EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2990-3003
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Results on Difference Paracompactness in
Topological Spaces

Rahmeh Alrababah1, Ala Amourah2,3,∗, Jamal Salah4,∗, Reyaz Ahmad5,
Ali A. Atoom1

1 Ajloun National Previate University, College of Science, Department of Mathematics,
Jordan
2 Mathematics Education Program, Faculty of Education and Arts,
Sohar University, Sohar 3111, Oman
3 Applied Science Research Center, Applied Science Private University, Amman, Jordan
4 College of Applied and Health Sciences, A’Sharqiyah University, Post Box No. 42,
Post Code No. 400 Ibra, Sultanate of Oman
5 Skyline University College of Sharjah, Department of General Education, United Arab
Emirates

Abstract. An interesting area of research in topology is D−paracompact spaces. It is a significant
type of topological spaces that retain compactness while benefiting from paracompactness, which is
considered a generalisation of compact spaces. The concept ofD−paracompactness was introduced,
and its basic characteristics were examined by the author in [18]. In this research, we introduce
and improve this concept further by using a special type of covering and the difference sets (called
D−sets), which contain new and impact properties. As a result, we obtained several new properties
and results. We discuss the concept, characteristics, and theorems that related of D−paracompact
space. We also studied different characterizations of D−paracompact spaces and discussed how
they relate to other topological characteristics. We also give numerous instances ofD−paracompact
spaces along with highlighting their applicability in different topological spaces.

2020 Mathematics Subject Classifications: 54B05, 54B10, 54C05, 54D05, 54D10, 54D30,
54E55

Key Words and Phrases: Topological spaces, Paracompact spaces, D−paracompact spaces,
Countably D−paracompact spaces

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5426

Email addresses: rrr.ra2020r@gmail.com (R. Alrababah), AAmourah@su.edu.om (A. Amourah),
damous73@yahoo.com (J. Salah), aliatoom@anu.edu.jo (A. Atoom)

https://www.ejpam.com 2990 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2991

1. Overview of the Historical Developments and Progress

Soon after the notion of paracompact spaces was introduced, several articles were pub-
lished that examine the idea from various angles and tools. In the year 1944, Dieudonnè
[7] first suggested the notion of paracompact space as a concept that is the extension of
compactness. Later, in 1969, Singal and Arya [20] provided concept a new type of para-
compactness named nearly paracompactness, weaker than the usual paracompactness, that
determines its basic topological properties. Based on that, in 1978, Steen and Seebach [21]
introduced the notion of metacompact in the topological space (X, τ). In 1982, Tong [22]
introduced the notion of difference sets, often known as D−sets. Also, in 1984, Pareek [16],
contributed the concept ofD−paracompact and studied their properties and relations with
other topological spaces. In 1998, Mukherjee and Debray [15] used paracompact to give,
the same definition. It has been investigated in terms of a certain type of cover, called
regular even cover. While, in 2006, the topologists Al-Ghour [13] defined the concepts
of ω−paracompactness and countable ω−paracompactness as generalizations of paracom-
pactness. He explained characterized each of them, which he studied dealt with subspaces,
products, and mappings of each. Also, in 2006, Al-Zoubi [2] as a generalization of paracom-
pact spaces, he uses the semi−open sets to developed the class of S−paracompact space
and define S−paracompact spaces, he also investigated their fundamental characteristics.
The investigations are made into the connections between S−paracompact spaces and
other well-known spaces. Later, in 2007, Al-Zoubi and Al-Ghour [1] provided and stud-
ied P3−paracompact, a weaker variant of paracompactness. With it and its interactions
with other spaces, they obtained a variety of characterizations, properties, instances, and
counterexamples. In 2013, Demir and Ozbakir [6] defined the concepts β−paracompact
spaces and β−expandable spaces as a weak variant of paracompact and expandable spaces,
respectively. These spaces’ basic characteristics were also provided. Additionally, it is es-
tablished that every β−paracompact space is a β−expandable space, and the connections
between these spaces and a few previously researched spaces are investigated. In 2019,
Turanli and Ozbakir [23], introduced and investigated β1 − L−paracompact space, which
is a stronger variant of L−paracompact space established on an ideal space. After that,
they looked into connections between β1−L−paracompact spaces and other paracompact-
nesses. Additionally, they discovered numerous characteristics, instances, and counterex-
amples of β1 − L−paracompactness. In 2021, Al Ghour [12], introduced both concepts of
σ − ω−paracompactness and feebly ω−paracompactness, with ω−paracompactness being
a weaker version of σ−ω−paracompactness. Furthermore, In 2021, Oudetallah, et al. [16]
introduced the notions of D−metacompact spaces and studied their properties. However,
for more studies, you can see [14], [3], and [10]. The present research is mainly a contin-
uation of the study of D−paracompact spaces from new viewpoints. We have aimed to
eventually arrive at new concepts as well as a develop several characteristic theorems and
instances, all of which we present in this research.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2992

2. Preliminary and Basic Notions

Within this section, we include some essential symbols, such as R to denote the col-
lection of real numbers. Also, ϑu, ϑcof , ϑl.r, ϑind, and ϑdis will denote the standard,
co-finite, left-ray, indiscreet, and discrete topologies, respectively. Now, we provide some
basic definitions as well as the major results that are required.

Definition 1. [7] If any open cover of the topological space (W,ϑ) has an open locally−finite
refinement, then the space is called paracompact space.

Definition 2. [8] If any countable open cover of the topological space (W,ϑ) has an open
locally−finite refinement, then the space is called countably paracompact.

Definition 3. [23] If any open cover of the topological space (W,ϑ) has an open locally-
countable refinement, then the space is called paralindelöf space.

Definition 4. [22] A subset W1 ⊆ (W,ϑ) is D−set if there are open sets A and B such
that A ̸= W and W1 = A−B. Furthermore, the subset W1 is D−set that generated by A
and B.

Definition 5. [19] A cover Ẽ = {Eρ : ρ ∈ Λ} of the space (W,ϑ) is called D−cover if any
Eρ is D−set, for every ρ ∈ Λ.

Definition 6. If any D−cover of the topological space (W,ϑ) has an open locally−countable
refinement, then the space is called D−paralindelöf space.

Definition 7. [19] If any D−cover of the topological space (W,ϑ) has a countable subcover,
then the space is called D−lindelöf.

Definition 8. [19] If any D−cover of the topological space (W,ϑ) has a finite−subcover,
then the space is called D−compact.

Definition 9. [11] A subset W1 ⊆ (W,ϑ) is called D−dense set, if for any y ∈ W , we
have Dy ∩W1 ̸= ϕ, and each D−set containing y.

Definition 10. [11] The space (W,ϑ) is called D−separable, if the space has a subset H,
which is D−dense countable.

Definition 11. [17] The space (W,ϑ) is called locally−indiscreet if any set ϑn−open is
ϑn−clopen, where n = 1, 2.

Definition 12. [9] If any cover of the topological space (W,ϑ) has a countable subcover,
then the space is called lindelöf space.

Definition 13. [11] Let (W,ϑ) and (Q, ι) be any topological spaces. If the inverse image
of any open subset in Q of φ : W −→ Q is open in W , then the function φ is called
continuous function. That is, if x ∈ ι, then φ−1(x) ∈ ϑ is its inverse image.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2993

Definition 14. [5] Let (W,ϑ) and (Q, ι) be any topological spaces. If the image φ(A) is
open in Q for each open set A in W , then the function φ : W −→ Q is called open.

Definition 15. [9] Let (W,ϑ) and (Q, ι) be any topological spaces. If the image φ(F ) is
closed in Q for each closed set F in W , then the function φ : W −→ Q is called closed.

Definition 16. [4] If the function φ : (W,ϑ) −→ (Q, ι) is continuous, D−compact and
closed for each q ∈ Q, then the function φ is called D−perfect.

Theorem 1. [19] Any open cover is D−cover.

3. New Results of D−Paracompact Spaces

This section presents the new notion of D−paracompact spaces, and explores their
connections to other spaces,and introduces a new set of properties.

Definition 17. A topological space (W,ϑ) is called D−paracompact if every D−cover of
the space (W,ϑ) has an open locally-finite refinement.

Figure 1: The space of Difference paracompact.

Figure 1 illustrates the space of D−paracompact such that Ẽ = {Eρ : ρ ∈ Λ} is
D−cover of the topological space (W,ϑ) has an open locally−finite refinement K̃ = {Kω :
ω ∈ Ω}. i.e. each Eρ is a D−set for all ρ ∈ Λ such that W = ∪ρ∈ΛEρ has an open
locally−finite refinement K̃, which the open refinement K̃ is a locally−finite if every point
xi of the space W has a neighborhood Ni such that Eρ ∩Ni ̸= ϕ is finite for all ρ,i ∈ Λ.

Theorem 2. Every D−paracompact space (W,ϑ) is paracompact.

Proof. Suppose that (W,ϑ) is a D−paracompact space and Ẽ = {Eρ : ρ ∈ Λ} is
an open-cover of (W,ϑ). By theorem 1, the cover Ẽ is a D−cover, and it has an open
locally−finite refinement.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2994

Figure 2: The relation of D−paracompact with paracompact spaces.

Figure 2 presents the basic relation between D−paracompact and paracompact spaces,
which represents a crucial fact that every D−paracompact space must be paracompact
such that Ẽ = {Eρ : ρ ∈ Λ} is D−cover of the topological space (W,ϑ) has an open
locally−finite refinement and we have G̃ = {Gω : ω ∈ Ω} is cover of the topological space
(Q, ι) has an open locally−finite refinement.

Example 1. Let (R, ϑu) be a D−paracompact topological space. Then by using the The-
orem 2 we get that the space (R, ϑu) is paracompact.

The explanation for why the previous theorem’s converse may not be true is illustrated
by the example that follows.

Example 2. The topological space (R, ϑcof ) is paracompact but not a D-paracompact.
This is because for all x ∈ R, each set of the form R − x is open in a topological space
(R, ϑcof ). Now, for all a, b ∈ R, if A = R − {a} and B = R − {b} be any two open sets
in (R, ϑcof ), then D = A − B = {b} is a D−set but not an open set. This is because a
D−cover D̃ = {{b} : b ∈ R} has not an open locally-finite refinement. Because, if D̃ has
an open locally-finite refinement {{b1, b2, . . . , bn}, then we have R ⊆ ∪n

i=1bi, which implies
that R is a finite set. Which is a contradiction.

The following example illustrates the contrapositive of the above theorem.

Example 3. The space (R, ϑl.r) is not paracompact, which is not D−paracompact space.

The next theorem has the purpose to indicate that, in under conditions, the converses
of the theorem above could be held.

Theorem 3. Every topological space (W,ϑ) that is a locally-indiscreet paracompact is
D−paracompact.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of (W,ϑ). Then Ẽ = {Eρ : ρ ∈ Λ} is
open cover, which has an open locally-finite refinement. Hence, the result.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2995

Figure 3: The relation of D−paracompact and paracompact spaces with locally indiscreet condition.

Figure 3 presents the complex relation between D−paracompact and paracompact
spaces under extra condition, which represents a significant fact that everyD−paracompact
space must be paracompact, while the converse is true if the paracompact space is locally
indiscreet such that Ẽ = {Eρ : ρ ∈ Λ} is D−cover of the topological space (W,ϑ) has an
open locally−finite refinement and we have G̃ = {Gω : ω ∈ Ω} is cover of the topological
space (Q, ι) has an open locally−finite refinement.

Example 4. (i) Notice that the space (R, ϑind) is locally-indiscreet and paracompact, then
it is D−paracompact space.
(ii) Let W = R and ϑ = {ϕ,R,R−{5}, {5}}. Then (W,ϑ) is locally-indiscreet paracompact
space. Thus, it must be D−paracompact.

Theorem 4. Given that A ⊆ W of the topological space (W,ϑ), then (A, ϑA) is D−paracompact
if and only if any D−cover of A by D−sets in W has an open locally-finite refinement.

Proof. ⇒) Let (A, ϑA) be D−paracompact and Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of A
in W . Let E∗

ρ = Eρ ∩A, for all ρ ∈ Λ is a D−set in A. Then, Ẽ∗ = {E∗
ρ : ρ ∈ Λ} must be

D−cover of A by D−sets in A. Since (A, ϑA) is D−paracompact space, then Ẽ∗ has an
open locally-finite refinement {E∗

ρ1, E
∗
ρ2, . . . , E

∗
ρn}. Thus, the family {Eρ1, Eρ2, . . . , Eρn}

must be an open locally-finite refinement of Ẽ in W for A, which E∗ = E ∩A. Hence, the
result.
⇐) Suppose that any D−cover of A by D−sets in W has an open locally-finite refinement.
Let à = {Aρ : ρ ∈ Λ} be a D−cover of A by D−sets in W . In consequence, there is
D−set of Eρ in W such as Aρ = Eρ ∩ A for all ρ ∈ Λ. So, Ẽ = {Eρ : ρ ∈ Λ} where
Ẽ is a D−cover of A by D−sets in W . Based on the assumption that Ẽ has an open
locally-finite refinement {Eρ1, Eρ2, . . . , Eρn}. Since Aρ ⊆ Eρ, for all ρ ∈ Λ, then the
family {Eρ1, Eρ2, . . . , Eρn} must be an open locally-finite refinement of à for A. Hence,
the result.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2996

Figure 4: The subspace topology of D−paracompact space.

Figure 4 demonstrates a significant fact between the topological space and its sub-
space, where the topological subspace (A, ϑA) of the D−paracompact space (W,ϑ) must
be D−paracompact space such that Ẽ = {Eρ : ρ ∈ Λ} is D−cover of the topological
subspace (A, ϑA) has an open locally−finite refinement, for all D−sets in the space (W,ϑ).

Observe the next corollaries such that each open cover is a D−cover.

Corollary 1. If the space (A, ϑA) is D−paracompact, then any open cover of A by open
sets in W has an open locally-finite refinement.

Corollary 2. A space (A, ϑA) is paracompact if any D−cover of A by D−sets in W has
an open locally-finite refinement.

Proof. Since each D−paracompact space is paracompact, then the second part of The-
orem 4 is this corollary’s direct cause.

Theorem 5. Let (W,ϑ1) and (W,ϑ2) be any topological spaces. If ϑ1 ⊆ ϑ2 and (W,ϑ2) is
D−paracompact, then (W,ϑ1) is a D−paracompact.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of (W,ϑ1). Since ϑ1 ⊆ ϑ2, we have Ẽ is
D−cover of (W,ϑ2), which it is has an open locally-finite refinement. Hence, the result.

Theorem 6. If the topological space (W,ϑ) is D−paracompact, then there are closed,
paracompact subspaces in (W,ϑ).

Proof. Let A be any closed subset of W and W be D−paracompact space. If Ẽ = {Eρ :
ρ ∈ Λ} is an open cover of A by open sets in W , so Ẽ ∪ {W − A} must be open cover of
W . Since W is a D−paracompact space, it has an open locally-finite refinement of Ẽ∗.
Moreover, Ẽ∗ − {W − A} is an open locally-finite refinement of Ẽ for A. Hence, we get
the result.

Theorem 7. Any D−separable, D−paracompact (W,ϑ) must be D−lindelöf.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2997

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of W . Believe that Ẽ has no a countable
subcover. Let K̃ = {Kε : ε ∈ Υ} is an open uncountable locally-finite refinement subcover
of Ẽ. Now, if D is a countable D−dense subsets of W , then Kε ∩D ̸= ϕ, for all ε ∈ Υ.
Thus, the set D is uncountable because K̃ is uncountable, that is a contradiction. Hence,
the result.

Figure 5: The relation of D−paracompact and D−lindelöf spaces with D−separable condition.

Figure 5 presents the basic relation between D−paracompact and D−lindelöf spaces
under extra condition, which represents a significant fact that if the space ofD−paracompact
is D−separable then must be D−lindelöf such that Ẽ = {Eρ : ρ ∈ Λ} is D−cover of the
topological space (W,ϑ) has an open locally−finite refinement and we have G̃ = {Gω :
ω ∈ Ω} is cover of the topological space (Q, ι) has an open locally−finite refinement.

With the same work, we can achieve the next corollary.

Corollary 3. Any D−separable, D−paracompact (W,ϑ) must be lindelöf.

Definition 18. The topological space (W,ϑ) is called countably D−paracompact if every
countably D−cover of (W,ϑ) has an open locally-finite refinement.

Figure 6: The space of countably D−paracompact.

Figure 6 illustrates the space of countablyD−paracompact such that Ẽ = {Eρ : ρ ∈ Λ}
is countably D−cover of the topological space (W,ϑ) has an open locally−finite refinement
K̃ = {Kω : ω ∈ Ω}. i.e. each countably of Eρ isD−set for all ρ ∈ Λ such thatW = ∪ρ∈ΛEρ

has an open locally−finite refinement K̃, which the open refinement K̃ is a locally−finite
if every point xi of the space W has a neighborhood Ni such that Eρ ∩Ni ̸= ϕ is finite for
all ρ,i ∈ Λ.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2998

Theorem 8. If any topological space (W,ϑ) is D−lindelöf and countably D−paracompact,
then it must be D−paracompact.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be a D−cover for W and let W is a D−lindelöf. Then Ẽ
has a countable subcover K̃ = {K}∞i=1. Since W is countably D−paracompact, then K̃ has
an open locally-finite refinement S̃ of Ẽ.Thus, the space (W,ϑ) must be D−paracompact.

Figure 7: The relation of D−paracompact and countably D−paracompact spaces with D−lindelöf
condition.

Figure 7 presents the relation betweenD−paracompact spaces countablyD−paracompact
under extra condition, which represents a significant fact that if the space of countably
D−paracompact is D−lindelöf then must be D−paracompact such that Ẽ = {Eρ : ρ ∈ Λ}
is the countably D−cover of the topological space (W,ϑ) has an open locally−finite re-
finement and we have G̃ = {Gω : ω ∈ Ω} is D−cover of the topological space (Q, ι) has
an open locally−finite refinement.

With the same work, we can achieve the next corollaries.

Corollary 4. If any topological space (W,ϑ) is D−lindelöf and countably D−paracompact,
then it must be paracompact.

Corollary 5. If any topological space (W,ϑ) is lindelöf and countably D−paracompact,
then it must be paracompact.

Example 5. Let the space (W,ϑdis) be a D−lindelöf and a countably D−paracompact
space. Then by using the Theorem 3.16, we get that (W,ϑdis) must be D−paracompact.

Theorem 9. If any topological space (W,ϑ) is D−paralindelöf and countably D−paracompact,
then it must be a D−paracompact space.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover for W and let W is D−paralindelöf
space. Then Ẽ has an open locally-countable refinement K̃ = {Kρi}∞i=1, where is also a
D−cover of (W,ϑ) . Now, since W is countably D−paracompact, then K̃ has an open
locally-finite refinement S̃ of Ẽ. Thus, the space (W,ϑ) must be D−paracompact.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 2999

Figure 8: The relation of D−paracompact and countably D−paracompact spaces with D−paralindelöf
condition.

Figure 8 presents the relation betweenD−paracompact spaces countablyD−paracompact
under extra condition, which represents a significant fact that if the space of countably
D−paracompact is D−paralindelöf then must be D−paracompact such that Ẽ = {Eρ :
ρ ∈ Λ} is the countably D−cover of the topological space (W,ϑ) has an open locally−finite
refinement and we have G̃ = {Gω : ω ∈ Ω} is D−cover of the topological space (Q, ι) has
an open locally−finite refinement.

The following corollaries can be obtained with the same work.

Corollary 6. If any topological space (W,ϑ) is D−paralindelöf and countably D−paracompact,
then it must be a paracompact space.

Corollary 7. If any topological space (W,ϑ) is paralindelöf and countably D−paracompact,
then it must be a paracompact space.

Figure 9: The flow chart of D−paracompact with some spaces.

Figure 9 presents the study’s results flow chart, which represents the relation of Differ-
ence paracompact spaces with common topological spaces, in which the space of Difference
paracompact represents a basic class characterized by special coverings.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 3000

4. Product of D-Paracompact Topological Spaces

This section presents several main theoretical results the concepts of maps with the
product for two D−paracompact spaces.

Theorem 10. Let (W,ϑ) and (Q, ι) be any topological spaces. If φ : W −→ Q is a
D−perfect function, and W is locally-indiscreet space, then W must be D−paracompact if
the space Q is so

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of W . Since φ is D−perfect, then for any
q ∈ Q, we have φ−1(q) is D−compact subsets of W . So, there is a finite subset ιq of Λ, such
that φ−1(q) ⊆ ∪ρ∈ιqHρ, and Ẽ is an open cover of W . That is, Pq = Q−φ(W −∪ρ∈ιqHρ)

is D−open subsets of Q and φ−1(Pq)∪ρ∈ιq Hρ, for any q ∈ Pq, therefore P̃ = {Pq : q ∈ Q}
is open D−cover of Q. Since Q is D−paracompact, then we get that P̃ has an open locally-
finite refinement P̃ ∗ = {P ∗

q : q ∈ Q}. Thus, the set P ∗
q is D−open subsets of W . Since

the function φ is D−perfect, that is means {φ−1(P ∗
q ) : q ∈ Q} is an open locally-finite

refinement of W . Hence, the space W is D−paracompact space.

Theorem 11. Let φ : W −→ Q as D−perfect function. Then, W is paracompact space if
the space Q is D−paracompact

Proof. As the proof of the above Theorem 10, this theory is also simply to prove.

Theorem 12. If φ : W −→ Q be D−perfect, where Q is a countable and W is locally-
indiscreet, then W must be countably D−paracompact, if the space Q is so.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be countable D−cover of W and φ is D−perfect function.
Then for any q ∈ Q, φ−1(q) is D−compact space subsets of W . It therefore obtains in
a finite subset of Λ such as φ−1(q) ⊆ ∪n

i=1Ai. Since W is locally indiscreet, then Ai is
D−open subset of W , for any i ∈ Λ. Presently, Bq = Q − φ(W − ∪n

i=1Ai) is D−set
containing q. Moreover, φ−1(Bq) ⊆ ∪n

i=1Ai. Thus, B̃ = {Bq : q ∈ Q} presents countable
D−cover for Q. Now, since Q is countably D−paracompact, then B̃ has an open locally-
finite refinement B̃∗ = {Bq1, Bq2, . . . , Bqn} and so B∗

q is D−open subset of W . Also, since
φ is D−perfect, so {φ−1(B∗

q ) : q ∈ Q} is an open locally-finite refinement of W . Thus,
the space W must be countably D−paracompact.

Theorem 13. Let (W,ϑ) and (Q, ι) be topological spaces, such that W is D-compact and
Q is D−paracompact spaces. Then W ×Q must be D−paracompact space.

Proof. Given the truth that the projection function T : W × Q −→ Q is continuous
and T−1{q} = W × {q} ≃ W is D−compact, for any q ∈ Q. Then T : W × Q −→ Q
is D−perfect function. So, since Q is D−paracompact space, then W × Q must be also
D−paracompact.



A. Amourah et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 2990-3003 3001

Theorem 14. Let (W,ϑ) and (Q, ι) be topological spaces. If W is paracompact, and Q is
D−paracompact spaces, then the projection function T : (W ×Q,ϑ× ι) −→ (Q, ι) must be
closed.

Proof. Let (W,ϑ) be paracompact and (Q, ι) be a D−paracompact. Then (W ×Q,ϑ×ι)
is D−paracompact, therefore the projection function T : (W ×Q,ϑ× ι) −→ (Q, ι) must be
closed function.

Theorem 15. Let φ : (W,ϑ) −→ (Q, ι) be a closed, continuous, onto function such that
Q is locally-indiscreet space. Then Q is D−paracompact, if the space W is so.

Proof. Let Ẽ = {Eρ : ρ ∈ Λ} be any D−cover of Q. Since φ is onto function and
continuous, that means Ẽ = {φ−1(Eρ) : ρ ∈ Λ} is an open cover of W . Since W is
D−paracompact, then there is an open locally-finite refinement of Ẽ as Ẽ∗ = {φ−1(E∗

ρ) :
ρ ∈ Λ}. Thus, Q must be a D−paracompact space.

With the same work, the corollary that follows can be achieved.

Corollary 8. Let φ : (W,ϑ) −→ (Q, ι) be a closed, onto, continuous function such that Q
is locally-indiscreet space. Then, Q is D−paracompact, if the space W is paracompact.

5. Conclusions

This research highlights that D−paracompact topological spaces have a key topolog-
ical characteristic. Their flexibility to provide D−covers with locally-finite refinements is
demonstrated by their conclusion. The research introduced several new properties and ex-
plained examples that relate to such superimposed spaces. The study’s additional objective
was to draw attention to some of the new notions and characteristics of D−paracompact
spaces, as well as some properties of the Cartesian multiplication of such spaces under spe-
cial conditions. Furthermore, certain illustrative examples and the main characteristics of
these concepts were carefully studied. We identified their principal characteristics along
with illustrated figures. We talked about their main traits and demonstrated how they
work together. The study lies at the interface of topology and other branches of mathe-
matics. As in fuzzy sets, researchers can generalize the fuzzy paracompact spaces to be
fuzzy D−paracompact spaces. Also, in the algebra field, there is a compact (topological)
group, which we can generalize to a D−paracompact group and provide mathematicians
with a way to expand their knowledge about both topological and group algebra. This
also acts as an effective basis for future research. In general topology, these spaces might
be generalized to bitopological and tritopological spaces.

Acknowledgments

We very much appreciate everyone who made a contribution to this research project.
Their help, guidance, and teamwork have been key to the completion of this research.



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