}
.

Now for each y ∈ Y , we have

f−1(y) ⊆
⋃{

intcoc
(
Fα

)
|α ∈ ∆(y), |∆(y)| < ω0

}
and

f−1(y) ⊆
⋃

α∈∆(y)

intcoc
(
Fα

)
⊆

⋃
α∈∆(y)

Uα.

Define
Gy = Y − f

(
X −

⋃{
intcoc

(
Fα

)
|α ∈ ∆(y), |∆(y)| < ω0

})
,

Hy = Y − intcoc
(
f
(
X −

⋃{
Fα|α ∈ ∆(y), |∆(y)| < ω0

})coc
,

and
Vy = V − f

(
X −

⋃{
Uα|α ∈ ∆(y), |∆(y)| < ω0

})
,

clearly Gy and Vy are coc-open subsets in Y , Hy is coc-regular closed with y ∈ Gy ⊆
Hy ⊆ Vy, therefore the family V = {Vy|y ∈ K} is a cover of K by coc-open subsets of Y
satisfies condition (∗∗), but K is a coc-weakly-compact relative to Y , so there exists set
y1, y2, ..., yn ∈ Y of such that

K ⊆
⋃

{Vyi
coc|i = 1, 2, ..., n

}
,

but f−1
(
Vy

coc) ⊆ f−1
(
Vy

)coc
for all y ∈ Y , so

f−1
(
Vy

)
⊆

⋃{
Uα|α ∈ ∆(y)

}
,

therefore
f−1

(
K
)
⊆

{
Uα

coc|α ∈ ∆(yi), i = 1, 2, ..., n
}
,

hence f−1
(
K
)
is coc-weakly-compact relative to X.

Corollary 8. The coc-open coc-perfect function of a coc-weakly-compact space is coc-
weakly-compact.

Theorem 7. If f : X → Y is a coc-continuous surjective function and X is coc-R-
compact, then Y is coc-R-compact.



F.A. Abushaheen, F. Alrimawi / Eur. J. Pure Appl. Math, 15 (1) (2022), 199-206 205

Proof. Let U =
{
Uα|α ∈ ∆

}
be a coc-regular cover of Y . For each α ∈ ∆, there exists

a coc-closed subset Fα ⊆ Uα and Y =
⋃{

intcoc(Fα)
}
. Now f−1

(
U
)
=

{
f−1(Uα)|α ∈ ∆

}
,

f−1(Fα) ⊆ f−1(Uα) and
X = f−1(Y ) = f−1

(⋃
intcoc(Fα)

)
⊆

⋃
f−1

(
Uα

)
= X, clearly f−1

(
U
)
is coc-regular cover

of X, so there exists a finite subcover such that

X =
⋃{

f−1(Uα)|α ∈ ∆0 ⊆ ∆, |∆0| < ω0

}
,

hence the result.

Theorem 8. Let f : X → Y be a coc-almost coc-open and coc-perfect function. If Y is
coc-R-compact space, then X is so.

Proof. Let U =
{
Uα|α ∈ ∆

}
be a coc-regular cover of X, so for each α ∈ ∆, there

exists a coc-closed subset Fα ⊆ Uα and X =
⋃{

intcoc(Fα)
}
, but f is coc-perfect function,

so for each y ∈ Y , there is a finite set ∆0 ⊆ ∆ such that

f−1(y) ⊆
⋃

α∈∆0

intcoc(Fα) ⊆
⋃

α∈∆0

Fα ⊆
⋃

α∈∆0

Uα.

Define
Ky = Y − f

(
X −

⋃{
intcoc

(
Fα

)
|α ∈ ∆0

})
,

Jy = Y − f
(
X −

⋃
{intcoc(Fα)|α ∈ ∆0

}coc)
,

and
Ly = Y − f

(
X −

⋃{
Uα|α ∈ ∆0

})
.

ClearlyKy ⊆ Jy ⊆ Ly, alsoKy and Ly are coc-open subsets of Y and Jy is coc-closed subset
in Y , so for each y ∈ Y ,there exists α ∈ ∆0 such that y ∈ Ky, therefore

{
Ky|y ∈ Y

}
is a

coc-regular cover of Y , but Y is coc-R-compact, so Y =
⋃{

Lyi |i = 1, 2, ..., n
}
, moreover

X = f−1
(
Y
)
= f−1

(
{Lyi |i = 1, 2, ..., n

})
⊆

⋃{
Uα|α ∈ ∆0

}
,

this complete the proof.

Corollary 9. Let f : X → Y be a coc-open coc-perfect coc-continuous, then X is coc-R-
compact space if and only if Y is coc-R-compact space.

Corollary 10. If X is coc-weakly-compact space and Y is coc-compact space, then X×Y
is coc-weakly-compact.

Proof. Notice that the projection PX : X × Y → X is coc-closed and the result comes
from Theorem 6.

Corollary 11. For a coc-almost-regular spaces X and Y , if X is coc-compact, then X×Y
is coc-nearly (coc-weakly-compact, coc-R-compact).

Corollary 12. If Y is coc-compact space, then X × Y is coc-almost-compact (coc-nearly-
compact, coc-weakly-compact) if and only if X is coc-R-compact space.



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Acknowledgements

The authors are grateful to the Middle East University, Amman, Jordan for the finan-
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