EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 280-286
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

A-paracompactness and Strongly A-screenability in
Topological Groups

Muhammad Kashif Maqbool1,∗, Awais Yousaf1, Muhammad Siddique Bosan2,
Saeid Jafari3

1 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100,
Pakistan
2 Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad,
Islamabad 44000, Pakistan
3 College of Vestsjaelland South, Slagelse 4200, Denmark

Abstract. A space is said to be strongly A-screenable if there exists a σ-discrete refinement for
each open cover. In this article, we have investigated some of the features of A-paracompact and
strongly A-screenable spaces in topological and semi topological groups. We predominantly show
that (i) Topological direct product of (countably) A-paracompact topological group and a compact
topological group is (countably) A-paracompact topological group. (ii) All the left and right cosets
of a strongly A-screenable subset H of a semi topological group (G, ∗, τ) are strongly A-screenable.

2020 Mathematics Subject Classifications: 22C05, 22A05, 22A10, 54C05

Key Words and Phrases: A-paracompactness, Semi δ-topological group, Strongly A-screenability,
N−capc disjoint set

1. Introduction and Background Results

It is always captivating to dig into relationship of the topological spaces with alge-
braic structures. To bring out some new results and explore several concepts, many
of the mathematicians make a relationship between these two structures by debilitating
or strengthening different conditions [3, 19, 20]. J. Dieudonne (1944) and P. Alexandrov
(1945) introduced the terms paracompactness and A-paracompactness respectively [5, 11].
L. Ivanovski and V. Kusminov discussed that each bicompact topological group is dyadic.
In 1962, J. Kister proved some properties of compactness in topological groups [17]. In
1972, O. T. Alas explored the properties of paracompactness in topological groups [4],
and L. G. Brown discussed some properties in topologically complete topological groups
[9]. In 1981, A. V. Arhangelskii discussed locally subparacompact, locally paracompact,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3637

Email addresses: kashifmaqbool9@gmail.com (M. K. Maqbool), awaisysf@gmail.com (A. Yousaf),
siddiquebosan@hotmail.com (M. S. Bosan), jafaripersia@gmail.com (S. Jafari)

http://www.ejpam.com 280 c© 2020 EJPAM All rights reserved.



M. K. Maqbool et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 280-286 281

locally strongly paracompact topological groups [8]. In 1989, D. B. Shakhmatov presented
strongly and completely paracompactness in topological groups [24]. In 1996, D. Buha-
giar and B. Pasynkov discussed uniform paracompactness in topological groups [10]. S.
Romaguera and M. Sanchis in 2000 investigated locally compactness in topological groups
[23]. In 2007, A. V. Arhangelskii discusses some properties concerning paracompactness in
topological groups [6]. Strong realcompactness is discussed in 2012 by M. G. Tkachenko in
topological groups [26]. In 2017, H. Juarez-Anguiano discussed strongly paracompactness
in topological groups [15].

To extend this work, we have discussed A-paracompactness and strongly A-screenability
in topological and semi topological groups. We introduced the term N -capc disjoint sets,
and presented the notion semi δ-topological group. Moreover, we prove that, (R,+, τ) is
an ultra-A-paracompact topological group.

2. Preliminaries

A topological space is said to be compact if there is a finite subcover for each open
cover [28]. An A-paracompact space is a space which contain a locally finite refinement
for every open cover. In paracompact space there exists a locally finite open refinement
for each open cover [11]. Unlike paracompact space, in A-paracompact space locally finite
refinement need not be necessarily open. Every closed subset of a paracompact space is
paracompact [5]. Closed continuous image of a paracompact space is paracompact [22].
Every regular strongly screenable topological space is paracompact [21]. Topological prod-
uct of metric and compact Hausdorff space is paracompact [25]. Katetov [16] and Dowker
[12] introduced countably paracompact spaces. A space in which there exists a locally
finite refinement for each countably open cover is said to be countably A-paracompact
space [12]. Moreover, in many results countable paracompactness occur with normality
[13, 14].

Let K ≤ G and g ∈ G, then Kg and gK are said to be the right and left cosets of K in G
respectively. Left (right) translation lt1 : G→ G (rt1 : G→ G) is defined as lt1(t2) = t1∗t2
(rt1(t2) = t2 ∗ t1). For a group (G, ∗), the multiplication mapping m : G × G → G is
defined as m(x, y) = x × y = z, for x, y, z ∈ G. A multiplication mapping is said to be
jointly continuous if the defined multiplication mapping is continuous and is separately
continuous if the left and the right translations are continuous. For a space τ and a
group G, a triplet (G, ∗, τ) is said to be a semi topological group if multiplication mapping
is separately continuous. A quasi topological group is a semi topological group with
continuous inverse mapping. In a paratopological group (G, ∗, τ) multiplication mapping
is jointly continuous. A paratopological group having continuous inverse mapping is said
to be a topological group [7]. In addition, many mathematicians have explored different
properties related to compactness [1, 2, 18]. Our notations are standard as used in [13, 27].



M. K. Maqbool et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 280-286 282

3. A-paracompactness and Strongly A-screenability

Definition 1. A space is said to be strongly A-screenable if there exists a σ-discrete
refinement for each open cover.

Theorem 1. All the left and right cosets of a strongly A-screenable subset H of a semi
topological group (G, ∗, τ) are strongly A-screenable.

Proof. For any a ∈ G, let Ω be an open cover of left coset aH. Then l−1
a (Ω) is an open

cover of subset H. Since H is strongly A-screenable, there exists σ-discrete refinement
U = ∪∞i=1µi. Thus, la(U) is σ-discrete refinement of open cover Ω of aH which asseverates
that for every a ∈ G, aH is strongly A-screenable. Similarly, all right cosets are strongly
A-screenable.

Corollary 1. A semi topological group (G, ∗, τ) is strongly A-screenable if it contains a
strongly A-screenable subset H such that |G|/|H| is countable.

Theorem 2. In a semi topological group free product of an A-paracompact (countably A-
paracompact) subset with any finite subset is A-paracompact (countably A-paracompact).

Proof. Suppose that (G, ∗, τ) is a semi topological group, where S and T are respec-
tively A-paracompact (countably A-paracompact) and finite subsets of G. For t1 ∈ T ,
lt1(S) = t1 ∗ S. Let {Aλ, λ ∈ ω} be an open (countably open) cover of t1 ∗ S. Then
{lt−1

1
(Aλ), λ ∈ ω} is an open (countably open) cover of S. So, there is a locally finite

refinement {lt−1
1

(A∗λ), λ ∈ ω∗} of S. Therefore, {A∗λ, λ ∈ ω∗} is locally finite refinement

of t1 ∗ S. Hence, t1 ∗ S is A-paracompact. Let Ω be an open (countably open) cover of
TS = ∪ti∈T ti ∗ S, then there is an open (countably open) cover Ω1 ⊆ Ω of t1 ∗ S. So,
there exists a locally finite refinement Ω∗1 of Ω1. Therefore, Ω∗ = ∪{Ω∗i , i = 1, 2, ..., |T |} is
locally finite refinement of TS.

Theorem 3. Let H be a Hausdorff paracompact subset of a semi topological group (G, ∗, τ),
then each right or left coset of H is a normal space if and only if each pair of closed disjoint
singleton subsets of a coset can be separated by its open sets.

Proof. For g ∈ G, let F1 and F2 be a pair of disjoint singleton closed subsets and M1

is an open set of a coset gH of a set H ⊆ G. As the set U = {h|lg(h) ∩ F1 ⊆ M1} is
open in H. Let W1 = H \ l−1

g (F1 ∩ (gH \M1)) and h∗ be an arbitrary point such that
lg(h

∗) ∩ F1 ⊆ M1, then W1 is open in H and h∗ ∈ W1, (lg(W1) ∩ F1) ∩ (gH \M1) = φ.
Hence, lg(W1)∩F1 ⊆M1. Thus, the set U is open in H. Moreover, UM1 = {h|lg(h)∩F1 ⊆
M1, lg(h)∩F2 ⊆ gH \Cl(M1)} is open in H. For each h∗ ∈ H, lg(h

∗)∩F1 and lg(h
∗)∩F2

are closed and disjoint sets of lg(h
∗). Therefore, there exists two open sets M∗1 and M∗2 of

gH such that lg(h
∗)∩F1 ⊆M∗1 , lg(h

∗)∩F2 ⊆M∗2 and M∗1 ∩M∗2 = φ. As Cl(M∗1 )∩M∗2 = φ,
M∗2 contained in complement of closure of M∗1 in gH, so y∗ ∈ U∗M1

. Hence, {UM1} for
all open sets M1 of gH is open cover of H. Thus, there exists an open locally finite
refinements {WM1 |M1 ∈ Ω}, Ω is collection of open sets of gH such that Cl(WM1) ⊆ UM1



M. K. Maqbool et al. / Eur. J. Pure Appl. Math, 13 (2) (2020), 280-286 283

for each M1 ∈ Ω. Suppose, M2 = ∪M1∈Ω(lg(WM1) ∩M1), then M2 is open in gH and
{lg((WM1) ∩M1),M1 ∈ Ω} is locally finite. Therefore, Cl(M2) = ∪M1∈Ω(Cl(lg(WM1) ∩
M1)) ⊆ ∪M1∈Ω(lg(Cl(WM1)) ∩ Cl(M1)). Also, as lg(WM1) ∩ F1 ⊆ lg(UM1) ∩ F1 ⊆ M1,
we have lg(WM1) ∩ F1 ⊆ lg(WM1) ∩M1 ⊆ M2. As {lg(WM1)|M1 ∈ Ω} is cover of gH,
so F1 ⊆ M2. Moreover, (lg(Cl(WM1)) ∩ F2) ∩ Cl(M1) ⊆ (lg(UM1) ∩ F2) ∩ Cl(M1) ⊆
(gH\Cl(M1))∩Cl(M1) = φ. Then F2∩Cl(M2) = φ. F2 contained in open set gH\Cl(M2).
Thus, open sets M2 and gH \ Cl(M2) separates F1 and F2. Hence, any left coset of H
is normal. Similarly, it can be prove that, any right coset of H is normal. Conversely, if
any left or right coset of H is a normal space, then each pair of disjoint singleton closed
subsets of coset can be separated by its open sets.

Theorem 4. Topological direct product of (countably) A-paracompact topological group
and a compact topological group is (countably) A-paracompact topological group.

Proof. Suppose that X is a (countably) A-paracompact topological group and Y be a
compact topological group. Suppose that {Uj}(j = 1, 2, 3, ...) is a (countable) covering of
X × Y . Let Vi consists of all points x of X satisfying x × Y ⊆ ∪j≤iUj . If x ∈ Vi, then
each (x, y) of x × Y has a neighbourhood N ×M contained in open set ∪j≤iUj . These
finite open sets M cover Y . Let Nx be the intersection of corresponding sets N . Then
x ∈ Nx, Nx is open and Nx × Y ⊆ ∪j≤iUj , and hence Nx ⊆ Vi. Therefore, Vi is open.
Moreover, for an arbitrary x ∈ X, x × Y is contained in some finite sets of the covering
{Uj}, because x×Y is compact. Therefore, x is in some Vi. Thus, {Vi} is a covering of X.
As {Vi} is (countable) open covering and X is (countably) A-paracompact, {Vi} possess a
locally finite refinement B. For every W ∈ B, suppose g(W ) is the first Vi that contain W
and let Gi is the union of all W for which g(W ) = Vi. Then Gi ⊆ Vi and {Gi} is locally
finite covering of X.

Let Gij = (Gi × Y ) ∩ Uj , for j ≤ i. If (x, y) is an arbitrary point of (X,Y ), then for
some i, x ∈ Gi, (x, y) ∈ Gi × Y . Also since x ∈ Gi ⊆ Vi, (x, y) ∈ x× Y ⊆ ∪j≤iUj . Hence,
for some i ≥ j, (x, y) ∈ Uj . It follows that, (x, y) ∈ Gij . Therefore, {Gij} is covering of
X × Y . Since, Gij ⊆ Uj , then {Gij} is a refinement of {Uj}. Also if (x, y) ∈ X × Y , x
belongs to an open set H(x) which meet only a finite sets of {Gi}. Then H(x)× Y is an
open set containing (x, y) which can meet Gij only if H(x) meet Gi. But for each i there is
only finite sets Gij . Hence, H(x)×Y meets only a finite sets of {Gij}. So, {Gij} is locally
finite. Also X×Y is a topological group. Therefore, X×Y is (countably) A-paracompact
topological group.

Definition 2. A triplet (G, ∗, τ), where τ is a space and G is a group, is called semi
δ-topological group if multiplication mapping is separately δ-continuous in (G, ∗, τ).

Definition 3. A space is called almost A-paracompact if its each open cover has a star-
finite refinement.

Definition 4. A space is called nearly almost A-paracompact if its every regular open
cover has a star-finite refinement.



REFERENCES 284

Theorem 5. (G, ∗, τs) is almost A-paracompact semi topological group if and only if
(G, ∗, τ) is nearly almost A-paracompact semi δ-topological group.

Proof. Separate continuity of multiplication mapping in (G, ∗, τs) is the same as sepa-
rate δ-continuity of multiplication mapping in (G, ∗, τ). Let Ω is a regular open cover of
(G, ∗, τ). Therefore, for each ω ∈ Ω, Ω = Int(Cl(ω)), so {Int(Cl(ω)), ω ∈ Ω} is an open
cover of (G, ∗, τs). Thus, there is a star finite refinement U = {µα, α ∈ J} of (G, ∗, τs).
Hence, U is star finite refinement of Ω. Conversely, every open cover Ω of (G, ∗, τs) is
regular open cover of (G, ∗, τ). So, there is a star finite refinement U = {µα, α ∈ J} of Ω.

Definition 5. In semi topological group (G, ∗, τ) having a closed A-paracompact (countably
A-paracompact) subset N . A subset S is said to be N−capc disjoint (N−ccapc disjoint),
if for each s1, s2 ∈ S, s1 /∈ s2 ∗N .

Theorem 6. Let (G, ∗, τ) be a semi topological group with B ⊆ G and H is closed A-
paracompact (countably A-paracompact) subset of G. Then there exists a subset N of H
such that B is N−capc disjoint (N−ccapc disjoint).

Proof. Since, H is closed A-paracompact (countably A-paracompact) therefore lb(H) =
b∗H for every b ∈ B is A-paracompact (countably A-paracompact). Moreover, b∗H being
inverse image of closed set under translation lb−1(b∗H) = H is closed. Each ai ∈ B has an
open neighbourhood Ui which intersects finite sets in the refinement of open (countably
open) cover of b ∗H. By removing all Ui from b ∗H a closed A-paracompact (countably
A-paracompact) set N = b ∗H \Ui is obtained such that B is N−capc disjoint (N−ccapc
disjoint).

Definition 6. A Hausdorff space is called an ultra-A-paracompact if there exits a closed
locally finite refinement for each open cover.

Example 1. (R,+, τ) is an ultra-A-paracompact topological group for a usual topology τ .

Proof. Let Ω be an open cover of (0,∞). If κ = sup{xα : α < β} < ∞ for an
ordinal β, then let there is a real number xβ satisfying κ < xβ and [κ, xβ] ⊆ U for
U ∈ Ω. Then {[xβ, xβ+1] : β} is a partition into closed sets of (0,∞) that refines Ω. As
R is homeomorphic to (0,∞) and (R,+, τ) is a topological group. Hence, (R,+, τ) is an
ultra-A-paracompact topological group.

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