@
Appl. Gen. Topol. 21, no. 1 (2020), 53-56

doi:10.4995/agt.2020.11976

c© AGT, UPV, 2020

Counterexample to theorems on star versions

of Hurewicz property

Manoj Bhardwaj

Department of Mathematics, University of Delhi, New Delhi-110007, India (manojmnj27@gmail.com)

Communicated by S. Romaguera

Abstract

In this paper, an example contradicting Theorem 4.5 and Theorem
5.3 [1] is provided and these theorems are proved under some extra
hypothesis.

2010 MSC: 54D20; 54B20.

Keywords: Hurewicz spac; star-Hurewicz space; strongly star-Hurewicz
space.

1. Introduction

In covering properties, Hurewicz property is one of the most important pro-
perty. In 1925, Hurewicz [4] (see also [5]) introduced Hurewicz property in
topological spaces and studied it. This property is stronger than Lindelöf and
weaker than σ-compactness. In 2004, the authors M. Bonanzinga, F. Cam-
maroto, Lj.D.R. Kočinac [1] introduced the star version of Hurewicz property.

For the terms and symbols that we do not define follow [2]. The basic
definitions are given.

Let A and B be collections of open covers of a topological space X.
In [6], Kočinac introduced star selection principles in the following way.
The symbol S?1 (A,B) denotes the selection hypothesis that for each sequence

< Un : n ∈ ω > of elements of A there exists a sequence < Un : n ∈ ω > such
that for each n, Un ∈Un and {St(Un,Un) : n ∈ ω}∈B.

The symbol S?fin(A,B) denotes the selection hypothesis that for each se-
quence < Un : n ∈ ω > of elements of A there exists a sequence < Vn : n ∈ ω >

Received 12 June 2019 – Accepted 24 November 2019

http://dx.doi.org/10.4995/agt.2020.11976


M. Bhardwaj

such that for each n, Vn is a finite subset of Un and
⋃

n∈ω{St(V,Un) : V ∈Vn}
is an element of B

The symbol U?fin(A,B) denotes the selection hypothesis that for each se-
quence < Un : n ∈ ω > of elements of A there exists a sequence < Vn : n ∈ ω >
such that for each n, Vn is a finite subset of Un and {St(

⋃
Vn,Un) : n ∈ ω}∈B

or there is some n such that St(
⋃
Vn,Un) = X.

Let K be a family of subsets of X. Then we say that X belongs to the
class SS?K(A,B) if X satisfies the following selection hypothesis that for every
sequence < Un : n ∈ ω > of elements of A there exists a sequence < Kn : n ∈
ω > of elements of K such that {St(Kn,Un) : n ∈ ω}∈B.

When K is the collection of all one-point [resp., finite, compact] subspaces of
X we write SS?1 (A,B) [resp., SS?fin(A,B), SS

?
comp(A,B)] instead of SS?K(A,B).

In this paper A and B will be collections of the following open covers of a
space X:

O : the collection of all open covers of X.
Ω : the collection of ω-covers of X. An open cover U of X is an ω-cover [3]

if X does not belong to U and every finite subset of X is contained in
an element of U.

Γ : the collection of γ-covers of X. An open cover U of X is a γ-cover [3] if
it is infinite and each x ∈ X belongs to all but finitely many elements
of U.

Ogp : the collection of groupable open covers. An open cover U of X is
groupable [7] if it can be expressed as a countable union of finite, pair-
wise disjoint subfamilies Un, such that each x ∈ X belongs to

⋃
Un for

all but finitely many n.

2. Main results

In [1], Bonanzinga, Cammaroto and Kočinac introduced the notion of SH≤n
in topological spaces.

A space X is said to have SH≤n if for each sequence < Un : n ∈ ω > of
open covers of X there is a sequence < Vn : n ∈ ω > such that for each n, Vn
is a finite subset of Un of cardinalilty atmost n and {St(

⋃
Vn,Un) : n ∈ ω} is

a γ-cover of X.

Theorem 2.1 ([1]). Let a space X satisfies SH≤n. Then X satisfies S
?
1 (O,Ogp).

According to definition of SH≤n, there does not exist any topological space
which satisfies SH≤n, take any topological space X and Un = {X} for each n.
Then for any finite subset Vn of Un, St(

⋃
Vn,Un) = X for each n. So {X} is

not a γ-cover of X since it is finite. To avoid this possibility, without loss of
generality if we consider infinite open covers for the Theorem then following
example shows that the above theorem is not correct.

Example 2.2. There is a space which satisfies SH≤n but does not satisfy
S?1 (O,Ogp).

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 54



Counterexample to theorems on star versions of Hurewicz property

Proof. Let N be set of natural numbers with discrete topology on it. Since N is
countable, it satisfies SH≤n. Now consider a sequence < Un = {{n} : n ∈ ω} >
of open covers for each n. Then it does not satisfy S?1 (O,Ogp) since {{n} : n ∈
ω} is not groupable. �

For the existence of SH≤n, we consider only infinite covers such that X
does not belong to each cover. In order to prove above theorem we need more
hypothesis on a space X, that is, we define CDR?sub(A,B).

Definition 2.3 ([8]). Let A and B be families of subsets of the infinite set S.
Then CDRsub(A,B) denotes the statement that for each sequence < An : n ∈
ω > of elements of A there is a sequence < Bn : n ∈ ω > such that for each n,
Bn ⊆ An, for m 6= n, Bm ∩Bn = ∅, and each Bn is a member of B.

Definition 2.4. Let A and B be families of subsets of the infinite set S. Then
CDR?sub(A,B) denotes the statement that for each sequence < An : n ∈ ω >
of elements of A there is a sequence < Bn : n ∈ ω > such that for each n,
Bn ⊆ An, for m 6= n, {St(B,Am) : B ∈ Bm}∩{St(B,An) : B ∈ Bn} = ∅, and
each Bn is a member of B.

Theorem 2.5. Let a space X satisfies SH≤n and CDR
?
sub(O,O). Then X

satisfies S?1 (O,Ogp).

Proof. The proof is similer as given in [1] with necessary modifications. �

In [1], Bonanzinga, Cammaroto and Kočinac consider the hypothesis : for
each sequence < Un : n ∈ ω > of open covers of X there is a sequence < Vn :
n ∈ ω > of finite subsets of X such that for each n, Vn has atmost n elements
and {St(Vn,Un) : n ∈ ω} is a γ-cover of X.

Theorem 2.6 ([1]). Let a space X satisfies the following condition : for each
sequence < Un : n ∈ ω > of open covers of X there is a sequence < Vn : n ∈ ω >
of finite subsets of X such that for each n, Vn has atmost n elements and
{St(Vn,Un) : n ∈ ω} is a γ-cover of X. Then X satisfies SS?1 (O,Ogp).

According to hypothesis, there does not exist any topological space which
satisfies the hypothesis considered in above Theorem, because take any topo-
logical space X and Un = {X} for each n. Then for any finite subset Vn of
Un, St(

⋃
Vn,Un) = X for each n. So {X} is not a γ-cover of X since it is

finite. To avoid this possibility, without loss of generality if we consider infinite
open covers for the Theorem then the following example shows that the above
theorem is not correct.

Example 2.7. There is a space which satisfies following condition : for each
sequence < Un : n ∈ ω > of open covers of X there is a sequence < Vn : n ∈ ω >
of finite subsets of X such that for each n, Vn has atmost n elements and
{St(Vn,Un) : n ∈ ω} is a γ-cover of X but does not satisfy SS?1 (O,Ogp).

Proof. Let N be set of natural numbers with discrete topology on it. Since N
is countable, it satisfies following condition : for each sequence < Un : n ∈ ω >

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 55



M. Bhardwaj

of open covers of X there is a sequence < Vn : n ∈ ω > of finite subsets of X
such that for each n, Vn has at most n elements and {St(Vn,Un) : n ∈ ω} is a
γ-cover of X. Now consider a sequence < Un = {{n} : n ∈ ω} > of open covers
for each n. Then it does not satisfy SS?1 (O,Ogp) since {{n} : n ∈ ω} is not
groupable. �

For the existence of above hypothesis, we consider only infinite covers such
that X does not belong to each cover. In order to prove above Theorem, we
need more hypothesis on a space X, that is, we define CDRF?sub(A,B).

Definition 2.8. Let A and B be families of subsets of the infinite set S. Then
CDRF?sub(A,B) denotes the statement that for each sequence < An : n ∈ ω >
of elements of A there is a sequence < Bn : n ∈ ω > such that for each n,
Bn ⊆ An, for m 6= n and for each finite subset F of S, {St(x,Bm) : x ∈
F}∩{St(x,Bn) : x ∈ F} = ∅, and each Bn is a member of B.

Theorem 2.9. Let a space X satisfies CDRF?sub(O,O) and the following con-
dition : for each sequence < Un : n ∈ ω > of open covers of X there is a
sequence < Vn : n ∈ ω > of finite subsets of X such that for each n, Vn has at-
most n elements and {St(Vn,Un) : n ∈ ω} is a γ-cover of X. Then X satisfies
SS?1 (O,Ogp).

Proof. The proof is similer as given in [1] with necessary modifications. �

Acknowledgements. The author acknowledges the fellowship grant of Uni-
versity Grants Commission, India.

References

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c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 56