sthuragt.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 5, No. 1, 2004

pp. 79- 89

Star-Hurewicz and related properties

M. Bonanzinga, F. Cammaroto and Lj.D.R. Kočinac ∗

Abstract. We continue the investigation of star selection principles
first considered in [9]. We are concentrated onto star versions of the
Hurewicz covering property and star selection principles related to the
classes of open covers which have been recently introduced.

2000 AMS Classification: 54D20.

Keywords: Selection principles, (strongly) star-Menger, (strongly) star-
Rothberger, (strongly) star-Hurewicz, groupability, weak groupability, ω-cover,
γ-cover.

1. Introduction

A number of the results in the literature show that many topological proper-
ties can be described and characterized in terms of star covering properties (see
[3], [13], [2], [12]). The method of stars has been used to study the problem
of metrization of topological spaces, and for definitions of several important
classical topological notions. We use here such a method in investigation of
selection principles for topological spaces.

Let A and B be collections of open covers of a topological space X.
The symbol S1(A, B) denotes the selection hypothesis that for each sequence

(Un : n ∈ N) of elements of A there exists a sequence (Un : n ∈ N) such that
for each n, Un ∈ Un and {Un : n ∈ N} ∈ B [18].

The symbol Sfin(A, B) denotes the selection hypothesis that for each se-
quence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N) such
that for each n ∈ N, Vn is a finite subset of Un and

⋃
n∈N Vn is an element of

B [18].

In [9], Kočinac introduced star selection principles in the following way.

∗The first and the second authors were supported by MURST - PRA 2000. The third
author (corresponding author) was supported by MSRS, grant N0 1233.



80 M. Bonanzinga, F. Cammaroto, Lj.D.R. Kočinac

Definition 1.1. Let A and B be collections of open covers of a space X. Then:
(a) The symbol S∗1(A, B) denotes the selection hypothesis: for each sequence

(Un : n ∈ N) of elements of A there exists a sequence (Un : n ∈ N) such that
for each n, Un ∈ Un and {St(Un, Un) : n ∈ N} is an element of B;

(b) The symbol S∗fin(A, B) denotes the selection hypothesis: for each se-

quence (Un : n ∈ N) of elements of A there is a sequence (Vn : n ∈ N) such that
for each n ∈ N, Vn is a finite subset of Un, and

⋃
n∈N{St(V, Un) : V ∈ Vn} ∈ B;

(c) By U∗fin(A, B) we denote the selection hypothesis: for every sequence

(Un : n ∈ N) of members of A there exists a sequence (Vn : n ∈ N) such that
for every n, Vn is a finite subset of Un and {St(∪Vn, Un) : n ∈ N} ∈ B or there
is some n ∈ N such that St(∪Vn, Un) = X.

Definition 1.2. Let A and B be collections of open covers of a space X and
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: for every sequence

(Un : n ∈ N) of elements of A there exists a sequence (Kn : n ∈ N) of elements
of K such that {St(Kn, Un) : n ∈ N} ∈ 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).

Here, as usual, for a subset A of a space X and a collection P of subsets of
X, St(A, P) denotes the star of A with respect to P, that is the set ∪{P ∈
P : A ∩ P 6= ∅}; for A = {x}, x ∈ X, we write St(x, P) instead of St({x}, P).

In [10] it was explained that selection principles in uniform spaces are actu-
ally a kind of star selection principles.

Let X be a space. If U and V are families of subsets of X we denote by U ∧V
the set {U ∩ V : U ∈ U, V ∈ V}. The symbols [X]<ω and [X]≤n denote the
collection of all finite subsets of X and all subsets of X having ≤ n elements,
respectively.

In this paper all spaces will be Hausdorff. 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 [5] 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 [5] 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 [11] if it can be expressed as a countable union of finite, pairwise
disjoint subfamilies Un, n ∈ N, such that each x ∈ X belongs to ∪Un for all
but finitely many n;

Owgp: the collection of weakly groupable open covers. A cover U of X is a
weakly groupable [1] if it is a countable union of finite, pairwise disjoint sets Un,
n ∈ N, such that for each finite set F ⊂ X we have F ⊂ ∪Un for some n.



Star-Hurewicz and related properties 81

We consider only spaces X whose each ω-cover contains a countable ω-
subcover (or equivalently, for each n ∈ N, every open cover of Xn has a count-
able subcover). Thus all considered covers are assumed to be countable.

Recall that a space X is said to have the Menger property [15], [6], [16],
[8] (resp. the Rothberger property [17], [16], [18] if the selection hypothesis
Sfin(O, O) (resp. S1(O, O)) is true for X.

The following terminology was introduced in [9]. A space X is said to have:

1. the star-Rothberger property SR,
2. the star-Menger property SM,
3. the strongly star-Rothberger property SSR,
4. the strongly star-Menger property SSM,

if it satisfies the selection hypothesis:
1. S∗1(O, O),
2. S∗fin(O, O) (or, equivalently, U

∗
fin(O, O)),

3. SS∗1(O, O),
4. SS∗fin(O, O),

respectively.

In 1925 in [6] (see also [7]), W. Hurewicz introduced the Hurewicz covering
property for a space X in the following way:

H: For each sequence (Un : n ∈ N) of open covers of X there is a sequence
(Vn : n ∈ N) of finite sets such that for each n Vn ⊂ Un, and for each
x ∈ X, for all but finitely many n, x ∈ ∪Vn.

Two star versions of this property are:

SH: A space X satisfies the star-Hurewicz property if for each sequence
(Un : n ∈ N) of open covers of X there is a sequence (Vn : n ∈ N) such
that for each n ∈ N Vn is a finite subset of Un and each x ∈ X belongs
to St(∪Vn, Un) for all but finitely many n.

SSH: A space X satisfies the strongly star-Hurewicz property if for each se-
quence (Un : n ∈ N) of open covers of X there is a sequence (An : n ∈ N)
of finite subsets of X such that each x ∈ X belongs to St(An, Un) for
all but finitely many n (i.e. if X satisfies SS∗fin(O, Γ)).

In this paper we study some properties of these spaces. We also consider
SM and SSM spaces, in particular in connection with new classes of covers that
appeared recently in the literature - groupable and weakly groupable covers.

2. Spaces related to SM spaces

Theorem 2.1. If each finite power of a space X is SM, then X satisfies
U∗fin(O, Ω).

Proof. Let (Un : n ∈ N) be a sequence of open covers of X and let N =
N1 ∪ N2 ∪ · · · be a partition of N into infinitely many infinite subsets. For
each k and each m ∈ Nk let Wm = {U1 × · · · × Uk : U1, · · · , Uk ∈ Um}. Then



82 M. Bonanzinga, F. Cammaroto, Lj.D.R. Kočinac

(Wm : m ∈ Nk) is a sequence of open covers of X
k, and since Xk is a star-

Menger space, one can choose a sequence (Hm : m ∈ Nk) such that for each m,
Hm ∈ [Wm]

<ω and
⋃

m∈Nk
{St(H, Wm) : H ∈ Hm} is an open cover of X

k. For

every m ∈ Nk and every H ∈ Hm we have H = U1(H) × · · · × Uk(H), where
Ui(H) ∈ Um for every i ≤ k. Put Vm = {Ui(H) : i ≤ k, H ∈ Hm}. Then for
each m ∈ Nk Vm is a finite subset of Um. We claim that {St(∪Vn, Un) : n ∈ N}
is an ω-cover of X. Let F = {x1, · · · , xs} be a finite subset of X. Then
x = (x1, · · · , xs) ∈ X

s so that there is an n ∈ Ns such that x ∈St(H, Wn) for
some H ∈ Hn. But H = U1(H) × · · · × Us(H), where U1(H), · · · , Us(H) ∈ Vn.
The point x belongs to some W ∈ Wn of the form V1 × · · · × Vs, Vi ∈ Un for
each i ≤ s, which meets Ui(H)×· · ·×Us(H). This means that for each i ≤ s we
have xi ∈ St(Ui(H), Un) ⊂ St(∪Vn, Un), i.e. F ⊂ St(∪Vn, Un). So, X satisfies
U∗fin(O, Ω). �

Now we shall see that the previous theorem can be given in another form.

Theorem 2.2. For a space X the following are equivalent:

(1) X satisfies U∗fin(O, Ω);

(2) X satisfies U∗fin(O, O
wgp).

Proof. Because each countable ω-cover is weakly groupable, (1) implies (2) is
trivial, so that we have to prove only (2) ⇒ (1). Let (Un : n ∈ N) be a
sequence of open covers of X. Let for each n, Hn :=

∧
i≤n Ui. Apply (2) to the

sequence (Hn; n ∈ N). There is a sequence (Wn : n ∈ N) such that for each n
Wn ∈ [Hn]

<ω and {St(∪Wn, Hn) : n ∈ N} is a weakly groupable cover of X.
There is, therefore, a sequence n1 < n2 < · · · in N such that for each finite set
F in X one has F ⊂ ∪{St(∪Wi, Hi) : nk ≤ i < nk+1} for some k. Consider the
sequence (Vn : n ∈ N) defined in the following way:

Vn =
⋃

i<n1
Wi, for n < n1;

Vn =
⋃

nk≤i<nk+1
Wi, for nk ≤ n < nk+1.

Then for each n Vn is a finite subset of Un satisfying: for each finite set
F ⊂ X there is an n such that F ⊂ St(∪Vn, Un). This means that X satisfies
U∗fin(O, Ω). �

Problem 2.3. Is it true that S∗fin(O, Ω) = S
∗
fin(O, O

wgp)?

3. Spaces related to SSM spaces

Theorem 3.1. If all finite powers of a space X are strongly star-Menger, then
X satisfies SS∗fin(O, Ω).

Proof. Let (Un : n ∈ N) be a sequence of open covers of X and let N =
N1 ∪ N2 ∪ · · · be a partition of N into infinite (pairwise disjoint) sets. For
every k ∈ N and every m ∈ Nk let Wm = U

k
m. Then (Wm : m ∈ Nk) is a

sequence of open covers of Xk. Applying to this sequence the fact that Xk

is SSM we find a sequence (Am : m ∈ Nk) of finite subsets of X
k such that

{St(Am, Wm) : m ∈ Nk} is an open cover of X
k. For each m take a finite set



Star-Hurewicz and related properties 83

Sm ⊂ X such that S
k
m ⊃ Am. We claim that {St(Sn, Un) : n ∈ N} is an ω-cover

of X. Let F = {x1, · · · , xp} be a finite subset of X. Then (x1, · · · , xp) ∈ X
p so

that there is n ∈ Np such that (x1, · · · , xp) ∈ St(An, Wn) ⊂ St(S
p
n, Wn), and

consequently F ⊂ St(Sn, Un). �

The following theorem shows that spaces whose all finite powers are SSM
can be also related to spaces defined in terms of weakly groupable covers.

Theorem 3.2. For a space X the following are equivalent:

(1) X satisfies SS∗fin(O, Ω);

(2) X satisfies SS
∗
fin(O, O

wgp).

Proof. (1) ⇒ (2): It is evident because countable ω-covers are weakly groupable
and SS

∗
fin is monotone in the second variable.

(2) ⇒ (1): Let (Un : n ∈ N) be a sequence of open covers of X and let for
each n ∈ N, Vn =

∧
i≤n Ui. For each n Vn is an open cover of X that refines

Ui for all i ≤ n. Apply (2) to the sequence (Vn : n ∈ N). There is a sequence
(Sn : n ∈ N) of finite subsets of X such that {St(Sn, Vn) : n ∈ N} is a weakly
groupable open cover of X. Therefore, there is a sequence n1 < n2 < · · · <
nk < · · · of natural numbers such that for every finite subset F of X we have
F ⊂

⋃
nk≤i<nk+1

St(Si, Vi) for some k ∈ N. For each n let

An =
⋃

i<n1
Si for n < n1, and

An =
⋃

nk≤i<nk+1
Si for nk ≤ n < nk+1.

Then (An : n ∈ N) is a sequence of finite subsets of X which witnesses for
(Un : n ∈ N) that X satisfies SS

∗
fin(O, Ω). �

4. Star-Hurewicz spaces

It is clear that each Hurewicz space is strongly star-Hurewicz and each
strongly star-Hurewicz space is star-Hurewicz.

The following proposition shows that in the class of paracompact spaces, the
Hurewicz, strongly star-Hurewicz and star-Hurewicz properties are equivalent.

Proposition 4.1. A paracompact Hausdorff space X has the star-Hurewicz
property if and only if it has the Hurewicz property.

Proof. We already noted that if a space X has the Hurewicz property then X
has also the star-Hurewicz property. So let X be a paracompact space having
star-Hurewicz property and let (Un : n ∈ N) be a sequence of open covers of
X. By the Stone characterization of paracompactness [4] for every n ∈ N, Un
has an open star-refinement, say Vn. Since X has the star-Hurewicz property
there exists a sequence (Wn : n ∈ N) such that for each n, Wn ∈ [Vn]

<ω and
each x ∈ X belongs to all but finitely many of the sets St(∪Wn, Vn). For each
W ∈ Wn, let UW be a member of Un such that St(W, Vn) ⊂ UW . For every
n ∈ N, we have that Hn := {UW : W ∈ Wn} ∈ [Un]

<ω. It is easily seen that
the sequence (Hn : n ∈ N) witnesses for (Un : n ∈ N) that X is a Hurewicz
space. �



84 M. Bonanzinga, F. Cammaroto, Lj.D.R. Kočinac

Proposition 4.2. The product X × Y of a star-Hurewicz space X and a com-
pact space Y is star-Hurewicz.

Proof. Let (Wn : n ∈ N) be a sequence of open covers of X × Y . Without
loss of generality, we can assume that for every n ∈ N, Wn = Un × Vn, where
Un is an open cover of X and Vn is an open cover of Y . Fix x ∈ X. Since
{x}×Y is a compact space and for each n ∈ N Wn is an open cover of {x}×Y ,
there exists (Un,x × Vn,x) ∈ [Wn]

<ω such that ∪(Un,x × Vn,x) ⊃ {x} × Y . For
every n ∈ N, the set Un,x = ∩Un,x is open in X, so that the family Gn =
{Un,x : x ∈ X} is an open cover of X. Since X has the star-Hurewicz property,
there exists a sequence (Hn : n ∈ N) such that for every n ∈ N Hn ∈ [Gn]

<ω

and each x ∈ X belongs to all but finitely many sets {St(∪Hn, Gn) : n ∈ N}.
For every n ∈ N, let Hn = {Un,x1, · · · , Un,xk(n)}. We have that for every n,

Kn := (Un,x1 × Vn,x1) ∪ · · · ∪ (Un,xk(n) × Vn,xk(n)) ∈ [Wn]
<ω. The sequence

(Kn : n ∈ N) is such that {St(∪Kn, Wn) : n ∈ N} is a γ-cover of X × Y . So the
space X × Y has the star-Hurewicz property. �

Let us mention that there is a SSH space X and a Lindelöf space Y such
that the product X × Y is not SSH. Take X = [0, ω1) with the usual order
topology and Y to be the one-point Lindelöfication of X.

Theorem 4.3. For a space X the following are equivalent:

(1) X is a star-Hurewicz space;
(2) X satisfies U∗fin(O, O

gp).

Proof. Each countable γ-cover is groupable and thus (1) implies (2). Let us
show (2) ⇒ (1). Let (Un : n ∈ N) be a sequence of open covers of X and
let for each n Gn be the open cover

∧
i≤n Ui. Apply (2) to the sequence of

these Gn’s and for each n ∈ N select a finite set Vn ⊂ Gn such that the set
{St(∪Vn, Gn) : n ∈ N} is a groupable cover of X. Let n1 < n2 < · · · be a
sequence of natural numbers which witnesses the last fact: for each x ∈ X x
belongs to ∪{St(∪Vi, Gi) : nk ≤ i < nk+1} for all but finitely many k. As in
the proof of Theorem 2.2 put

Wn =
⋃

i<n1
Vi, for n < n1;

Wn =
⋃

nk≤i<nk+1
Vi, for nk ≤ n < nk+1.

Then the sequence (Wn : n ∈ N) shows that X satisfies (1), because, evidently,
each x ∈ X belongs to all but finitely many sets St(∪Wn, Un). �

Problem 4.4. Is it true that S∗fin(O, Γ) = S
∗
fin(O, O

gp)?

Consider now what happens if a space X satisfies the following condition
closely related to the star-Hurewicz property:

SH≤n : For each sequence (Un : n ∈ N) of open covers of X there is a
sequence (Vn : n ∈ N) such that for each n Vn ∈ [Un]

≤n and the set
{St(∪Vn, Un) : n ∈ N} is a γ-cover of X.

The answer to this question is given by the next theorem.



Star-Hurewicz and related properties 85

Theorem 4.5. Let a space X satisfies SH≤n. Then X satisfies S
∗
1(O, O

gp).

Proof. Let (Un : n ∈ N) be a sequence of open covers of X. We define new
covers Vn, n ∈ N, of X putting for each n

Vn = ∧{Ui : (n − 1)n/2 < i ≤ n(n + 1)/2}.

Apply SH≤n to the sequence (Vn : n ∈ N). We find a sequence (Wn : n ∈ N)
such that for each n, |Wn| ≤ n, Wn ⊂ Vn and {St(∪Wn, Vn) : n ∈ N} is a γ-
cover of X. Write Wn = {Wi : (n− 1)n/2 < i ≤ n(n+ 1)/2}. For each Wi take
also the set Ui ∈ Ui which is a term in the representation of Wi given above. We
shall prove now that the set {St(Un, Un) : n ∈ N} is an open groupable cover
of X. Consider the sequence n1 < n2 < · · · < nk < · · · of natural numbers
defined by nk = k(k − 1)/2. Then, by construction, for each x ∈ X we have
x ∈

⋃
nk<i≤nk+1

St(Wi, Ui) for all but finitely many k which implies that the

cover {St(Un, Un) : n ∈ N} is groupable. �

5. Strongly star-Hurewicz spaces

Recall that a space X is strongly starcompact [3], [13] if for each open cover
U of X there is a finite set F ⊂ X such that St(F, U) = X. Clearly, every
strongly starcompact space is SSH. But we have a bit more. Call a space X
σ-strongly starcompact if it is a union of countably many strongly starcompact
spaces.

Theorem 5.1. Every σ-strongly starcompact space is SSH.

Proof. Let X =
⋃

n∈N Xn, where each Xn is strongly starcompact, and let
(Un : n ∈ N) be a sequence of open covers of X. One may suppose that
X1 ⊂ X2 ⊂ · · · ⊂ Xn ⊂ · · · , because the union of finitely many strongly
starcompact spaces is also strongly starcompact. For each n let An be a finite
subset of X such that St(An, Un) ⊃ Xn. It follows that each point of X belongs
to all but finitely many sets St(An, Un), i.e. the sequence (An : n ∈ N) shows
that X is a strongly star-Hurewicz space. �

The following theorem gives a characterization of SSH spaces in terms of
groupable covers.

Theorem 5.2. For a space X the following are equivalent:

(1) X has the strongly star-Hurewicz property;
(2) X satisfies the selection principle SS∗fin(O, O

gp).

Proof. (1) ⇒ (2): It follows from the fact that countable γ-covers are groupable
and monotonicity of SS∗fin in the second variable.

(2) ⇒ (1): Let (Un : n ∈ N) be a sequence of open covers of X and let
for each n ∈ N, Vn = ∧i≤nUn. Applying (2) to the sequence (Vn : n ∈ N)
of open covers of X find a sequence (Tn : n ∈ N) of finite subsets of X such
that {St(Tn, Vn) : n ∈ N} is a groupable open cover of X. Let n1 < n2 <
· · · < nk < · · · be a sequence of natural numbers such that for each x ∈ X,



86 M. Bonanzinga, F. Cammaroto, Lj.D.R. Kočinac

x ∈
⋃

nk≤i<nk+1
St(Ti, Vi) for all but finitely many k. For each n ∈ N put

Sn =
⋃

i<n1
Ti for n < n1, and Sn =

⋃
nk≤i<nk+1

Ti for nk ≤ n < nk+1. Each

Sn is a finite subset of X. We prove that the family {St(Sn, Un) : n ∈ N} is a
γ-cover of X. Let x ∈ X. There exists k0 ∈ N such that x ∈

⋃
nk≤i<nk+1

Ti for

all k > k0. Since St(Ti, Vi) ⊂ St(Si, Ui) for all i with nk ≤ i < nk+1, we have
that for each k > k0, x ∈ St(Sk, Uk), i.e. {St(Sn, Un) : n ∈ N} is a γ-cover of
X. �

The previous theorem suggests to consider also the selection principle SS∗1(O, O
gp)

that is naturally related to the SSH property. We have the following result.

Theorem 5.3. Let a space X satisfies the following condition: for each se-
quence (Un : n ∈ N) of open covers of X there is a sequence (An : n ∈ N)
of subsets of X such that for each n |An| ≤ n and {St(An, Un) : n ∈ N} is a
γ-cover of X. Then X satisfies SS∗1(O, O

gp).

Proof. Let (Un : n ∈ N) be a sequence of open covers of X. For each n let
Vn =

∧
(n−1)n/2<i≤n(n+1)/2 Ui. We apply the assumption of the theorem to the

sequence (Vn : n ∈ N) to find a sequence (An : n ∈ N) of subsets of X such that
for each n, |An| ≤ n and {St(An, Vn) : n ∈ N} is a γ-cover of X: for each x ∈ X
there exists n0 such that x ∈ St(An, Vn) for all n > n0. For each n write An as
An = {xi : (n − 1)n/2 < i ≤ n(n + 1)/2}. Then {St(xi, Ui) : i ∈ N} is an open
groupable cover of X. Indeed, consider the sequence n1 < n2 < · · · < nk < · · ·
of natural numbers defined by nk = k(k − 1)/2. Then for each point x ∈ X we
have x ∈

⋃
nk<i≤nk+1

St(xi, Ui) for all but finitely many k. �

To each selection principle mentioned above one can associate a game [18],
[8], [11], [9]. A number of selection principles of the Sfin and S1 form have
been characterized by the corresponding games.

The following game G is naturally associated to the SSH property. We shall
call this game the strongly star-Hurewicz game.

Let X be a space. Two players, ONE and TWO, play a round per each
natural number n. In the n–th round ONE chooses an open cover Un of X and
TWO responds by choosing a finite set An ⊂ X. A play U1, A1; · · · ; Un, An; · · ·
is won by TWO if {St(An, Un) : n ∈ N} is a γ-cover of X; otherwise, ONE wins.

Evidently, if ONE has no winning strategy in the strongly star-Hurewicz
game, then X is an SSH space.

Conjecture 5.4. The strongly star-Hurewicz property of a space X need not
imply ONE does not have a winning strategy in the strongly star-Hurewicz game

played on X.

Notice that similar situations might be expected also for the other star se-
lection principles and corresponding games.



Star-Hurewicz and related properties 87

6. Relativization

In this section we define a relative version of the SSH property and show
that there is a subspace Y of a SSM space X which is relatively SSH in X but
Y with the subspace topology is not SSH.

Let Y be a subspace of a space X. We say that Y is strongly star-Hurewicz
in X (or Y is relatively strongly star-Hurewicz in X) if for each sequence (Un :
n ∈ N) of open covers of X there is a sequence (An : n ∈ N) of finite subsets of
X such that each point y ∈ Y belongs to all but finitely many sets St(An, Un).

Remark 6.1. The following two definitions also give relative versions of the
SSH property. Below Y is a subspace of X.

(R1) For each sequence (Un : n ∈ N) of covers of Y by sets open in X there
is a sequence (An : n ∈ N) of finite subsets of X such that each point y ∈ Y
belongs to all but finitely many sets St(An, Un).

(R2) For each sequence (Un : n ∈ N) of open covers of X (or of Y by sets
open in X) there is a sequence (An : n ∈ N) of finite subsets of Y such that
each point y ∈ Y belongs to all but finitely many sets St(An, Un).

We do not consider these two properties, but they can be also investigated.

Denote by NN the set of all functions from N into N. For f, g ∈ NN we write
f ≺∗ g if and only if there is n0 ∈ N such that f(n) < g(n) for each n ≥ n0. A
family F ⊂ NN is called bounded if there exists a function g in NN such that
f ≺∗ g for each f ∈ F. The symbol b denotes the minimal cardinality of a
unbounded subset of NN.

A family A of infinite subsets of N is called almost disjoint if the intersection
of any two distinct sets in A is finite.

Let A be an almost disjoint family. The symbol Ψ(A) denotes the space N∪A
with the following topology: all points of N are isolated; a basic neighborhood
of a point a in A is of the form {a} ∪ (N \ F), where F is a finite subset of N.

Example 6.2. Let A be an almost disjoint family of cardinality < b. Then A
is relatively SSH in Ψ(A), but A with the subspace topology is not SSH.

Let (Un : n ∈ N) be a sequence of open covers of Ψ(A). One can suppose
that for each n elements of Un are basic sets:

Un = {Un(a) : a ∈ A} ∪ {{n} : n ∈ N \ ∪a∈AUn(a)}.

Also, we can suppose that to each a ∈ A only one neighborhood Un(a) ∈ Un
is assigned. For each a ∈ A define a function f : N → N by fa(n) = min{k ∈
N : k ∈ Un(a)}. By assumption the family {fa : a ∈ A} is bounded, so that
there exists a function f ∈ NN such that fa ≺

∗ f for each a ∈ A. This means
that for each a ∈ A there is na ∈ N with f(n) > fa(n) for every n ≥ na.
Further, consider for each n ∈ N the finite set An := {1, 2, · · · , f(n)} subset
of N. We claim that the sequence (An : n ∈ N) witnesses for (Un : n ∈ N)
that A is relatively SSH in Ψ(A). Indeed, for each a ∈ A the intersection



88 M. Bonanzinga, F. Cammaroto, Lj.D.R. Kočinac

Un(a) ∩ An 6= ∅ (because fa(n) ∈ An ∩ Un(a)) for each n ≥ na, i.e. each point
a ∈ A belongs to all but finitely many sets St(An, Un).

On the other hand, the subspace A of Ψ(A) is the discrete space of cardi-
nality b and thus it can not be SSH.

Let us remark that according to a result from [14] this space Ψ(A) is SSM.

Acknowledgements. The third author thanks INDAM for the support
and the Dipartimento di Matematica of the Università di Messina and F. Cam-
maroto for the hospitality he enjoyed during his visit in November/December
2002.

References

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Received December 2002

Accepted February 2004



Star-Hurewicz and related properties 89

M. Bonanzinga, F. Cammaroto (milena@dipmat.unime.it, camfil@unime.it)
Dipartimento di Matematica, Università di Messina, 98166, Messina, Italia

Lj.D.R. Kočinac (lkocinac@ptt.yu)
Faculty of Sciences and Mathematics, University of Nǐs, 18000, Nǐs, Serbia