@
Appl. Gen. Topol. 23, no. 1 (2022), 189-199

doi:10.4995/agt.2022.14846

© AGT, UPV, 2022

Selection principles: s-Menger and

s-Rothberger-bounded groups

Muhammad Asad Iqbal and Moiz ud Din Khan

Department of Mathematics, COMSATS University Islamabad, Pakistan. (m.asadiqbal494@gmail.com,

moiz@comsats.edu.pk)

Communicated by O. Valero

Abstract

In this paper, selection principles are defined and studied in the realm
of irresolute topological groups. Especially, s-Menger-bounded and s-
Rothberger-bounded type covering properties are introduced and stud-
ied.

2020 MSC: 22A05; 54D20; 03E75.

Keywords: irresolute topological group; s-Menger-bounded group; s-
Rothberger-bounded group; selection principle.

1. Introduction

Many topological properties are defined or characterized in terms of the
following two classical selection principles.

Let P and Q be sets consisting of families of subsets of an infinite set X.
Then:

Sfin(P,Q) denotes the selection hypothesis: for each sequence (Pn)n∈N of
elements of P there is a sequence (Qn)n∈N of finite sets such that for each n,
Qn ⊂ Pn, and

⋃
n∈N Qn ∈Q.

S1(P,Q) is the selection hypothesis: for each sequence (Pn)n∈N of elements
of P there is a sequence (pn)n∈N such that for each n, pn ∈ Pn, and {pn : n ∈
N} is an element of Q (see [28]).

Received 25 December 2020 – Accepted 17 January 2022

http://dx.doi.org/10.4995/agt.2022.14846
https://orcid.org/0000-0002-9732-1198


M. A. Iqbal and M. Khan

Let O denote the family of all open covers of a space X. The property
Sfin(O,O) (resp. S1(O,O)) is called the Menger (resp. Rothberger) covering
property. For more information about selection principles theory and its rela-
tions with other fields of mathematics we refer the reader to see [16, 27, 29, 30].

A topological group is a group with a topology, such that the group opera-
tions are continuous. If the group operations are irresolute mappings instead of
continuous mappings then we obtain the irresolute topological groups (ITG).

In the recent years many papers about selection principles and topological
groups have appeared in the literature. o-bounded topological groups were
introduced by O.Okunev. (This notion was also given by Kocinac by the same
definition but under the name of Menger-bounded in an unpublished work of
Kocinac in [15]. Let us now recall [10].

Definition 1.1. A topological group (G,∗,τ) is M-bounded (R-bounded) if
there is for every sequence (Pn)n∈N of neighborhoods (nbd) of 1G, a sequence
(Qn)n∈N of finite subsets of G (a sequence (Pn)n∈N of elements of G) such that
G =

⋃
n∈N Qn ∗Pn (resp. G =

⋃
n∈N pn ∗Pn).

ITGs was first studied by Khan, Siab and Kocinac in [13] where their prop-
erties were investigated and their differences from topological groups were es-
tablished. Although many papers on topological groups were published there
are very few papers which deal with ITGs.

Our main aim in considering selection principles is to link this with earlier
work on irresolute topological groups. Hence, Section 2 contains several defini-
tions and results which will be needed later on. In Section 3 s-Menger-bounded,
s-Rothberger-bounded and s-Hurewicz-bounded type covering properties are
introduced.

2. Preliminaries

In this section we recall some basic definitions and results that will enable
the casual reader to follow the general ideas presented here.

If (G,∗) is a group, and τ a topology on G, then we say that (G,∗,τ) is a
topologized group with multiplication mapping µ : G ×G → G, (p,q) 7→ p∗ q
and the inverse mapping i : G → G,p 7→ p−1. The identity element of G is
denoted by e, or eG when it is necessary,

Throughout the paper X and Y denote topological spaces. For a subset P
of X, Cl(P) and Int(P) will denote the closure and interior of P . We denote
f←(Q) to define the preimage of a subset Q ⊂ Y for a mapping f : X 7→ Y . The
reader is refereed to [7] for undefined topological terminology and notations.
A subset P of a topological space X is said to be semi-open [20] if there is an
open set R in X such that R ⊂ P ⊂ Cl(R). If a semi-open set P contains a
point p ∈ X we say that P is a semi-open nbd of p. If X satisfies Sfin(sO,sO)
(resp. S1(sO,sO)), then we say that X has the s-Menger (resp. s-Rothberger)
covering property [18, 26], where sO denotes the family of all semi-open covers

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 190



Selection principles: s-Menger and s-Rothberger-bounded groups

of X. Throughout SO(X) represents the collection of all semi-open sets in X.
For terms not defined here we refer the reader to see [26].

Definition 2.1. A mapping f : X → Y between spaces X and Y is called
irresolute [6] (resp. pre-semi-open) if for each semi open set Q ⊂ Y (resp.
P ⊆ X), the set f←(Q) is semi open in X (resp. f(P) is semi open in Y ).

Definition 2.2. A triplet (G,∗,τ) is called an ITG [13] if for each p,q ∈G and
each semi-open nbd R of p∗q−1 in G there exist semi-open nbds P of p and Q
of q such that P ∗Q−1 ⊂ R.

We note that the union of any family of semi-open sets is semi-open whereas
the intersection of two semi-open sets need not be semi-open, thus the family
of semi-open sets in a topological space need not be a topology. However in
[13] the authors pointed out that if (G,∗,τ) is an ITG such that the family
SO(G) is a topology on G with SO(G) 6= τ, then (G,∗,SO(G)) is a topological
group. (see, Observation [13]).

Lemma 2.3 ([13]). If (G,∗,τ) is an ITG, then

(1) P ∈ SO(G) if and only if P−1 ∈ SO(G).
(2) If P ∈ SO(G) and Q ⊂G, then P ∗Q and Q∗P are both in SO(G).

Lemma 2.4 ([12]). A space X is extremely disconnected if and only if the
intersection of any two semi-open subsets of X is semi-open.

Lemma 2.5 ([23]). Let P ⊂ X0 ∈ SO(X), then P ∈ SO(X) if and only if
P ∈ SO(X0).

Lemma 2.6 ([25]). Let X0 be a subspace of X and P ∈ SO(X0), then P=Q∩
X0 for some Q ∈ SO(X).

Recall the following notations for collection of covers of a space X.

• sω-cover : A semi open cover P of X is semi-ω-cover (sω-cover) [26]
if for each finite subset Q of X there exists P ∈ P such that Q ⊂ P
and X is not the member of P. The symbol sΩ denotes the family of
sω-covers of X.

• sγ-cover : A semi open cover P of X is a sγ-cover [26] if it is infinite
and for every p ∈ X the set {P ∈P = p /∈ P} is finite. The collection
of sγ-covers of X will be denoted by sΓ.

We are particularly interested here in the case where P and Q are open covers of
topological spaces or topological groups. Specifically, let H and G be topological
spaces with G a subspace of H.

• sOH : The collection of semi-open covers of H.
• sOHG : The collection of covers of G by sets semi-open in H.
• sΩH : The collection of sω-covers of H.
• sΩHG : The collection of sω-covers of G by sets semi-open in H.
• sOH(P) : Let (H,∗,τ) be an ITG with neutral element eH, if P is a

semi-open nbd of eH, then p∗P =: {p∗ q : q ∈ P} is a semi-open nbd

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 191



M. A. Iqbal and M. Khan

of p. Thus, {p∗P : p ∈H} is semi-open cover of H and will be denoted
by sOH(P).
• sOnbd(H) : = {P ∈ sO : (∃semi-open nbd P of eH) (P = {p∗P : p ∈
H})}.
• sΩH(P) : For each semi open nbd P of eH, sΩH(P) = {Q ∗ P : Q ⊂
H finite} is an sω-cover of H, where Q∗P := {q ∈ Q and p ∈ P} when
H is not an element of this set.
• sΩnbd(H) : = {P ∈ sΩ : (∃ semi-open nbd P of eH)(U = {Q∗P : Q ⊂H

finite})}.
• sOnH(P) : =For each semi open nbd P of eH, sO

n
H(P) = {Q∗P : Q ⊂H

and 1 ≤| Q |≤ n} is an n-cover of group H.

Lemma 2.7. If (H,∗) is an extremely disconnected ITG with the neutral ele-
ment e, then for each semi-open nighborhood P of e, there exists a symmetric
semi-open nbd W of e such that: Q = Q−1 ⊂ P.

3. s-Menger-bounded, s-Rothberger-bounded and
s-Hurewicz-bounded groups

Babinkostova, Kocinac and Scheepers in [4] investigated Menger-bounded (o-
bounded [9]) and Rothberger-bounded groups in the area of selection principles.
On analogues to the Menger-bounded (o-bounded) and Rothberger-bounded
groups we examine s-Menger-bounded and s-Rothberger-bounded groups. We
also investigate the internal characterizations of groups having these properties
in all finite powers (Theorem 3.8, Theorem 3.9, and Theorem 3.13). To intro-
duce this new concept we use covering properties by semi open sets instead of
open sets and the ITG properties. Semi-Menger spaces have been investigated
in [26]. We recall that a space X is said to have the semi-Menger property (or
s-Menger property) if it satisfies Sfin(sO,sO). Specifically from [26, Theorem
3.8] X is s-Menger if and only if X satisfies Sfin(sΩ,sO).

Definition 3.1. An ITG (G,∗,τ) is:
(1) s-Menger-bounded if for each sequence (Pn)n∈N of semi-open nbds of

the neutral element e ∈ G, there exists a sequence (Qn)n∈N of finite
subsets of G such that G =

⋃
n∈N Qn ∗Pn.

(2) s-Rothberger-bounded if for each sequence (Pn)n∈N of semi-open nbds
of the neutral element e ∈ G, there exists a sequence (pn)n∈N of ele-
ments of G such that G =

⋃
n∈N pn ∗Pn.

(3) s-Hurewicz-bounded if there is for each sequence (Pn)n∈N of semi open
nbdss of neutral element e ∈ G, there exists a sequence (Qn)n∈N of
finite subsets of G such that each x ∈G belongs to all but finitely many
Qn ∗Pn.

Let (G,∗) be a subgroup of group (H,∗). Then G is s-Menger-bounded if
the selection principle Sfin(Onbd(H),OHG) holds, s-Rothberger-bounded if the
selection principle S1(Onbd(H),OHG) holds and s-Hurewicz-bounded if the se-
lection principle S1(Ωnbd(H),O

gp
HG) holds.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 192



Selection principles: s-Menger and s-Rothberger-bounded groups

In subsections 3.1, 3.2 and 3.3 we have verified various properties of each se-
lection principle by taking three types of covering semi open, s-gamma and
s-omega and using relation with one another.

3.1. s-Menger-bounded groups. In this subsection we have verified some
results on s-Menger-bounded groups.

Theorem 3.2. Let (H,∗,τ) be an ITG and G ≤ H. Then Sfin(sOH,sOHG)
implies Sfin(sΩnbd(H),sOHG).

Proof. Since sΩnbd(H) is a subclass of sOH. Therefore, the proof follows im-
mediately. �

Remark 3.3. Converse of the Theorem 3.2 is not true in general.

Example 3.4. Real line (R, +,τ) with usual topology τ is an ITG under the
binary operation of addition. It is known [26], that (R, +) is a Menger space
but not the s-Menger space. Therefore, there is at least one collection of semi
open covers say P1, P2, ..., Pn,... for which there exists no collection Vn of finite
subsets of Pn which satisfy

⋃
Qn = H. Thus Sfin(sOH,sOHG) fails to hold.

In order to show that Sfin(sΩnbd(H),sOHG) holds, we follow as under: Let
(Pn)n∈N be a sequence from sΩnbd(H). Then for each n, Pn={Q∗Pn : Q ⊂H
finite} and Pn ∈ SO(H,eH). Since Q is finite set therefore each Pn is finite.
Then for each n ∈ N we can choose Rn of finite subsets of Pn such that⋃
Rn = G. This proves that, Sfin(sΩnbd(H),sOHG) holds.

Theorem 3.5. Let (H,∗,τ) be an ITG and G ≤H. Then the following state-
ments are equivalent:

(1) S1(sΩnbd(H),sOHG).
(2) Sfin(sOnbd(H),sOHG).
(3) Sfin(sΩnbd(H),sOHG).

Proof. (1) ⇒ (2) is straightforward.
(2) ⇒ (3) : Since sΩnbd(H) is a subclass of sOnbd(H). Therefore, the proof
follows immediately.
(3) ⇒ (1) : Let (Pn)n∈N ∈ sΩnbd(H). Select a semi-open nbd Pn of eH for each
n such that Pn = sΩH(Pn). Now, Apply Sfin(sΩnbd(H),sOHG) to (Pn)n∈N :
For each n let a finite set Rn ⊂ Pn such that

⋃
n∈N Rn is semi-open cover of

G. Then each Qn is a finite subset of H. Put Rn = Qn ∗Pn. Then for each n
we have Rn ∈Pn, and, thus {Rn}n∈N is a semi-open cover of G.

Indeed, by writing N =
⋃

n∈N Yn here union is disjoint, and applying
S1(sΩnbd(H),sOHG) to each sequence (sΩ(Pk) : k ∈ Yn) independently, one
finds a sequence (Qn : n ∈ N) of subsets of H also finite such that for each
p ∈G there are infinitely many with p ∈ Qn ∗Pn. �

Theorem 3.6. Let (H,∗,τ) be an ITG and a semi open set G ≤H. Then the
following statements are equivalent:

(1) S1(sΩnbd(H),sOHG).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 193



M. A. Iqbal and M. Khan

(2) S1(sΩnbd(G),sOG).

Proof. (1) ⇒ (2) : Let (sΩ(Pn) : n ∈ N) ∈ sΩnbd(G), where every Pn is a
semi-open nbd in G containing the group neutral element. Then by Lemma
2.6, select Qn ∈ SO(e,H) for each n such that Pn = Qn ∩G. Now, select
Rn ∈ SO(e,H) for each n such that R−1n ∗Rn ⊆ Qn. Apply S1(sΩnbd(G),sOG)
to (sΩ(Rn) : n ∈ N) ∈ sΩ(H): We find for each n a set Qn ⊂H which is finite
such that G ⊆

⋃
n∈N Sn ∗Rn. Since Rn is semi open therefore Sn ∗Rn is semi

open. For each n, and for each m ∈ Sn, choose a pm ∈G as follows:

pm

{
∈G∩m∗Rn if nonempty,
= e if otherwise.

Then put a finite set Tn = {pm : m ∈ Sn}⊂G. For each n we have Tn ∗Pn ∈
sΩ(Pn) ∈ sΩnbd(G). Now only remaining to show that G =

⋃
n∈N Tn ∗ Pn.

For let l ∈ Tn be given. Choose n so that g ∈ Sn ∗ Rn, and choose m ∈ Sn
so that l ∈ m ∗ Sn. Then obviously G ∩ m ∗ Rn 6= φ, and so pm ∈ G is
belonging to G∩m∗Rn. Since pm ∈ m∗Rn, We have m ∈ pm ∗R−1n , and so
l ∈ pm ∗R−1n ∗Rn ⊆ pm ∗Qn. Now p−1m ∗ l ∈G∩Qn = Pn, and so we have that
l ∈ pm ∗Pn ⊂ Tn ∗Pn.

(2) ⇒ (1) : By Lemma 2.5 the proof is evident. �

Corollary 3.7. Let (H,∗,τ) be an ITG and G ≤H. If Sfin(sOH,sOHG) holds
then S1(sΩnbd(G),sOG).

Proof. By Theorem 3.2, Theorem 3.5 and Theorem 3.6, we have
Sfin(sOH,sOHG) ⇒ Sfin(sΩnbd(H),sOHG) ⇒ S1(sΩnbd(H),sOHG)
⇒ S1(sΩnbd(G),sOG). �

Theorem 3.8. Let (H,∗,τ) be an extremely disconnected ITG and G ≤ H.
Then the following statements are equivalent:

(1) S1(sΩnbd(H),sO
wgp
HG ).

(2) S1(sΩnbd(H),sΩHG).

Proof. (1) ⇒ (2) : Let (Pn : n ∈ N) ∈ sΩnbd(H), and select Pn ∈ SO(eH,H)
for each n with Pn = sΩH(Pn). Then define, Qn =

⋂
j≤n Pj. Each Qn =

sΩH(Qn) ∈ sΩnbd(H). Apply S1(sΩnbd(H),sO
wgp
HG ) to (Qn : n ∈ N), then

there is a sequence Rn ∈ Qn for each n, such that {Qn : n ∈ N} is a cover
of G and cover is weakly groupable. Suppose an increasing sequence p1 <
p2 < p3 < ... < pk < ... such that there is for each finite S ⊂ G, a k with
S ⊂

⋃
pk≤j≤pk+1 Rj. Further, select a finite Sn ⊂H with Rn = Sn ∗Qn. Since

Qn is semi open so is Rn. For i < p1 set Ti =
⋃

j<p1
Sj, and for pk ≤ i < pk+1,

set Ti =
⋃

pk≤j<pk+1 Sj. So each finite Ti ⊂ H, and for each i if we put
Ui = Ti ∗Pi, then Pi ∈Pi for each i, and {Pi}i∈N in sΩHG.

(2) ⇒ (1) :This is evident. �

Theorem 3.9. Let (H,∗,τ) be an ITG and G ≤H. Then the following state-
ments are equivalent:

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 194



Selection principles: s-Menger and s-Rothberger-bounded groups

(1) For each n the selection principle S1(sΩnbd(Hn),sOHnGn ).
(2) S1(sΩnbd(H),sΩHG).

Proof. (1) ⇒ (2) : Suppose (sΩ(Pn))n∈N ∈ sΩnbd(H). Let Pn be a semi open
nbd eH . Let natural number N =

⋃
n∈N Rn, and Rn is infinite for each n, and

for m is not equal to n we have Rm∩Rn is empty. For each k, (sΩ(Pkn ) : n ∈ Rk)
is in sΩnbd(Hk). By (1), for each k, and for each n ∈ Rk, choose a finite
Qn ⊂ H such that {Qkn ∗ Pkn : n ∈ Rk} is a cover of Gk by sets semi open in
Hk. Now we show that {Qn ∗ Pn : n ∈ N} is in sΩHG. For let a finite set
Q = {q1,q2, ...,qj} ⊂ G. Then q = (q1,q2, ...,qj) ∈ Gj. Select an n ∈ Rj with
h ∈ Qjn ∗Pjn. Then Q ⊆ Qn ∗Pn ∈ sΩH(Pn).

(2) ⇒ (1) : Set m ∈ N, and consider (sΩH(Pn))n∈N ∈ sΩnbd(Hm). For each
k choose a semi open nbd Sk of eH such that S

m
k ⊆ Pk. Then (sΩH(Sk))k∈N ∈

sΩnbd(H). Apply S1(sΩnbd(H),sΩHG) and for each k, a finite Qk ⊂ H with
{Qk ∗Sk}k∈N ∈ sΩHG. Then {Qmk ∗S

m
k }k∈N is cover of G

m by semi open sets.
For each k choose Tk ∈ sΩH (Pk) with Qmk ∗S

m
k ⊂ Tk. Then {Tk : k ∈ N} is a

cover of Gm by semi open sets. �

Corollary 3.10. Let (H,∗,τ) be an ITG and G ≤ H. Then the following
statements are equivalent:

(1) S1(sΩnbd(H),sO
wgp
HG ).

(2) For each n, S1(sΩnbd(Gn),sOGn ).
(3) S1(sΩnbd(G),sΩG).
(4) S1(sΩnbd(G),sO

wgp
G ).

Proof. (1) ⇒ (2) : From Theorem 3.8, we have S1(sΩnbd(H), sO
wgp
HG ) ⇒

S1(sΩnbd(H), sΩHG) and from Theorem 3.9, for each n, we have S1(sΩnbd(H),
sΩHG) ⇒ S1(sΩnbd(Hn), sOHnGn ). Finally, from Theorem 3.6, S1(sΩnbd(Hn),
sOHnGn ) ⇒ S1(sΩnbd(Gn), sOGn ).
(2) ⇒ (1) : If we take H = G then from Theorem 3.9, for each n,
S1(sΩnbd(Gn),sOGn ) ⇒ S1(sΩnbd(G),sOG).
(3) ⇒ (4) : This is obvious because sω-cover is always weakly groupable cover.
(4) ⇒ (1) : This is obvious. �

3.2. s-Rothberger-bounded groups. In this subsection we have verified
some results on s-Rothberger-bounded groups.

Theorem 3.11. Let (H,∗,τ) be an ITG and a semi open set G ≤ H. Then
the following statements are equivalent:

(1) S1(sOnbd(H),sOHG).
(2) S1(sOnbd(G),sOG).

Proof. The proof is similar to the proof of Theorem 3.6. �

Theorem 3.12. Let (H,∗,τ) be an extremely disconnected ITG and G ≤ H.
Then the following statements are equivalent:

(1) S1(sOnbd(H),sO
wgp
HG ).

(2) S1({sOnnbd(H)}n∈N,sΩHG).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 195



M. A. Iqbal and M. Khan

(3) For each (Tn)n∈N ∈ N diverging to ∞, S1({sOTnnbd(H) : n ∈ N},sΩHG).

Proof. (1) ⇒ (2) : Let sOnnbd(H) be a collection of all semi open covers of
the form sOnH(P). For each n, put Pn = sO

n(Pn). Here, Pn is a semi open
nbd of the neutral element and Pn = sOn(Pn) = {Q ∗ Pn : Q ⊂ H, and
1 ≤ |Qn| ≤ n}. For each n put Vn =

⋂
j≤n Uj and for each n put Rn =

sO(Rn). Apply S1(sOnbd(H),sO
wgp
HG ) to the sequence (Rn : n ∈ N). For each

n choose an pn such that {pn ∗ Rn : n ∈ N} is in sO
wgp
HG . Choose a sequence

q1 < q2 < q3 < ... < qk < qk+1 <... such that: For each finite set Q ⊂G there
is an n with Q ⊆

⋃
mn≤j≤qn+1pj ∗Rj. For each such i define:

Qi =

{
{pj : j ≤ i} if i ≤ q1
{pj : qn ≤ j ≤ i} if qn ≤ i ≤ qn+1

Then put Sn = Qn ∗Pn. For each n we have Sn ∈ sOnH(Pn), and {Sn : n ∈ N}
is an sω-cover of G.

(2) ⇒ (3) : Choose 1 < r1 < r2 < ... < rn < ... such that (∀j ≥ rn)(Tj ≥ n).
Let Pn = sOTnH (Pn), n ∈ N be given. For each sequence (sO

Tn
nbd(H) : n ∈ N),

where each Pn is a semi open nbd of eH, there exist a sequence (Rn : n ∈ N)
such that for each n, Rn ∈ sOTn (Pn) and {Rn : n ∈ N} ∈ sΩHG. Define
R1 =

⋂
j≤r1 Pj, and for each n put Rn+1 =

⋂
rn≤j≤rn+1 Pj. To apply (2)

to (sOnH(Rn) : n ∈ N). For each n, choose Qn ⊂ G with |Qn| ≤ n with
Sn = Qn ∗Gn, n ∈ N, the set {Sn : n ∈ N}∈ sΩHG.

(3) ⇒ (1) : Let (Tn : n ∈ N) be given. Choose 1 ≤ q1 < q2 < ... <
qk < qk+1 < ... such that T1 ≤ q1 and for each k, Tk+1 ≤ (qk+1 − qk). Now,
let Pn = sOH(Pn) be given for each n. Define R1 =

⋂
j≤n Pj and Rk+1 =⋂

qk≤j≤qk+1 Pj. Then put Rn = sO
Tn
H (Rn), n ∈ N. Apply (3) to the sequence

(Rn : n ∈ N) and choose for each n an Sn ∈Rn so that {Sn : n ∈ N}∈ sΩHG.
For each n choose finite set Qn ⊆ H such that |Qn| ≤ Tn, and Sn = Qn ∗Rn.
For each m write Qm = {pqm+1, ...,pqm+1} with repetitions, if necessary. Then
(pk ∗Pk : k < ∞) is the sequence with pk ∗Pk ∈Pk for each k, the sequence of
nj’s witness the weak groupability {pk ∗Pk : k < ∞}. �

Theorem 3.13. Let (H,∗,τ) be an extremely disconnected ITG and G ≤ H.
Then the following statements are equivalent:

(1) S1(sOnbd(H),sO
wgp
HG )

(2) For each n, S1(sOnbd(Hn),sO
wgp
HnGn )

(3) For each n, S1(sOnbd(Hn),sOHnGn )

Proof. (1) ⇒ (2) : Put n > 1 and suppose Hn = H×H× ...×H (n copies).
Let Pp = sOp

p

(Pp,1 ×Pp,2 × ...×Pp,n) for each p and define Qp =
⋂

j≤nUp,j,

a semi open nbd of eH. For select a finite set Qp ⊂ H such that |Qp| ≤ p,
and such that {Rp ∗ Qp : p < ∞} ∈ sΩHG. Since S1(sOnbd(H),sO

wgp
HG ) →

Sfin(sOnbd(H),sO
wgp
HG ) → S1(sΩnbd(H),sΩHG), as we saw in Theorem 3.8.

Then for each m put Gp = Rp × Rp ×... × Rp(n copies), p < ∞. Then put
Sp = Gp∗(Pp,1×Pp,1× ...×Pp,n). For each p we have Sp ∈ sOp

p

(Up,1×Up,2×

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Selection principles: s-Menger and s-Rothberger-bounded groups

...×Up,n) and we have {Sp : p < ∞}∈ sΩHG. By (3) ⇒ (1) of Theorem 3.12,
S1(sOnbd(Hn),sO

wgp
HnGn ) holds.

(2) ⇒ (3) : This is obvious.
(3) ⇒ (1) : Let Pn = sOn(Pn) for each p. Write N =

⋃
k<∞Bk where

for each k, k ≤ min(Bk) and Bk is infinite, and for p 6= n, Bp ∩ Bn = ∅.
For each k : For p ∈ Bk put Qm = sO(Pkm). Then (Qp : p ∈ Bk) is a
sequence from sOnbd(Hk). Applying (3) choose for each p ∈ Bk an qp ∈ Hk
such that {qp ∗ Pkp : p ∈ Bk} is a semi open cover of Gk. For each p in Bk
write qp = (qp(1), ...,qp(k)), and then set φ(qp) = {qp(1), ...,qp(k)}. Note that
for each p ∈ Bk we have |φ(qp)| ≤ k ≤ p, and so φ(qp) ∗ Pp is in sOp(Pp).
Set Qp = φ(qp) ∗ Pp for each p, a member of sOp(Pp) = Qp. Claim that
{Qp : p < ∞} is in sΩHG. For let R ⊂ G be a finite and put k = |R|. Write
R = {r1, ...,rk}. Suppose q = (r1, ...,rk) ∈ Gk. For some p ∈ Bk we have
q ∈ qp ∗ Pkp , and so R ⊂ φ(qp) ∗ Pp = Qp. Now (2) ⇒ (1) of Theorem 3.12
implies that S1(sOnbd(H),sO

wgp
HG ) holds. �

Theorem 3.14. Let (H,∗,τ) be an extremely disconnected ITG and a semi
open set G ≤H. Then the following statements are equivalent:

(1) S1(sOnbd(H),sO
wgp
HG )

(2) S1(sOnbd(G),sO
wgp
G )

Proof. The proof is similar to the proof of Theorem 3.6. �

3.3. s-Hurewicz-bounded groups. In this subsection we have verified some
results on s-Hurewicz-bounded groups.

Theorem 3.15. Let (H,∗,τ) be an extremely disconnected ITG and G ≤ H.
Then the following statements are equivalent:

(1) S1(sΩnbd(H), sO
gp
HG).

(2) S1(sΩnbd(H),sΓHG).

Proof. (1) ⇒ (2) : For each n ∈ N let Pn ∈ sΩnbd(H), and select Pn ∈
SO(eH,H) with Pn = sΩ(Pn). Put Qn =

⋂
j≤n Pj. For each n put Qn =

sΩ(Qn) is in sΩnbd(H). Then apply S1(sΩnbd(H),sO
gp
HG) to (Qn : n ∈ N).

Choose Rn ∈ Qn such that {Rn : n ∈ N} is a groupable semi open cover of
G. Choose a sequence p1 < p2 < p3 < ... < pk < ... such that x belongs to G,
for all but finitely many n, x ∈

⋃
pn≤j≤pn+1 Rj. Select finite set Sn ⊂ H with

Rn = Sn ∗Qn. So Rn is also semi open because of ITG. Now define, for each
k, the finite set Tk by,

Tk =

{ ⋃
i≤m1 Si if k ≤ p1⋃
pn≤i≤pn+1 Si if pn ≤ k ≤ pn+1

For each n put An = Tn ∗Pn an element of sΩ(Pn). Claim that {An : n ∈ N}
is a sγ-cover of G. For consider g is an element of G. Select M ∈ N in such a
way for all n ≥ M we have g ∈

⋃
pn<i≤pn+1 Ri. But for pn < i ≤ pn+1 we have

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M. A. Iqbal and M. Khan

Ri = Si ∗Qi ⊂ Ak = Tk ∗Pk for pn < i ≤ pn+1. Thus for all k > pM we have
g ∈ Ak. It follows that {Ak : k ∈ N} is sγ-cover of G.

(2) ⇒ (1) : This is evident. �

Theorem 3.16. Let (H,∗,τ) be an ITG and a semi open set G ≤ H. Then
the following statements are equivalent:

(1) S1(sΩnbd(H),sO
gp
HG).

(2) S1(sΩnbd(G),sO
gp
G ).

Proof. The proof is similar to the proof of Theorem 3.6. �

Corollary 3.17. If (H,∗,τ) has property S1(sΩnbd(H),sΓH), then for each
G ≤H, S1(sΩnbd(G),sΓG) holds.

4. Conclusions

We have introduced three new types of selection principles in the realm of
irresolute topological groups. We have also proved that these new notions are
well defined, by means of studying their internal characterizations. Kocinac
introduced several types of selection principles available in the literature. For
future work one can see selection principle in the domain of soft sets.

References

[1] A. Arhangelskii and M. Tkachenko, Topological groups and related structures, Atlantis

Studies in Mathematics, Atlantis Press, 2008.

[2] K. H. Azar, Bounded topological groups, arXiv:1003.2876.
[3] L. Babinkostova, Metrizable groups and strict o-boundedness, Mat. Vesnik. 58 (2006),

131–138.

[4] L. Babinkostova, Lj. D. R. Kocinac and M. Scheepers, Combinatorics of open covers
(VIII), Topology Appl. 140 (2004), 15–32.

[5] T. Banakh and S. Ravsky, On subgroups of saturated or totally bounded paratopological

groups, Algebra Discrete Math. 4 (2003), 1–20.
[6] S. G. Crossley and S. K. Hildebrand, Semi-topological properties, Fund. Math. 74 (1972),

233–254.

[7] R. Engelking, General Topology, Heldermann-Verlag, Berlin, 1989.
[8] I. I. Guran, On topological groups close to being Lindelöf, Dokl. Akad. Nauk. 256 (1981),

1305–1307.

[9] C. Hernández, Topological groups close to being σ-compact, Topology Appl. 102 (2000),
101–111.

[10] C. Hernández, D. Robbie and M. Tkachenko, Some properties of o-bounded groups and
strictly o-bounded groups, Appl. Gen. Topol. 1 (2000), 29–43.

[11] W. Hurewicz, Uber die verallgemeinerung des borelschen theorems, Math. Z. 24 (1925),

401–425.
[12] D. S. Janković, On locally irreducible spaces, Ann. Soc. Sci. Bruxelles 2 (1983), 59–72.

[13] M. Khan, A. Siab and Lj. D. R. Kočinac, Irresolute-topological groups, Math. Morav.

19 (2015), 73–80.
[14] D. Kocev, Almost Menger and related spaces, Mat. Vesnik. 61 (2009), 172–180.
[15] Lj. D. R. Kočinac, On Menger, Rothberger and Hurewicz topological groups, Unpub-

lished note, (1998).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 198



Selection principles: s-Menger and s-Rothberger-bounded groups

[16] Lj. D. R. Kočinac, Selected results on selection principles, Proc. Third Seminar Geo.

Topo. (2004), 71–104 .

[17] Lj. D. R. Kočinac, Star selection principles: A survey, Khayyam J. Math. 1 (2015),
82–106.

[18] Lj. D. R. Kočinac, A. Sabah, M. Khan and D. Seba, Semi-Hurewicz spaces, Hacet. J.

Math. Stat. 46 (2017), 53–66.
[19] J. P. Lee, On semi-homeomorphisms, Internat. J. Math. 13 (1990), 129–134.

[20] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.

Monthly. 70 (1963), 36ÔÇô-41.

[21] F. Lin and S. Lin, A note on pseudobounded paratopological groups, Topological Algebra
Appl. 2 (2014), 11–18.

[22] K. Menger, Einige Uberdeckungssatze der Punktmengenlehre, Sitzungsberichte. Abt.

2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie,
Wein). 13 (1924), 421–444.

[23] T. Noiri and B. Ahmad, Semi-contiuous mappings functions, Accad. Nazionale Dei Lin-

cei. 10 (1982).
[24] R. Noreen and M. Khan, Quasi-boundedness of irresolute paratopological groups, Cogent

Math. & Stat. 5 (2018), 1–8.

[25] V. Pipitone and G. Russo, Spazi semi connessi e spazi semiaperti, Rend. Circ. Mat.
Palermo. 2 (1975), 273–287.

[26] A. Sabah, M. Khan and Lj. D. R. Kočinac, Covering properties defined by semi-open

sets, J. Nonlinear Sci. Appl. 9 (2016), 4388–4398.
[27] M. Sakai and M. Scheepers, The combinatorics of open covers, in: K. Hart, J. van Mill

and P. Simon (eds), Recent Progress in General Topology III, Atlantis Press (2014),
751–800.

[28] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69 (1996),

31–62.
[29] M. Scheepers, Selection principles and covering properties in topology, Note Mat. 2

(2003), 3–41.

[30] B. Tsaban, Some new directions in infinite-combinatorial topology, in: J. Bagaria, S.
Todorcevic (eds), Set Theory. Trends in Mathematics, Birkhäuser Basel (2006), 225–255.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 199