Absolutely strongly star-Hurewicz spaces

Abstract A space X is absolutely strongly star-Hurewicz if for each sequence (Un :n ∈ℕ/ of open covers of X and each dense subset D of X, there exists a sequence (Fn :n ∈ℕ/ of finite subsets of D such that for each x ∈X, x ∈St(Fn; Un) for all but finitely many n. In this paper, we investigate the relationships between absolutely strongly star-Hurewicz spaces and related spaces, and also study topological properties of absolutely strongly star-Hurewicz spaces.


Introduction
By a space we mean a topological space. Let us recall that a space X is countably compact if every countable open cover of X has a finite subcover. Fleischman [9] defined a space X to be starcompact if for every open cover U of X , there exists a finite subset F of X such that S t .F; U/ D X , where S t .F; U/ D S fU 2 U W U \ F 6 D ;g. He proved that every countably compact space is starcompact. van Douwen et al. in [6] showed that every T 2 starcompact space is countably compact, but this does not hold for T 1 -spaces (see [15,Example 2.5]). Matveev [14] defined a space X to be absolutely countably compact (=acc) if for each open cover U of X and each dense subset D of X , there exists a finite subset F of D such that S t .F; U/ D X . It is clear that every T 2 absolutely countably compact space is countably compact.
In [6], a starcompact space is called strongly starcompact. Kocinac [11,12] defined a space X to be strongly star-Menger if for each sequence .U n W n 2 N/ of open covers of X , there exists a sequence .F n W n 2 N/ of finite subsets of X such that fS t .F n ; U n / W n 2 Ng is an open cover of X .
Bonanzinga et al. in [2] (see also [3]) defined a space X to be strongly star-Hurewicz if for each sequence .U n W n 2 N/ of open covers of X , there exists a sequence .F n W n 2 N/ of finite subsets of X such that for each x 2 X , x 2 S t .F n ; U n / for all but finitely many n. It is clear that every strongly star-Hurewicz space is strongly star-Menger.
Caserta, Di Maio and Kocinac [5] gave the selective version of the notion of acc spaces and introduced the classes of the following spaces (as special cases of a more general definition).

Definition 1.1 ([5]).
A space X is said to be absolutely strongly star-Menger if for each sequence .U n W n 2 N/ of open covers of X and each dense subset D of X , there exists a sequence .F n W n 2 N/ of finite subsets of D such that fS t .F n ; U n / W n 2 Ng is an open cover of X . . A space X is said to be absolutely strongly star-Hurewicz if for each sequence .U n W n 2 N/ of open covers of X and each dense subset D of X , there exists a sequence .F n W n 2 N/ of finite subsets of D such that for each x 2 X , x 2 S t .F n ; U n / for all but finitely many n. The purpose of this paper is to investigate the relationships between absolutely strongly star-Hurewicz spaces and related spaces, and study topological properties of absolutely strongly star-Hurewicz spaces. Throughout this paper, the extent e.X / of a space X is the smallest cardinal number Ä such that the cardinality of every discrete closed subset of X is not greater than Ä. Let ! denote the first infinite cardinal, ! 1 the first uncountable cardinal, c the cardinality of the set of real numbers. For each ordinals˛,ˇwith˛<ˇ, we write OE˛;ˇ/ D f W˛Ä <ˇg, .˛;ˇ D f W˛< Äˇg, .˛;ˇ/ D f W˛< <ˇg and OE˛;ˇ D f W˛Ä Äˇg. As usual, a cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. Every cardinal is often viewed as a space with the usual order topology. Other terms and symbols that we do not define follow [8].

On absolutely strongly star-Hurewicz spaces
In this section, first we give some examples showing relationships between absolutely strongly star-Hurewicz spaces and related spaces. The results and examples extend and improve some results from [16].
Example 2.1. There exists a Tychonoff absolutely strongly star-Hurewicz space X which is not acc.
Proof. Let X D .OE0; ! OE0; !/ n fh!; !ig be the subspace of the product space OE0; ! OE0; !. Clearly, X is a Tychonoff space. But it is not countably compact, since fh!; ni W n 2 !g is a countable discrete closed subset of X . Hence X is not acc. Now we show that X is absolutely strongly star-Hurewicz. To this end, let fU n W n 2 Ng be a sequence of open covers of X . Let D D OE0; !/ OE0; !/: Then D is a dense subspace of X and every dense subset of X includes D, since every point of D is isolated. Thus it suffices to show that there exists a sequence .F n W n 2 N/ of finite subsets of D such that for each x 2 X , x 2 St.F n ; U n / for all but finitely many n. For each n 2 N, let K n D .OE0; ! OE0; n 1/ [ .OE0; n 1 OE0; !/: Then K n is the union of finitely many compact subsets. For each n 2 N, we can find a finite subset F n of D such that K n Â St.F n ; U n /. Thus the sequence .F n W n 2 N/ witnesses for .U n W n 2 N/ that X is absolutely strongly star-Hurewicz. In fact, S n2N K n D X . For any x 2 X , there exists n 0 2 N such that x 2 K n 0 , thus x 2 S t .F n ; U n / for each n > n 0 , which shows that X is absolutely strongly star-Hurewicz.
We show that X is not absolutely strongly star-Hurewicz. For each˛< ! 1 , let U˛D OE0;˛/ .˛; ! 1 and D D OE0; ! 1 / OE0; ! 1 /: Let us consider the sequence .U n W n 2 N/ of open covers of X and the dense subset D of X. Let .F n W n 2 N/ be any sequence of finite subsets of D. We only show that there exists a point x 2 X such that x … S t .F n ; U n / for all n 2 N. For each n 2 N, let˛n D supf˛W˛2 .F n /g, where W OE0; ! 1 / OE0; ! 1 ! OE0; ! 1 is the projection. Then n < ! 1 , since F n is finite. LetˇD supf˛n W n 2 Ng. Thenˇ< ! 1 . If we pick˛0 >ˇ, then h˛0; ! 1 i … S t .F n ; U n / for all n 2 N, since, for every Uˇ2 U n , if h˛0; ! 1 i 2 Uˇ, thenˇ>˛0; for eachˇ>˛0, Uˇ\ F n D ;, which shows that X is not absolutely strongly star-Hurewicz.
Next we give an example of a Tychonoff absolutely strongly star-Menger space which is not absolutely strongly star-Hurewicz by using the following result from [4]. Recall that a family of sets is almost disjoint (a.d., for short) if the intersection of any two distinct elements of the family is finite. Let A be an a.d. family of infinite subsets of !. Put « .A/ D A [ ! and topologize « .A/ as follows: the points of ! are isolated and a basic neighborhood of a point a 2 A takes the form fag [ .a n F /, where F is a finite set of !. « .A/ is called a « -space (see [8,10]). It is well known that A is a maximal almost disjoint family (m.a.d. family, for short) iff « .A/ is pseudocompact.
We make use of two of the cardinals defined in [7]. Define ! ! as the set of all functions from ! to itself. For all f; g 2 ! !, we say f Ä g if and only if f .n/ Ä g.n/ for all but finitely many n. The unbounding number, denoted by b, is the smallest cardinality of an unbounded subset of . ! !; Ä /. The dominating number, denoted by d, is the smallest cardinality of a cofinal subset of   Remark 2.6. From the proof of Proposition 3 in [4], it is not difficult to see that the above conditions are equivalent to (3) « .A/ is absolutely strongly star-Hurewicz.
Example 2.7. There exists a Tychonoff absolutely strongly star-Menger space X which is not absolutely strongly star-Hurewicz.
Proof. Let X D « .A/ D ! [A be the Isbell-Mrówka space, where A is the almost disjoint family of infinite subsets of ! with jAj D b. Then X is absolutely strongly star-Menger by Lemma 2.3 and Remark 2.4 above. However X is not absolutely strongly star-Hurewicz by Lemma 2.5 and Remark 2.6 above. Thus we complete the proof.
Remark 2.8. Assuming ! 1 < b D c, the space X D « .A/ with jAj D ! 1 is absolutely strongly star-Hurewicz by Lemma 2.5 and Remark 2.6 above. This space shows that there exists a Tychonoff absolutely strongly star-Hurewicz space X such that e.X / D ! 1 , since A is a discrete closed subset of X with jAj D ! 1 . However the author does not know if there exists an example in ZFC showing that there exists a Tychonoff absolutely strongly star-Hurewicz space X such that e.X / c. Quite recently, M. Sakai proved that the answer to this question in negative.
In the following, we study topological properties of absolutely strongly star-Hurewicz spaces. Assuming ! 1 < b D c, the space X D « .A/ with jAj D ! 1 is absolutely strongly star-Hurewicz. This space shows that a closed subspace of a Tychonoff absolutely strongly star-Hurewicz space X need not be absolutely strongly star-Hurewicz, since A is a discrete closed subset of X with jAj D ! 1 . Next we give a stronger example. Example 2.9. There exists a Tychonoff absolutely strongly star-Hurewicz space having a regular-closed G ı -subspace which is not absolutely strongly star-Hurewicz.
Recall the Alexandorff duplicate A.X / of a space X . The underlying set A.X / is X f0; 1g; each point of X f1g is isolated and a basic neighborhood of hx; 0i 2 X f0g is a set of the form .U f0g/ [ ..U f1g/ n fhx; 1ig/; where U is a neighborhood of x in X . It is well known that a T 2 space X is countably compact iff A.X / is acc (see [17,18]). In the following, we give two examples to show that the result can not be generalized to the absolutely strongly star-Hurewicz.
Example 2.10. Assuming ! 1 < b D c, there exists a Tychonoff absolutely strongly star-Hurewicz space X such that A.X/ is not absolutely strongly star-Hurewicz.
Proof. Assuming ! 1 < b D c, let X D ! [ A be the Isbell-Mrówka space with jAj D ! 1 . Then X is absolutely strongly star-Hurewicz by Lemma 2.5 and Remark 2.6. However A.X / is not absolutely strongly star-Hurewicz. In fact, the set A f1g is an open and closed subset of A.X / with jA f1gj D ! 1 , and for each a 2 A, the point ha; 1i is isolated in A.X/. Hence A.X / is not absolutely strongly star-Hurewicz, since every open and closed subset of an absolutely strongly star-Hurewicz space is absolutely strongly star-Hurewicz, and A f1g is not absolutely strongly star-Hurewicz.
Example 2.11. There exists a Tychonoff strongly star-Hurewicz space X such that X is not absolutely strongly star-Hurewicz, but A.X / is absolutely strongly star-Hurewicz.
Proof. Let X D OE0; ! 1 / OE0; ! 1 be the space X of Example 2.2. Then X is not absolutely strongly star-Hurewicz. But X is strongly star-Hurewicz being starcompact. Since X is countably compact, then A.X / is acc (see [17,18]), hence A.X / is absolutely strongly star-Hurewicz. Thus we complete the proof.
Proof. Suppose that e.X / ! 1 . Then there exists a discrete closed subset B of X such that jBj ! 1 . Hence B f1g is an open and closed subset of A.X / and every point of B f1g is an isolated point. Thus A.X/ is not absolutely strongly star-Hurewicz, since every open and closed subset of an absolutely strongly star-Hurewicz space is absolutely strongly star-Hurewicz and B f1g is not absolutely strongly star-Hurewicz. Question 1. Is the space A.X / of an absolutely star-Hurewicz space X with e.X / < ! 1 also absolutely star-Hurewicz?
The following example shows that the continuous image of an absolutely strongly star-Hurewicz space need not be absolutely strongly star-Hurewicz.
Example 2.13. There exists a continuous mapping f W X ! Y such that X is absolutely strongly star-Hurewicz, but Y is not absolutely strongly star-Hurewicz.
Let Y D OE0; ! 1 / OE0; ! 1 be the space X of Example 2.2. Then Y is not absolutely strongly star-Hurewicz. Let f W X ! Y be a mapping defined by f .h˛;ˇi/ D h˛;ˇi for each h˛;ˇi 2 OE0; ! 1 / OE0; ! 1 / and f .˛/ D h˛; ! 1 i for each˛2 OE0; ! 1 /: Then f is a continuous one-to-one mapping, which completes the proof.
Recall from [13] or [14] that a continuous mapping f W X ! Y is varpseudoopen provided i nt Y f .U / 6 D ; for every nonempty open set U of X . In [14], it was proved that a continuous varpseudoopen image of an acc space is acc. Similarly, we may show the following result.
Theorem 2.14. A continuous varpseudoopen image of an absolutely strongly star-Hurewicz space is absolutely strongly star-Hurewicz.
Proof. Suppose that X is an absolutely strongly star-Hurewicz. space and f W X ! Y is a continuous varpseudoopen onto map. Let .U n W n 2 N/ be a sequence of open covers of Y and D a dense subset of Y . For each n 2 N, let V n D ff 1 .U / W U 2 U n g. Then .V n W n 2 N/ is a sequence of open covers of X , and f 1 .D/ a dense subset of X , since f is varpseudoopen. Hence there exists a sequence .E n W n 2 N/ of finite subsets of f 1 .D/ such that for each x 2 X , x 2 S t .E n ; V n / for all but finitely many n. For each n 2 N, let F n D f .E n /. Then .F n W n 2 N/ is a sequence of finite subsets of D such that for each y 2 Y , y 2 S t .F n ; U n / for all but finitely many n, which shows that Y is absolutely strongly star-Hurewicz.
Question 2. Find an inner characterization of those spaces X for which f .X / is absolutely strongly star-Hurewicz for each continuous mapping f .
Since an open map is varpseudoopen, we have the following result by Theorem 2.14.
Theorem 2.15. Let X and Y be two spaces. If X Y is absolutely strongly star-Hurewicz, then X and Y are absolutely strongly star-Hurewicz.
The following remark shows that the converse of Theorem 2.15 need not be true even if the product of an absolutely strongly star-Hurewicz space and a compact space.
Remark 2.16. The product of an absolutely strongly star-Hurewicz space and a compact space need not be absolutely strongly star-Hurewicz. In fact, the space X D OE0; ! 1 / OE0; ! 1 of Example 2.2 is not absolutely strongly star-Hurewicz. The first factor is acc by [14,Theorem 1.8], hence it is absolutely strongly star-Hurewicz, and the second is compact. Matveev showed that the product of a T 2 acc space with a first countable compact space is acc (see [14,Theorem 2.3]). However, the author does not know if the product of an absolutely strongly star-Hurewicz space and a first countable compact space is absolutely strongly star-Hurewicz.
Next we turn to consider preimages. We show that the preimage of an absolutely strongly star-Hurewicz space under a closed 2-to-1 continuous map need not be absolutely strongly star-Hurewicz, Example 2.17. Assuming ! 1 < b D c, there exists a closed 2-to-1 continuous map f W X ! Y such that Y is an absolutely strongly star-Hurewicz space, but X is not absolutely strongly star-Hurewicz.
Proof. Let Y D « .A/ D ! [ A be the space X of Example 2.10. Then Y is absolutely strongly star-Hurewicz. Let X be the space A.Y / of Example 2.10. Then X is not absolutely strongly star-Hurewicz. Let f W X ! Y be the projection. Then f is a closed 2-to-1 continuous map, which completes the proof.
Remark 2.18. The space OE0; ! 1 / OE0; ! 1 in Remark 2.16 also shows that the preimage of an absolutely strongly star-Hurewicz space under an open perfect map need not be absolutely strongly star-Hurewicz.