EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5899
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On gµ-Paracompact Sets

Heyam H. Al-Jarrah1, Amani Rawshdeh2,∗, Khalid Y. Al-Zoubi1, Shefa A.
Bani Melhem1

1 Department of Mathematics, Faculty of Science, Yarmouk University, Irbid, Jordan
2 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Alsalt,
Jordan

Abstract. In this work, we use the notion of the gµ-paracompact space [6] to introduce two
types of gµ-paracompact sets called α-gµ-paracompact and β-gµ-paracompact. We show that
every α-gµ-paracompact set is β-gµ-paracompact and if a generalized topological space (S, µ) is
gµ-paracompact, then every µ-closed subset in (S, µ) is α-gµ-paracompact while every µg-closed
subset in (S, µ) is β-gµ-paracompact. Finally, we introduce the notion of co-α-gµ-paracompact set
as an application of α-gµ-paracompact and study some of its features.

2020 Mathematics Subject Classifications: 54A05, 54C08, 54D10.

Key Words and Phrases: generalized topological space, gµ-paracompact, α-gµ-paracompact,
β-gµ-paracompact.

1. Introduction

In 1944, Dieudonné [7] introduced a broader class of compact spaces, namely para-
compact spaces. Sorgenfrey [15] and Stone [16] investigated the behavior of paracompact
spaces within the product space. Michail [10] defined paracompactness in the sense of
regular topological spaces and demonstrated how metrizability implies paracompactness.
Therefore, paracompactness is one of the most essential concepts and possibly the most
successful generalization of compactness, which is introduced not only in general topology
but also in other structures, such as generalized topological spaces (see [6, 9]).

The study of generalized topological spaces (briefly, GTS) was first initiated by Császár
[4]. The pair (S, µ) is called GTS if µ ⊆ P (S) with ϕ ∈ µ and µ is closed under the ar-
bitrary union where P (S) denotes the power set of S. A GTS in turn motivated other
researchers to generalize the topological concepts including covering properties of gener-
alized topology. For instance, in [6] the authors defined µ-paracompact spaces and gµ-
paracompact which are a generalization of paracompactness in GTS, where a GTS (S, µ)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5899

Email addresses: heyam@yu.edu.jo (H. H. Al-Jarrah), amanirawshdeh@bau.edu.jo (A. Rawshdeh),
khalidz@yu.edu.jo (K. Y. Al-Zoubi), shefa.bm@yu.edu.jo (Sh. A. Bani Melhem)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 2 of 11

is called µ-paracompact (resp. gµ-paracompact) if every (S, µ)-cover of S has a µ−LF(S,µ)

(resp. gµ−LF(S,µ)) (S, µ)-refinement. In [11], Qahis and Noiri investigated the concept of
µ−paracompact spaces with respect to hereditary class H, which is a generalization for a
µ-paracompact space.

In classical topology, studying the covering definitions for sets after studying the con-
cept of space is an area that has found interest among some authors, such as: based on
the definition of I-Lindelöf space [1] the author presented the definition of I-Lindelöf sets
[2]. In addition, the authors used the concept of paracompactness to study and introduce
different notions of paracompact sets such as α-paracompact and β-paracompact [see [3]].
Therefore, in this work, we employ the definition of gµ-paracompact spaces that are de-
fined in [6] to introduce the notions of α-gµ-paracompact and β-gµ-paracompact subsets
and provide some illustrative examples to elucidate the relationship between them and
demonstrate the results that were achieved.

For a GTS (S, µ) the elements of µ are called µ-open sets and the collection of all
µ-open sets containing s ∈ S will be denoted by µ(s). The complement of a µ-open set is
called a µ−closed set, and the intersection of all µ-closed sets containing E will be denoted
by cµ(E). A subset E of GTS (S, µ) is called a generalized closed set [13], denoted by
µg-closed set, if cµ(E) ⊆ G whenever E ⊆ G and G is µ-open. A GTS (S, µ) is called
µ-T2-space [14] if for each s, e ∈ S with s ̸= e, there are G ∈ µ(s) and H ∈ µ(e) with
G ∩H = ϕ. For E ⊆ S, the subspace of (S, µ) in E is denoted by (E,µE).

2. Preliminaries

In this section, we recall the main concepts and properties which will be needed in this
work.

Definition 1. [6] Let (S, µ) be a GTS. Then:

(i) µ∗(s) = {∩n
i=1Gi : Gi ∈ µ(s), ∀i = 1, ..., n ∈ N} for each s ∈ S.

(ii) γµ(E) = {s ∈ S : H ∩ E ̸= ϕ for all H ∈ µ∗(s)} for each E ⊆ S.

In [6], the authors show that the operator γµ(S) created a topology on S defined by
µ∗ = {E ⊆ S : γµ(S − E) = S − E} that is finer than µ, and if µ is a topology on S
then µ = µ∗. The elements of µ∗ are called µ∗-open sets and their complements are called
µ∗-closed sets. For each E ⊆ S the subspace of (S, µ∗) on E is denoted by (E,µ∗E).

Definition 2. [6] A GTS (S, µ) is called γµ-regular if for each s ∈ S and G ∈ µ(s), there
is H ∈ µ(s) with γµ(H) ⊆ G.

Definition 3. [6] Let (S, µ) be a GTS. Then a collection G = {Gα : α ∈ ∆} is called:

(i) µ-locally finite in (S, µ) (resp. (E,µE)), denoted by µ−LF(S,µ) (resp. µ−LF(E,µE)),
if for each s ∈ S (resp. s ∈ E) there is H ∈ µ(s) (resp. H ∈ µE(s) ) with the set
{η : H ∩Gη ̸= ϕ} is finite.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 3 of 11

(ii) gµ-locally finite in (S, µ) (resp. (E,µE)), denoted by gµ−LF(S,µ) (resp. gµ−LF(E,µE)),
if for each s ∈ S (resp. s ∈ E) there is H ∈ µ∗(s) (resp. H ∈ µ∗E(s)) with the set
{η : H ∩Gη ̸= ϕ} is finite.

It follows from the above definition that each µ−LF(S,µ) is gµ−LF(S,µ) but the converse
need not be true in general (see Example 2.2 of [6]).

Theorem 1. [6] If G = {Gα : α ∈ ∆} is a gµ−LF(S,µ). Then:

(i) γµ(G) = {γµ(Gα) : α ∈ ∆} is gµ−LF(S,µ).

(ii) G is γµ-closure preserving, i.e. γµ(∪α∈∆Gα) = ∪α∈∆γµ(Gα).

Definition 4. [11] Let (S, µ) be a GTS and E ⊆ Z ⊆ S. Then:

(i) A collection G = {Gα : α ∈ ∆} is cover of E if E ⊆ ∪α∈∆Gα and if Gα ∈ µ (resp.,
Gα ∈ µZ) for each α ∈ ∆, then G is called (S, µ)-cover (resp., (Z, µZ)-cover) of E.

(ii) If G and H are covers of E, then H is called a refinement of G if there is G ∈ G with
H ⊆ G for each H ∈ H. Moreover, if H is a refinement of G with H ∈ µ (resp.,
H ∈ µZ), then H is called (S, µ)-refinement (resp., (Z, µZ)-refinement) of G.

(iii) A subset E is called µ-paracompact relative to S (or µ-paracompact subset) if each
(S, µ)-cover of E has µ−LF(S,µ) (S, µ)-refinement and E is called µE-paracompact
(or µE-paracompact subspace) if (E,µE) is µE-paracompact as a subspace.

Definition 5. Let (S1, µ1) and (S2, µ2) be GTS. A mapping ψ : (S1, µ1) → (S2, µ2) is
called:

(i) (µ1, µ2)-continuous [4] if ψ−1(H) ∈ µ1 for each H ∈ µ2.

(ii) (µ1, µ2)-open [12] if ψ(G) ∈ µ2 for each G ∈ µ1.

(iii) (µ1, µ2)-closed [13] if ψ(M) is µ2-closed in S2 for each µ-closed set M of S1.

Proposition 1. [8] A function ψ : (S1, µ1) → (S2, µ2) is (µ1, µ2)-closed iff for each e ∈ S2
and G ∈ µ1 with ψ−1(e) ⊆ G, there is H ∈ µ2(e) with ψ

−1(H) ⊆ G.

3. α-gµ-paracompact and β-gµ-paracompact sets

In this section, the concepts of α-gµ-paracompact and β-gµ-paracompact subsets are
illustrated and the relationship between them with some of their properties are investi-
gated.

Proposition 2. Let (S, µ) be a GTS with E ⊆ S and G = {Gα : α ∈ ∆, Gα ⊆ E}. Then:

(i) G is gµ−LF(E,µE) if it is gµ−LF(S,µ).

(ii) G is gµ−LF(S,µ) if it is gµ−LF(E,µE) provided that E is µ-closed.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 4 of 11

Proof. (i) It follows from Definition 3.
(ii) Let G be gµ−LF(E,µE) . If s ∈ S, then either s ∈ E or s /∈ E. If s ∈ E, then there is
H ∈ µ∗E(s) with the set {η : H ∩Gη ̸= ϕ} is finite. Now H =W ∩E for some W ∈ µ∗(s).
Since G is a collection of subsets of E, then {η : W ∩ Gη ̸= ϕ} is finite. If s /∈ E then
S − E ∈ µ∗(s) which intersects no member of G.

Example 1. Let (S, µ) be a GTS where S = R and µ = {G : 0 /∈ G}. Put E = Q− {0}.
Then the collection {{e} : e ∈ E} is gµ−LF(E,µE) while it is not gµ−LF(S,µ).

Definition 6. Let (S, µ) be a GTS and E ⊆ S. Then:

(i) E is called α-gµ-paracompact in (S, µ) (simply, α-gµ-paracompact) if each (S, µ)-
cover of E has a gµ−LF(S,µ) (S, µ)-refinement.

(ii) E is called β-gµ-paracompact in (S, µ) (simply β-gµ-paracompact) if each (E,µE)-
cover of E has a gµ−LF(E,µE) (E,µE)-refinement.

Proposition 3. Each α-gµ-paracompact is β-gµ-paracompact.

Proof. It follows from Definition 3 and Proposition 2.

Note that the converse of Proposition 3 is not true in general. In Example 1, E is β-gµ-
paracompact, since each (E,µE)-cover of E has a gµ−LF(E,µE) (E,µE)-refinement (that
is {{e} : e ∈ E}). On the other hand, G = {{e} : e ∈ E} is an (S, µ)-cover of E and it has
no gµ−LF(S,µ)(S, µ)-refinement since µ∗(0) = ϕ. Therefore, E is not α-gµ-paracompact.

Theorem 2. Let (S, µ) be a gµ-paracompact GTS and E ⊆ S. Then E is α-gµ-paracompact
if one of the following holds:

(i) E is µg-closed in (S, µ).

(ii) E is µ-closed in (S, µ).

Proof. (i) Let G = {Gα : α ∈ ∆} be an (S, µ)-cover of E. Since cµ(E) ⊆ ∪α∈∆Gα, then
G1 = G ∪ {S − cµ(E)} is an (S, µ)-cover of S. So G1 has a gµ−LF(S,µ) (S, µ)-refinement,
say H = {Hβ : β ∈ Λ}. Therefore, the collection H1 = {Hβ ∈ H : Hβ ⊆ Gα for some
Gα ∈ G, α ∈ ∆ and β ∈ Λ} is gµ−LF(S,µ) (S, µ)-refinement for G.
(ii) The proof is obvious since every µ-closed set is µg-closed.

Corollary 1. Let (S, µ) be a gµ-paracompact GTS and E ⊆ S. Then E is β-gµ-
paracompact if one of the following holds:

(i) E is µg-closed in (S, µ).

(ii) E is µ-closed in (S, µ).

Proof. It follows from Proposition 3 and Theorem 2.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 5 of 11

Theorem 3. Let (S, µ) be a GTS. If each µ-open subset of (S, µ) is α-gµ-paracompact,
then each subset E of S is β-gµ-paracompact.

Proof. Let GE = {Gα ∩ E : Gα ∈ µ, α ∈ ∆} be (E,µE)-cover of E. Then G = {Gα :
α ∈ ∆} is an (S, µ)-cover of ∪α∈∆Gα and so G has a gµ−LF(S,µ) (S, µ)-refinement, say
H = {Hβ : β ∈ Λ}. Define HE = {Hβ ∩ E : β ∈ Λ}. Then HE is gµ−LF(E,µE)(E,µE)-
refinement for GE . Note that if s ∈ E there is G ∈ µ∗(s) with the set {η : G ∩Hη ̸= ϕ} is
finite which implies that the set {η : (G ∩ E) ∩ (Hη ∩ E) ̸= ϕ} is finite. Finally, since for
each Hβ ∩E ∈ HE , there is some Gα ∈ G with Hβ ⊆ Gα so we obtain Hβ ∩E ⊆ Gα ∩E.
Therefore, E is β-gµ-paracompact.

Theorem 4. Let (S, µ) be a GTS and E ⊆ S. If for each µ-open set G containing E,
there is β-gµ-paracompact Z with E ⊆ Z ⊆ G, then E is β-gµ-paracompact .

Proof. Let GE = {E∩Hα : Hα ∈ µ, α ∈ ∆} be an (E,µE)-cover of E. Then there is β-
gµ-paracompact Z with E ⊂ Z ⊂ ∪α∈∆Hα. Since GZ = {Hα∩Z : α ∈ ∆} is a (Z, µZ)-cover
of Z, then GZ has a gµ−LF(Z,µZ) (Z, µZ)-refinement, say HZ = {Hβ ∩Z : Hβ ∈ µ, β ∈ Λ}.
Put HE = {Hβ∩E : β ∈ Λ}. Then HE is gµ−LF(E,µE)(E,µE)-refinement for GE , since for
s ∈ E there isH∩Z ∈ µ∗Z(s) withH ∈ µ∗(s) and the set {η :(H∩Z)∩(Hη∩Z) ̸= ϕ} is finite
which implies that the set {η : [(H∩Z)∩(Hη∩Z)]∩E ̸= ϕ} = {η : (H∩E)∩(Hη∩E) ̸= ϕ}
is finite. Now, for each Hβ ∩E ∈ HE there is Hα ∩Z ∈ GZ with Hβ ∩Z ⊆ Hα ∩Z and so
Hβ ∩ E ⊆ Hα ∩ E. Therefore, E is β-gµ-paracompact.

Theorem 5. Let (S, µ) be a GTS and E ⊆ Z ⊆ S. If E is α-gµ-paracompact in (S, µ),
then E is α-gµ-paracompact in (Z, µZ).

Proof. Let GZ = {Z ∩ Hα : Hα ∈ µ, α ∈ ∆} be a (Z, µZ)-cover of E. Then G1 =
{Hα : α ∈ ∆} is an (S, µ)-cover of E and so it has a gµ−LF(S,µ) (S, µ)-refinement, say
H = {Hβ : β ∈ Λ}. Put H1 = {Hβ ∩ Z : β ∈ Λ}. As in the proof of Theorem 3, we can
show H1 is a gµ−LF(Z,µZ)(Z, µZ)-refinement of GZ . Therefore, E is α-gµ-paracompact in
(Z, µZ).

Example 2. Let (S, µ) be a GTS where S = R and µ = {G : Q ⊆ G} ∪ {ϕ}. Put
E = Z = R−Q. Then µZ = P(Z) and Z is µ-closed in (S, µ). Note that, E is α-gµ-
paracompact in (Z, µZ). On the other hand, E is not α-gµ-paracompact in (S, µ) since
{Q ∪ {x} : x ∈ E} is (S, µ)-cover of E has no gµ−LF(S,µ) (S, µ)-refinement.

Theorem 6. Let (S, µ) be a GTS and E ⊆ S with E is α-gµ-paracompact in a µ-closed
subspace (Z, µZ). If there is a µ∗-open set G with E ⊆ G ⊆ Z, then each (S, µ)-cover of
E has a gµ−LF(S,µ) (S, µ

∗)-refinement.

Proof. Let G = {Gα : α ∈ ∆} be an (S, µ)-cover of E. Then, the collection {Z ∩Gα :
α ∈ ∆} is a (Z, µZ)-cover of E and so it has a gµ − LF(Z,µZ) (Z, µZ)-refinement, say H.
Now, for each s ∈ E there is Hs ∈ H and Ws ∈ µ∗(s) with s ∈ Hs = Ws ∩ Z. Since
G is µ∗-open, then WE = {Ws ∩ G : s ∈ E} is a gµ-LF(S,µ) (S, µ∗)-refinement of G. At



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 6 of 11

first, since Z is µ-closed and for all s ∈ E, Ws ∩ G ⊆ Ws ∩ Z = Hs and the collection
{Ws∩Z : s ∈ E} is gµ−LF(Z,µZ) and so, by Proposition 2, WE is gµ−LF(S,µ). Moreover,
for each s ∈ E, there is Gα(s) ∈ G with s ∈ Ws ∩ G ⊆ Ws ∩ Z ⊆ Gα(s) ∩ Z ⊆ Gα(s) and
hence WE is (S, µ∗)-refinement of G.

Proposition 4. Let (S, µ) be a GTS and E ⊆ Z ⊆ S. Then E is β-gµ-paracompact in
(S, µ) iff E is β-gµ-paracompact in (Z, µZ).

Proof. Note that, (µZ)E = {E ∩G : G ∈ µZ} = {E ∩Z ∩H : H ∈ µ} = {E ∩H : H ∈
µ} = µE . Also, for each s ∈ E, (µZ)

∗
E(s) = µ∗E(s). Then, the result becomes obvious.

Lemma 1. Let (S, µ) be a GTS and E ⊆ S. If E is a µ∗-open set, then there is H ∈ µ
with E ⊆ H.

Proof. For each s ∈ E, there is Gs ∈ µ∗(s) with E = ∪
s∈E

Gs. Now Gs =
ns∩
i=1
Hi(s) where

Hi(s) ∈ µ(s) for each 1 ≤ i ≤ ns. Finally, for each s ∈ E, choose 1 ≤ i ≤ ns with s ∈ Hi(s).

Therefore, E ⊆
ns∪
i=1
Hi(s) = H and H ∈ µ.

Theorem 7. Let (S, µ) be an GTS and E,Z ⊆ S. Then:

(i) E ∩ Z is α-gµ-paracompact if E is µ∗-closed in (S, µ) and Z is α-gµ-paracompact.

(ii) E ∩ Z is β-gµ-paracompact if E is µ∗-closed in (S, µ) and Z is β-gµ-paracompact.

(iii) E ∩Z is α-gµ-paracompact (resp., β-gµ-paracompact) if E is µ-closed in (S, µ) and
Z is α-gµ-paracompact (resp., β-gµ-paracompact).

Proof. (i) Let G = {Gα : α ∈ ∆} be an (S, µ)-cover of E ∩ Z. Since S −E is µ∗-open,
by Lemma 1, there is W ∈ µ with S − E ⊆ Wand hence G1 = {Gα : α ∈ ∆} ∪ {W} is
an (S, µ)-cover of Z. So G1 has a gµ−LF(S,µ) (S, µ)-refinement, say H = {Hβ : β ∈ Λ}.
Hence the family H1 = {Hβ ∈ H : Hβ ⊆ Gα for some Gα ∈ G, α ∈ ∆ and β ∈ Λ} is
gµ−LF(S,µ) (S, µ)-refinement of G. Therefore, E ∩ Z is α-gµ-paracompact.
(ii) Let G = {Gα ∩ (E ∩ Z) : Gα ∈ µ, α ∈ ∆} be an (E ∩ Z, µE∩Z)-cover of E ∩ Z.
Then G1 = G∗ ∪ {W ∩ Z} is a (Z, µZ)-cover of Z, where G∗ = {Gα ∩ Z : α ∈ ∆} and
W is an µ-open set with S − E ⊆ W . So G1 has a gµ−LF(Z,µZ) (Z, µZ)-refinement, say
H = {Hβ ∩Z : Hβ ∈ µ, β ∈ Λ}. Hence, the family H1 = {Hβ ∩ (E ∩Z) : Hβ ∩Z ⊆ Gα ∩Z
for some Gα ∩ Z ∈ G∗, α ∈ ∆ and β ∈ Λ} is gµ−LF(E∩Z,µE∩Z) (E ∩ Z, µE∩Z)-refinement
of G. Therefore, E ∩ Z is β-gµ-paracompact.
(iii) Follows from (i) and (ii).

Theorem 8. Let (S, µ) be γµ-regular GTS and E ⊆ S. If E is α-gµ-paracompact, then
γµ(E) is α-gµ-paracompact.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 7 of 11

Proof. Let G = {Gα : α ∈ ∆} be (S, µ)-cover of γµ(E). Since G is (S, µ)-cover of E,
then G has a gµ-LF(S,µ)(S, µ)-refinement, say H = {Hβ : β ∈ Λ}. To show that H is
a cover for γµ(E), let Hβ ∈ H. Since (S, µ) is a γµ-regular GTS, then for each s ∈ Hβ

there is Wβs ∈ µ(s) with γµ(Wβs) ⊆ Hβ. Now, W = {Wβs : β ∈ Λ, s ∈ Hβ} is an
(S, µ)-cover of E and so it has a gµ-LF(S,µ)(S, µ)-refinement, say T = {Tλ : λ ∈ θ}. By
Theorem 1, we have γµ(E) ⊆ γµ(∪Tλ) = ∪γµ(Tλ) ⊆ ∪γµ(Wβs) ⊆ ∪Hβ. Therefore, γµ(E)
is α-gµ-paracompact.

Theorem 9. Let (S, µ) be γµ-regular GTS and E ⊆ S. If E is α-gµ-paracompact, then
each (S, µ)-cover of E has a µ∗-closed gµ-LF(S,µ) refinement.

Proof. Let G = {Gα : α ∈ ∆} be (S, µ)-cover of E. For each s ∈ E pick Gs ∈ G with
Gs ∈ µ(s). Since (S, µ) is γµ-regular, then there is Hs ∈ µ(s) with s ∈ Hs ⊆ γµ(Hs) ⊆ Gs.
Then, the collection H = {Hs : s ∈ E} is (S, µ)-cover of E and so it has a gµ-LF(S,µ)(S, µ)-
refinement, say W = {Wβ : β ∈ Λ}. Therefore, by Theorem 1, γµ(W) = {γµ(Wβ) : β ∈ Λ}
is µ∗-closed gµ-LF(S,µ) refinement.

Theorem 10. Let (S, µ) be a GTS and E ⊆ S. If E is α-gµ-paracompact subset of a
µ-T2-space, then E is µ∗-closed.

Proof. Let s /∈ E. Since (S, µ) is µ-T2-space, then for each e ∈ E there is Ge ∈ µ(e)
and s /∈ cµ(Ge). Therefore, G = {Ge : e ∈ E} is an (S, µ)-cover of E and hence it has
a gµ−LF(S,µS) (S, µ)-refinement, say W. Put H = ∪{W : W ∈ W}, then γµ(H) =
∪{γµ(W ) : W ∈ W}. Finally, take H∗ = S − γµ(H). Since H∗ is µ∗-open with s ∈ H∗

and H∗ ∩ E = ϕ, then s /∈ γµ(E) and hence E is µ∗-closed.

The converse of Theorem 10 is not true in general (see Example 20.11, page 148 of
[17]). Then (S, τ) is a T2-space such that (S, τ) is not paracompact and S is µ∗-closed
(µ = τ) while S it is not α-gµ-paracompact.

Theorem 11. Let (S, µ) be a GTS. If {Eα : α ∈ ∆} is a gµ-LF(S,µ) collection such that
Eα is α-gµ-paracompact of (S, µ) for each α ∈ ∆, then E = ∪α∈∆Eα is α-gµ-paracompact
of (S, µ).

Proof. Let G be an (S, µ)-cover of E. For each α ∈ ∆, G is an (S, µ)-cover of Eα

and hence it has a gµ-LF(S,µ) (S, µ)-refinement, say Hα = {Hβ : β ∈ Λα}. It is clear
that the collection H={Hβ : β ∈ Λα, α ∈ ∆} is an (S, µ)-refinement of G. To show that
H is gµ-LF(S,µ), let s ∈ S. Then there is Gs ∈ µ∗(s) and a finite subset ∆s of ∆ with
Gs ∩ Eα = ϕ for each α ∈ ∆ − ∆s. Now, for each α ∈ ∆s, there is Wα(s) ∈ µ∗(s) that
intersects at most finitely many members of Hα. Define Ts = Gs ∩ (∩α∈∆sWα(s)). Then
Ts ∈ µ∗(s) which intersects at most finitely many members of H.

Definition 7. Let (S, µ) be a GTS. If each µ∗-open cover of S has a finite subcover, then
(S, µ) is called µ∗-compact.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 8 of 11

Lemma 2. Let ψ : (S1, µ1) → (S2, µ2) be a (µ1, µ2)-continuous function. If G = {Gα :
α ∈ ∆} is a gµ−LF(S2,µ2), then ψ

−1(G) = {ψ−1(Gα) : α ∈ ∆} is gµ−LF(S1,µ1).

Proof. Let s ∈ S1 with e = ψ(s). Then there isH ∈ µ∗2(e) with the set {η : H∩Gη ̸= ϕ}
is finite. Since H = ∩n

i=1Wi where Wi ∈ µ2(e), then ψ−1(H) = ∩n
i=1ψ

−1(Wi) where
ψ−1(Wi) ∈ µ1(s). Therefore, ψ

−1(H) ∈ µ∗1(s) with the set {η : ψ−1(H)∩ψ−1(Gη) ̸= ϕ} is
finite. Hence ψ−1(G) is gµ−LF(S1,µ1).

Lemma 3. Let ψ : (S1, µ1) → (S2, µ2) be a surjection (µ1, µ2)-closed function with ψ−1(e)
is µ∗1-compact for each e ∈ S2. If G = {Gα : α ∈ ∆} is a gµ−LF(S1,µ1), then ψ(G) =
{ψ(Gα) : α ∈ ∆} is gµ−LF(S2,µ2).

Proof. Let e ∈ S2. For each s ∈ ψ−1(e) choose Hs ∈ µ∗1(s) with the set {η : Hs ∩Gη ̸=
ϕ} is finite. Therefore, the collection {Hs : s ∈ ψ−1(e)} is a µ∗1-open cover of the µ∗1-
compact subset ψ−1(e) and so there is a finite number of points s1, s2, ...sn in ψ−1(e) with
ψ−1(e) ⊆ ∪n

i=1Hsi . Note that, for each 1 ≤ i ≤ n, Hsi is a finite intersection of members
of µ1(si). Therefore, ∪n

i=1Hsi = ∩m
j=1Kj where Kj is a finite union of µ1-open sets and

so Kj is µ1-open for each 1 ≤ j ≤ m. By Proposition 1, there is We(j) ∈ µ2(e) with

ψ−1(We(j)) ⊆ Kj . Put W =
m
∩
j=1

We(j) . Then W ∈ µ∗2(e) with the set {η : W ∩ψ(Gη) ̸= ϕ}

is finite. Since if t ∈ W ∩ ψ(Gα) then there is r ∈ S1 with r ∈ ψ−1(t) ⊆ ψ−1(W ) =
∩m
j=1ψ

−1(We(j)) ⊆ ∩m
j=1Kj = Ks, this means Hsi ∩Gα ̸= ϕ for some i.

Theorem 12. Let ψ : (S1, µ1) → (S2, µ2) be a (µ1, µ2)-continuous mapping, (µ1, µ2)-
open and (µ1, µ2)-closed surjective with ψ−1(e) is µ∗1-compact for each e ∈ S2. If E is
α-gµ-paracompact in (S1, µ1), then ψ(E) is α-gµ-paracompact in (S2, µ2).

Proof. Let H = {Hα : α ∈ ∆} be (S2, µ2)-cover of ψ(E). Since ψ is a (µ1, µ2)-
continuous mapping, then the collection G = {ψ−1(Hα) : α ∈ ∆} is an (S1, µ1)-cover of E
and so G has a gµ−LF(S1,µ1) (S1, µ1)-refinement, say W = {Wβ : β ∈ Λ}. Therefore, by
Lemma 3, ψ(W) = {ψ(Wβ) : β ∈ Λ} is gµ−LF(S2,µ2) (S2, µ2)-refinement of H in (S2, µ2).

4. Some application on α-gµ-paracompact sets

In this section, we introduce the notion of co-α-gµ-paracompact set as an application
of α-gµ-paracompact and study some of its properties.

Definition 8. Let (S, µ) be a GTS. A subset E ⊆ S is called co-α -gµ-paracompact if
for each s ∈ E there is a pair (H,G) with H ∈ µ and G is α-gµ-paracompact such that
s ∈ H −G ⊆ E. The collection of all co-α-gµ-paracompact sets will be denoted by µαgµ.

Theorem 13. Let (S, µ) be a GTS. Then:

(i) (S, µαgµ) is a GTS with µ ⊆ µαgµ.

(ii) B(µαgµ) = {H −G : H ∈ µ and G is α-gµ-paracompact } generate a base for µαgµ.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 9 of 11

Proof. (i) Let E = {Eα : α ∈ ∆} be a collection of µαgµ subset. If s ∈ ∪
α∈∆

Eα, then

there is α◦ ∈ ∆ and a pair (H,G) with H ∈ µ and G is α -gµ-paracompact such that
s ∈ H − G ⊆ Eα◦ ⊆ ∪

α∈∆
Eα. Since ϕ is α -gµ-paracompact then µαgm is a GTS on S.

Moreover, if H ∈ µ then H = H − ϕ ∈ µαgm.
(ii) It follows from Definition 8.

The following example will show that the reverse inclusion of Theorem 13 is not true
in general.

Example 3. Consider S = (0, 1) and B = {ϕ} ∪ {(0, a), (a, 1) : a ∈ (0, 1)}. Assume that
(S, µ(B)) is the GTS generated on S by the base B. Then S ∈ µ(B) and hence S−{1

3 ,
1
2} ∈

µαgm − µ(B).

Theorem 14. Let (S, µ) be a GTS. Then the following are equivalent:

(i) µ = {H −G : H ∈ µ and G is α-gµ-paracompact };

(ii) µEc ⊆ µ for each E is α-gµ-paracompact;

(iii) µ = µαgµ.

Proof. (i ⇒ ii) Let H ∈ µEc . Then H = G ∩ Ec = G − E with G ∈ µ and E is
α-gµ-paracompact. By part (i), H ∈ µ.

(ii ⇒ iii) Let E ∈ µαgµ. Then, for each s ∈ E there is a pair (H,G) with H ∈ µ and
G is α-gµ-paracompact such that s ∈ H −G ⊆ E. Since H −G ∈ µGc ⊆ µ, then E ∈ µ.

(iii⇒ i) From Definition 8, the collection {H−G :H ∈ µ andG is α-gµ-paracompact}⊆
µαgµ = µ. Now, let E ∈ µ, then E − ϕ ∈ {H − G : H ∈ µ and G is α-gµ-paracompact}
and hence the result follows.

Theorem 15. Let (S, µ) be a GTS. If E is µ∗ closed, then (µαgµ)E ⊆ (µE)
αgµ.

Proof. Let H ∈ (µαgµ)E with s ∈ H. Then H = G ∩ E with G ∈ µαgµ. Since there is
a pair (Z,W ) with Z ∈ µ and W is α-gµ-paracompact such that s ∈ Z −W ⊆ G, then
s ∈ (Z ∩ E) − (W ∩ E) ⊆ H. Now Z ∩ E ∈ µE and by Theorems 5 and 7, W ∩ E is α
-gµ-paracompact in (E,µE). Therefore, H ∈ (µE)

αgµ.

Proposition 5. Let ψ : (S1, µ1) → (S2, µ2) be (µ1, µ2)-homeomorphism with ψ−1(e) is
µ∗-compact for each e ∈ S2. Then ψ : (S1, µ

αgµ1
1 ) → (S2, µ

αgµ2
2 ) is open mapping.

Proof. Let E ∈ B(µαgµ1
1 ). Then by Theorem 12, ψ(E) ∈ µαgµ2

2 and hence ψ is open.

Question: Let (S, µ) be aGTS. What are the conditions to become µαgµ = (µαgµ)αgµ?

The following consequence is a partial answer to this question.

Recall that a subset E is called α-paracompact of (S, µ) [3] if each (S, µ)-cover of E
has a µ−LF(S,µ) (S, µ)-refinement.



H. H. Al-Jarrah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5899 10 of 11

Theorem 16. Let (S, µ) be a T2-topological space. Then µαgµ = (µαgµ)αgµ.

Proof. At first, note that a subset E of (S, µ) is α-gµ-paracompact iff it is α-paracompact
in (S, µ) since µ = µ∗. As in the proof of Theorem 10 each α-paracompact set in (S, µ) is
closed and so by Theorem 14, µ = µαgµ. Therefore, (µαgµ)αgµ = µαgµ = µ.

5. Conclusion

One major area of study in topological studies is the exploration of topological notions
and topics through extensions of classical topology. Generalized topology is one of the
recent extensions of topology and hence we investigate the definition of gµ-paracompact
space that is defined in [6], to study the main characteristics of two types of gµ-paracompact
subsets, namely, α-gµ-paracompact and β-gµ-paracompact and we examine the relation-
ship between them. In future work, we intend to study gµ-paracompact spaces in other
structures such as supra and infra-topological spaces. Furthermore, we can study and de-
fine other forms of gµ-paracompact spaces by using µ-semi-open or µ-preopen sets which
are defined in [5].

Acknowledgements

The publication of this paper was supported by the Yarmouk University Research
Council.

Availability of data and material: No data were used to support this study.
Conflicts of interest: The authors declare no conflict of interest.

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