aa r X i v : . [ m a t h . GN ] J un STAR VERSIONS OF HUREWICZ SPACES SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
Abstract.
A space X is said to have the set star Hurewicz property if foreach nonempty subset A of X and each sequence ( U n : n ∈ N ) of sets openin X such that for each n ∈ N , A ⊂ ∪U n , there is a sequence ( V n : n ∈ N )such that for each n ∈ N , V n is a finite subset of U n and for each x ∈ A , x ∈ St( ∪V n , U n ) for all but finitely many n . In this paper, we investigatethe relationships among set star Hurewicz, set strongly star Hurewicz andother related covering properties and study the topological properties of thesetopological spaces. Introduction and Preliminaries
In [1], Arhangel’skii defined a cardinal function sL , and spaces X such that sL ( X ) = ω we call s-Lindel¨of: a space X is s-Lindel¨of if for each subset A of X and each cover U of A by sets open in X there is a countable set V ⊂ U such that A ⊂ ∪V . Motivated by this definition, and modifying it, Koˇcinac and Konca [14]considered new types of selective covering properties called set covering properties.Later on, Koˇcinac, Konca and Singh in [15] studied set star covering properties usingthe star operator, and, in particular, defined set star Hurewicz and set strongly starHurewicz properties.In this paper, we investigate the relationship among set star Hurewicz, setstrongly star Hurewicz and other related properties. Further, we study the topo-logical properties of these two classes of spaces.Throughout the paper we use standard topological terminology and notation asin [7]. By “a space” we mean “a topological space”, N denotes the set of naturalnumbers, and an open cover U of a subset A ⊂ X means elements of U are openin X and A ⊂ ∪U = ∪{ U : U ∈ U} . The cardinality of a set A is denoted by | A | . Let ω denote the first infinite cardinal, ω the first uncountable cardinal, c thecardinality of the set of real numbers. For a cardinal κ , κ + denotes the smallestcardinal greater than κ . As usual, a cardinal is an initial ordinal and an ordinalis the set of smaller ordinals. A cardinal is often viewed as a space with the usualorder topology. If A is a subset of a space X and U is a collection of subsets of X ,then the star of A with respect to U is the set St( A, U ) := S { U ∈ U : U ∩ A = ∅} ;St( x, U ) = St( { x } , U ). A set A in a space X is said to be regular open if A = Int( A ).Complements of regular open sets are called regular closed .We first recall the classical notions of spaces which are used in this paper.In 1925, Hurewicz [9, 10] introduced the Hurewicz covering property for a space X in the following way: Mathematics Subject Classification.
Key words and phrases.
Hurewicz, star Hurewicz, strongly star Hurewicz, set star Hurewicz,set-SSH. SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
A space X is said to have the Hurewicz property if each sequence ( U n : n ∈ N )of open covers of X there is a sequence ( V n : n ∈ N ) such that for each n ∈ N , V n is a finite subset of U n and for each x ∈ X , x ∈ ∪V n for all but finitely many n .Koˇcinac [11, 12, 5], introduced the star versions of the Hurewicz covering prop-erty using the star operator in the following way:(1) A space X is said to have the star Hurewicz property (shortly, SH property) ifeach sequence ( U n : n ∈ N ) of open covers of X , there is a sequence ( V n : n ∈ N ) suchthat for each n ∈ N , V n is a finite subset of U n and for each x ∈ X , x ∈ St( ∪V n , U n )for all but finitely many n .(2) A space X is said to have the strongly star Hurewicz property (shortly, SSH property) if for each sequence ( U n : n ∈ N ) of open covers of X , there is a sequence( F n : n ∈ N ) of finite subsets of X such that for each x ∈ X , x ∈ St( F n , U n ) for allbut finitely many n .In what follows we will use Theorem 1.1 below.Recall that a collection A of infinite subsets of ω is said to be almost disjoint ifthe sets A ∩ B are finite for all distinct elements A, B ∈ A . For an almost disjointfamily A , put ψ ( A ) = A ∪ ω and topologize ψ ( A ) as follows: all points in ω areisolated, and for each A ∈ A and each finite set F ⊂ ω , { A } ∪ ( A \ F ) is a basicopen neighborhood of A . The spaces of this type are called Isbell-Mr´owka ψ -spaces[3, 7, 17] or ψ ( A ) spaces. Theorem 1.1. ([3])
Let A be an almost disjoint family of infinite subsets of ω andlet ψ ( A ) = ω ∪ A be the Isbell-Mr´owka space. Then:(1) ψ ( A ) is strongly star Hurewicz if and only if |A| < b ;(2) If |A| = c , then X is not star Hurewicz. Recently, Koˇcinac and Konca [14] defined the set selection properties (and theirweak versions). See also the paper [13, 18] related to these properties.In [15], Koˇcinac, Konca and Singh defined (general versions of) set star selectionproperties, in particular the set star Hurewicz and set strongly star Hurewicz spaces.
Definition 1.2.
A space X is said to have the(1) set star Hurewicz property (shortly, set - SH property) if for each nonemptyset A of X and each sequence ( U n : n ∈ N ) of sets open in X such that A ⊂ ∪U n , n ∈ N , there is a sequence ( V n : n ∈ N ) such that for each n ∈ N , V n is a finite subset of U n and for each x ∈ A , x ∈ St( ∪V n , U n ) for all butfinitely many n .(2) set strongly star Hurewicz property (shortly, set - SSH property) if for eachnonempty subset A of X and each sequence ( U n : n ∈ N ) of sets open in X such that A ⊂ ∪U n , n ∈ N , there is a sequence ( F n : n ∈ N ) of finitesubsets of X such that for each x ∈ A , x ∈ St( F n , U n ) for all but finitelymany n . Definition 1.3. ([6, 16]) A space X is said to be:(1) starcompact (shortly, SC ) if for each open cover U of X , there is a finitesubset V of U such that X = St( ∪V , U ).(2) strongly starcompact (shortly, SSC ) if for each open cover U of X , there isa finite subset F of X such that X = St( F, U ).In a similar way, Koˇcinac, Konca and Singh [15] considered the following spaces. TAR VERSIONS OF HUREWICZ SPACES 3
Definition 1.4.
A space X is said to be:(1) set starcompact (shortly, set - SC ) if for each nonempty subset A of X andeach open cover U of A , there is a finite subset V of U such that A =St( ∪V , U ) ∩ A .(2) set strongly starcompact (shortly, set - SSC ) if for each nonempty subset A of X and each open cover U of A , there is a finite subset F of X such that A = St( F, U ) ∩ A .It is clear, by the definition, that every set strongly starcompact space is setstarcompact. Theorem 1.5. ([15])
Every countably compact space is set strongly starcompact.
Corollary 1.6. ([15])
Every countably compact space is set starcompact. Examples
In this section, we give some examples showing the relationships between set starHurewicz and set strongly star Hurewicz spaces and other related spaces. Some ofthese examples can be found in the literature, and we establish their additionalproperties.From the definitions we have the following diagram. set − SSC → set − SC ↓ ↓ Hurewicz → set − SSH → set − SH ↓ ↓ SSH → SHDiagram 1
However, the converse of the implications may not be true as we show by exam-ples.The following example shows that the implication
Hurewicz ⇒ set - SSH in Dia-gram 1 is not reversible.
Example 2.1.
Every countably compact non-Lindel¨of space is such an example.Such (Tychonoff) spaces are, for example, the ordinal space [0 , ω ), the long line[24, Example 45], the Novak space [24, Example 112].The following example shows that the implications set - SSC ⇒ set - SSH and set - SC ⇒ set - SH in Diagram 1 are not reversible. Example 2.2.
There exists a Tychonoff set strongly star Hurewicz (hence, set starHurewicz) space which is not set starcompact (hence, not set strongly starcompact).Indeed, let X = D ( ω ) be the countable discrete space. Since X is σ -compact, itis Hurewicz and thus set strongly star Hurewicz. But X is not set starcompact.The following example shows that the implication set - SH ⇒ SH in Diagram 1is not reversible. (Mention that the following space X was considered in severalpapers to obtain various counterexamples.) Example 2.3.
There exists a Hausdorff star Hurewicz space which is not set starHurewicz. SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
Proof.
Let X = Y ∪ A ∪ { p } , where A = [0 , c ), B = [0 , ω ), Y = A × B , p / ∈ Y .Topologize X as follows:(i) every point of Y is isolated;(ii) a basic neighborhood of α ∈ A is of the form U α ( n ) = { α } ∪ {h α, m i : n < m } (iii) a basic neighborhood of p takes the form U ( F ) = { p } ∪ S {h α, n i : α ∈ A \ F, n ∈ ω } for a countable subset F of A . In [21, Example 2.7], Song proved that X is starHurewicz. In [15] it was shown that X is not set star Menger, so that it cannot beset star Hurewicz. (cid:3) The following example shows that the implication set - SSH ⇒ set - SH in Diagram1 is not reversible.Recall that a space X is strongly star Lindel¨of (see [6] or [16] under differentname) if for every open cover U of X there exists a countable subset A of X suchthat St( A, U ) = X . Clearly, every strongly star Hurewicz space is strongly starLindel¨of. Example 2.4.
There exists a Tychonoff set star Hurewicz space X which is notset strongly star Hurewicz. Proof.
Let D ( κ ) be the discrete space of uncountable cardinality κ , and let Y = D ( κ ) ∪ {∞} be the one-point compactification of D ( κ ). Let X = ( Y × [0 , κ + )) ∪ ( D ( κ ) × { κ + } )be the subspace of the product space Y × [0 , κ + ].In [4], it was shown that X is not strongly star Lindel¨of. Hence, X is not setstrongly star Hurewicz (since every set strongly star Hurewicz space is strongly starLindel¨of, being star Hurewicz).On the other hand, in [15] it was shown that X is a set starcompact space. Itfollows that X is set star Hurewicz. (cid:3) Remark 2.5. (1) Example 2.4 shows that there exists a Tychonoff set starcompactspace X that is not set strongly starcompact (since every set strongly starcompactspace is set strongly star Hurewicz). This shows that the implication set - SSC ⇒ set - SC in Diagram 1 is not reversible.(2) It is known that there are star Hurewicz spaces which are not strongly starHurewicz (see [22]). Also this shows that the implication SSH ⇒ SH in Diagram 1is not reversible.We do not know the answer to the following problem. Problem 2.6.
Does there exist a (Tychonoff ) strongly star Hurewicz space whichis not set strongly star Hurewicz?
TAR VERSIONS OF HUREWICZ SPACES 5 Results
In some classes of spaces certain properties from Diagram 1 coincide. In [5] thefollowing theorem was proved.
Theorem 3.1. ([5, Proposition 4.1]) If X is a paracompact Hausdorff spaces, then X is star Hurewicz if and only if X is Hurewicz. From Theorem 3.1 and Diagram 1, we have following.
Theorem 3.2. If X is a paracompact Hausdorff space, then the following state-ments are equivalent:(1) X is Hurewicz;(2) X is set strongly star Hurewicz;(3) X is strongly star Hurewicz;(4) X is set star Hurewicz;(5) X is star Hurewicz. A space X is metacompact (resp., meta-Lindel¨of ) if each open cover of X has apoint-finite (resp., point-countable) open refinement. Theorem 3.3.
Every set strongly star Hurewicz hereditarily metacompact space X is a (set) Hurewicz space.Proof. Let A be a subset of X and ( U n : n ∈ N ) be a sequence of covers of A bysets open in X . For every n ∈ N the set S U n is metacompact. Let V n be a point-finite open refinement of U n , n ∈ N . As X is set strongly star Hurewicz, there is asequence ( F n : n ∈ N ) of finite subsets of X such that for each x ∈ A , x ∈ St( F n , V n )for all but finitely many n . Elements of each F n belongs to finitely many members V n, , ..., V n,k ( n ) of V n . Let W n = { V n, , ..., V n,k ( n ) } . Then St( F n , V n ) = S W n , sothat we have for each x ∈ A , x ∈ S W n for all but finitely many n . For each W ∈ W n take an element U W of U n such that W ⊂ U W . Then, for each n , H n = { U W : W ∈ W n } is a finite subset of U n and for each x ∈ A , x ∈ S H n forall but finitely many n . It is easy to prove that this fact actually gives that X is aHurewicz space. (cid:3) Theorem 3.4.
The following conditions are equivalent:(1) ψ ( A ) is a strongly star Hurewicz space;(2) ψ ( A ) is a set strongly star Hurewicz space;(3) |A| < b .Proof. (1) ⇔ (3) follows from Theorem 1.1, and (2) ⇒ (1) follows by definitions. Weonly have to prove (3) ⇒ (2). Let B ⊂ ψ ( A ) be any nonempty set and ( U n : n ∈ N )be a sequence of open covers of B . For each n ∈ N and for each a ∈ B ∩ A , pick anelement U n,a of U n that contains a . For each a ∈ B ∩ A define a function g a ∈ N N by g a ( n ) = min { k ∈ N : k ∈ U n,a } .Since |A| < b , the set of functions { g a : a ∈ A} is bounded by some function;so there is an f ∗ ∈ N N such that g a ≤ ∗ f ∗ for every a ∈ B ∩ A . Put A n =[0 , max { f ∗ ( n ) , n } ]. Then for each y ∈ B , y ∈ St( A n , U n ) for all but finitely many n . (cid:3) SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
Since every set strongly star Hurewicz space is strongly star Hurewicz, we havethe following corollary from [19, Theorem 2.11] and [19, Corollary 2.12].
Corollary 3.5. If X is a set strongly star Hurewicz space, then the followingstatements are equivalent:(1) X is a meta-Lindel¨of space.(2) X is a para-Lindel¨of space.(3) X is a Lindel¨of space. Theorem 3.6.
If each nonempty set A ⊂ X is dense in X and X is star Hurewicz(resp., strongly star Hurewicz) space, then X is set star Hurewicz (resp., set stronglystar Hurewicz).Proof. Because the proofs of two cases are quite similar, we prove only the set starHurewicz case.Let A be any nonempty subset of X and ( U n : n ∈ N ) be a sequence of opencovers of A . Since A is dense in X , ( U n : n ∈ N ) is a sequence of open covers of X . Since X is star Hurewicz, there is a sequence ( V n : n ∈ N ) such that for each n ∈ N , V n is a finite subset of U n and for each x ∈ X , x ∈ St( ∪V n , U n ) for all butfinitely many n . Thus for each x ∈ A , x ∈ St( ∪V n , U n ) for all but finitely many n ,which shows that X is set star Hurewicz. (cid:3) We now explore preservation of set star Hurewicz and set strongly star Hurewiczspaces under basic topological constructions.Observe that set star Hurewicz and set strongly star Hurewicz are not hereditaryproperties. The space X in Example 2.4, shows that a closed subset of a Tychonoffset star Hurewicz space X need not be set star Hurewicz. Indeed, the set D ( κ ) ×{ κ + } is a discrete closed subset of the space X in Example 2.4 of uncountablecardinality κ , so that it cannot be set star Hurewicz.We saw that the ordinal space X = [0 , ω ) is set strongly star Hurewicz. However,the subspace Y = { α + 1 : α is a limit ordinal } of X is not set strongly starHurewicz.Assuming ω < b = c (which is consistent with the ZFC system of axioms ofset theory), let ψ ( A ) = ω ∪ A be the Isbell-Mr´owla space generated by an almostdisjoint family of infinite subsets of ω with |A| = ω . Then by Theorem 3.4, ψ ( A )is a Tychonoff set strongly star Hurewicz space having a closed discrete subspace A which is not set strongly star Hurewicz.Let us mention that a regular-closed subset of a Tychonoff set star Hurewiczspace X which is not set star Hurewicz. It is the space X in [15, Example 2.3].However, we have the following result about preservation of set star Hurewiczand set strongly star Hurewicz spaces. Theorem 3.7.
A clopen subspace of a set star Hurewicz (resp., sert steongly starHureqicz) space is also set star Hurewicz (resp., set strongly star Hurewicz).Proof.
We prove only the set strongly star Hurewicz case because the proof for setstar Hurewicz case is quite similar.Let X be a set strongly star Hurewicz space and Y ⊂ X be a clopen subspace.Let A be any subset of Y and ( U n : n ∈ N ) be any sequence of open sets in ( Y, τ Y )such that for each n ∈ N , Cl Y ( A ) ⊂ ∪U n . Since Y is open, then ( U n : n ∈ N ) isa sequence of open sets in X , and since Y is closed, Cl Y ( A ) = Cl X ( A ). Applyingthe fact that X is set strongly star Hurewicz, we find a sequence ( F n : n ∈ N ) of TAR VERSIONS OF HUREWICZ SPACES 7 finite subsets of X such that for each x ∈ A , x ∈ St( F n , U n ) for all but finitelymany n . Set K n = F n ∩ Y , n ∈ N . Then the sequence ( K n : n ∈ N ) witnesses for( U n : n ∈ N ) that Y is set strongly star Hurewicz. (cid:3) We now consider (non)preservation of the set star Hurewicz and set strongly starHurewicz properties under some sorts of mappings.
Theorem 3.8.
A continuous image of a set star Hurewicz space is set star Hurewicz.Proof.
Let X be a set star Hurewicz space and f : X → Y be a continuous mappingfrom X onto Y . Let B be any nonempty subset of Y and ( V n : n ∈ N ) be asequence of open covers of B . Let A = f ← ( B ). Since f is continuous, for each n ∈ N , U n := { f ← ( V ) : V ∈ V n } is the collection of open sets in X with A = f ← ( B ) ⊂ f ← ( B ) ⊂ f ← ( ∪V n ) = ∪U n .As X is set star Hurewicz, there exists a sequence ( U ′ n : n ∈ N ) such that for each n ∈ N , U ′ n is a finite subset of U n and for each x ∈ A , x ∈ St( ∪U ′ n , U n ) for all butfinitely many n . Let V ′ n = { V : f ← ( V ) ∈ U ′ n } . Then for each n ∈ N , V ′ n is a finitesubset of V n . Let y ∈ B . Then there exists x ∈ A such that f ( x ) = y . Thus y = f ( x ) ∈ f (St( ∪U ′ n , U n )) ⊂ St( ∪ f ( { f ← ( V ) : V ∈ V ′ n } ) , V n ) = St( ∪V ′ n , V n )for all but finitely many n . Thus Y is a set star Hurewicz space. (cid:3) We can prove the following theorem similarly to the proof of Theorem 3.8.
Theorem 3.9.
A continuous image of a set strongly star Hurewicz space is setstrongly star Hurewicz.
In the sequel we need one known topological construction.The Alexandorff duplicate AD ( X ) of a space X is the set X × { , } with thefollowing topology. The basic neighborhood of a point h x, i ∈ X × { } is of theform ( U × { } ) S ( U × { } \ {h x, i} ), where U is a neighborhood of x in X ; allpoints h x, i ∈ X × { } are isolated points.Many topological properties of a space X are preserved by passing to the Alexan-droff duplicate of X . Such properties are, for instance, compactness, countablecompactness, Lindel¨ofness, and paracompactness.However, we have the following facts about the set star Hurewicz and set stronglystar Hurewicz properties. Lemma 3.10.
There exists a Tychonoff set star Hurewicz space X such that AD ( X ) is not set star Hurewicz.Proof. Let X be the space in Example 2.4. Then X is set star Hurewicz. Let A = {hh d α , κ + i , i : α < κ } , d α ∈ D ( κ ). Then A is an open and closed subset of AD ( X ) with | A | = κ and each point hh d α , κ + i , i is isolated. Therefore, AD ( X ) isnot set star Hurewicz by Theorem 3.7. (cid:3) Lemma 3.11.
Assuming ω < b = c , there exists a Tychonoff set strongly starHurewicz space X such that AD ( X ) is not set strongly star Hurewicz. SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
Proof.
Let X = ω ∪ A be the Isbell-Mr´owka space generated by an almost disjointfamily of infinite subsets of ω with |A| = ω . Then, by Theorem 3.4, X is a setstrongly star Hurewicz space. Let A = {h a, i : a ∈ A} . Then A is an open andclosed subset of AD ( X ) with | A | = ω and each point h a, i is isolated. Hence A is not set strongly star Hurewicz, and thus by Theorem 3.7, AD ( X ) cannot be setstrongly star Hurewicz. (cid:3) Remark 3.12.
We do not know if there is a space X such that AD ( X ) is set starHurewicz (resp., set strongly star Hurewicz), but X is not set star Hurewicz (resp.,set strongly star Hurewicz).Lemma 3.10 actually shows that the preimage of a set star Hurewicz space undera closed 2-to-1 continuous map need not be set star Hurewicz. Indeed, let X bethe space in Lemma 3.10 and let f : AD ( X ) → X be a natural projection (definedby f ( h x, i ) = f ( h x, i ) = x for each x ∈ X ). Then f is a closed 2-to-1 continuousmapping witnessing the above statement.Lemma 3.11 shows that the preimage of a set strongly star Hurewicz space undera closed 2-to-1 continuous map need not be set strongly star Hurewicz. Take thespace X in Lemma 3.11, the Alexandroff duplicate AD ( X ) and f : AD ( X ) → X tobe the projection. Then we have the statement.Example 3.18 below shows that the preimage of a set strongly star Hurewiczspace under a perfect open mapping need not be set strongly star Hurewicz.Now we give a positive result on preimages of set strongly star Hurewicz spaces.For this we need a new concept defined as follows. Call a space X nearly set stronglystar Hurewicz if for each A ⊂ X and each sequence ( U n : n ∈ N ) of open covers of X there is a sequence ( F n : n ∈ N ) of finite subsets of X such that for each x ∈ A , x ∈ St( F n , U n ) for all but finitely many n . Theorem 3.13.
Let f : X → Y be an open and closed, finite-to-one continuousmapping from a space X onto a set strongly star Hurewicz space Y . Then X isnearly set strongly star Hurewicz.Proof. Let A ⊂ X be any nonempty set and ( U n : n ∈ N ) be a sequence of opencovers of X . Then B = f ( A ) is a nonempty subset of Y . Let y ∈ B . Then f ← ( y ) isfinite subset of X , and thus for each n ∈ N , there is a finite subset U n y of U n suchthat f ← ( y ) ⊂ S U n y and U ∩ f ← ( y ) = ∅ for each U ∈ U n y . Since f is closed, thereexists an open neighborhood V n y of y in Y such that f ← ( V n y ) ⊂ ∪{ U : U ∈ U n y } .Since f is open, we can assume that V n y ⊂ ∩{ f ( U ) : U ∈ U n y } .For each n ∈ N , V n = { V n y : y ∈ B } is an open cover of B . Since Y is set stronglystar Hurewicz, there exist a sequence ( F n : n ∈ N ) of finite subsets of Y such thatfor each y ∈ B , y ∈ St( F n , V n ) for all but finitely many n .Since f is finite-to-one, the sequence ( f ← ( F n ) : n ∈ N ) is the sequence of finitesubsets of X . Now we have to show that for each x ∈ A , x ∈ St( f ← ( F n ) , U n ) for all but finitely many n .Let x ∈ A . Then there exist n ∈ N and y ∈ B such that y = f ( x ) ∈ V n,y and V n,y ∩ F n = ∅ for all n ≥ n . Since TAR VERSIONS OF HUREWICZ SPACES 9 x ∈ f ← ( V n,y ) ⊂ S { U : U ∈ U n y } , we can choose U ∈ U n y with x ∈ U . Then V n y ⊂ f ( U ). Thus U ∩ f ← ( F n ) == ∅ for all n ≥ n . Hence x ∈ St( f ← ( F n ) , U n ) for all n ≥ n .Thus X is nearly set strongly star Hurewicz. (cid:3) Following the above definition of nearly set strongly star Hurewicz spaces wewill call a space X nearly set star Hurewicz if for each nonempty A ⊂ X and eachsequence ( U n : n ∈ N ) of open covers of X there is a sequence ( V n : n ∈ N ) suchthat for each n ∈ N , V n is a finite subset of U n and for each x ∈ A , x ∈ St( V n , U n )for all but finitely many n .Similarly to the proof of Theorem 3.13, with necessary small modifications, wecan prove the following. Theorem 3.14. If f : X → Y is an open perfect mapping and Y is a set starHurewicz space, then X is nearly set star Hurewicz. From Theorem 3.14 we have the following corollary.
Corollary 3.15. If X is a set star Hurewicz space and Y is a compact space, then X × Y is nearly set star Hurewicz. Remark 3.16.
The product of two set star Hurewicz spaces need not be set starHurewicz. In fact, there exist two countably compact spaces X and Y such that X × Y is not set star Hurewicz (even not set star Menger; see [15]). Moreover,there exist a countably compact (hence, set star Hurewicz) space X and a Lindel¨ofspace Y such that X × Y is not set star Hurewicz (see [15]).The following theorem is a version of Corollary 3.15. Call the product X × Y rectangular set star Hurewicz if for each set A × B ⊂ X × Y and each sequence( U n : n ∈ N ) of covers of A × B by sets open in X × Y there are finite sets V n ⊂ U n , n ∈ N , such that for each z ∈ A × B , z ∈ St( ∪V n , U n ) for all but finitely many n . Theorem 3.17. If X is a set star Hurewicz space and Y is a compact space, then X × Y is rectangular set star Hurewicz.Proof. Let A = B × C be any nonempty rectangular subset of X × Y and ( U n : n ∈ N ) be a sequence open sets in X × Y such that A = B × C ⊂ S U n , n ∈ N .For each x ∈ B , { x } × C is a compact subset of X × Y . Therefore, for each n ∈ N , there is a finite subset { U xn, × V xn, , ..., U xn,m ( x ) × V xn,m ( x ) } of U n such that { x } × C ⊂ S m ( x ) i =1 ( U xn,i × V xn,i ). For each n ∈ N , define W xn = T m ( x ) i =1 U xn,i . Each W xn is an open subset of X containing x and { x } × C ⊂ S { W xn × V xn,i : 1 ≤ i ≤ m ( x ) } ⊂ S { U xn,i × V xn,i : 1 ≤ i ≤ m ( x ) } .Then for each n ∈ N , W n = { W xn : x ∈ B } is an open cover of B . Since X is setstar Hurewicz, for each n ∈ N , there is finite set W ′ n = { W x j : 1 ≤ j ≤ r n } of W n such that for each b ∈ B , b ∈ St( ∪W ′ n , W n ) for all but finitely many n . For each n ∈ N , let U ′ n = { U xn,i × V xn,i : 1 ≤ i ≤ n ( x j ) , ≤ j ≤ r n } . SUMIT SINGH AND LJUBIˇSA D.R. KOˇCINAC
Then U ′ n is a finite subset of U n . Hence for each a ∈ A , a ∈ (St( ∪W ′ n , W n ) ∩ B ) × C ⊂ St( ∪U ′ n , U n ) for all but finitely many n . Thus X × Y is rectangular set starHurewicz. (cid:3) The product X × Y is called rectangular set strongly star Hurewicz if for eachset A × B ⊂ X × Y and each sequence ( U n : n ∈ N ) of covers of A × B by sets openin X × Y there are finite sets F n ⊂ X × Y , n ∈ N , such that for each z ∈ A × B , z ∈ St( F n , U n ) for all but finitely many n .The following example shows that the result similar to Theorem 3.17 is not truein the case of set strongly star Hurewicz space X . Example 3.18.
Assuming ω < b = c , there exists a set strongly star Hurewiczspace X and a compact space Y such that X × Y is not rectangular set stronglystar Hurewicz. Proof.
Let X = ω ∪ A be the space of Lemma 3.11 with |A| = ω . Then X is setstrongly star Hurewicz (Theorem 3.4). Let D ( ω ) = { d α : α < ω } be the discretespace of cardinality ω and let Y = D ( ω ) ∪ {∞} be the one-point compactificationof D ( ω ). We show that X × Y is not set strongly star Hurewicz. Since |A| = ω ,we can enumerate A as { a α : α < ω } . Let B = A × D ( ω ) be a subset of X × Y .For each n ∈ N , let U n = { X × { d α } : α < ω } ∪ { ω × Y } .Then ( U n : n ∈ N ) is a sequence of open sets in X × Y such that for each n ∈ N , B ⊂ ∪U n . It suffices to show that there exists a point x ∈ B such that x / ∈ St( K n , U n ) for all n ∈ N ,for any sequence ( K n : n ∈ N ) of finite subsets of X × Y . Let ( K n : n ∈ N ) be sucha sequence. For each n ∈ N , since K n is finite, there exists α n < ω such that K n ∩ ( X × { d α } ) = ∅ for each α > α n .Let β = sup { α n : n ∈ N } . Then β < ω and( S n ∈ N K n ) ∩ ( X × { d α } ) = ∅ for each α > β .Pick α > β ; then h a α , d α i / ∈ St( K n , U n ), since X × { d α } is the only element of U n containing the point h a α , d α i for each n ∈ N . Thus h a α , d α i / ∈ St( K n , U n ) for each n ∈ N .This shows that X × Y is not rectangular set strongly star Hurewicz. (cid:3) Now a natural question arises: under which conditions the product of a setstrongly star Hurewicz space and a compact space is rectangular set strongly starHurewicz. The following theorem gives a partial answer to this question. Its proofcan be obtained by a simple modification in the proof of Theorem 3.17.
Theorem 3.19. If X is a set strongly star Hurewicz space and Y is a compactseparable space, then X × Y is rectangular set strongly star Hurewicz. TAR VERSIONS OF HUREWICZ SPACES 11
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