Further observations on bornological covering properties and selection principles
aa r X i v : . [ m a t h . GN ] M a y FURTHER OBSERVATIONS ON BORNOLOGICAL COVERINGPROPERTIES AND SELECTION PRINCIPLES
DEBRAJ CHANDRA † , PRATULANANDA DAS ∗ AND SUBHANKAR DAS ∗ Abstract.
This article is a continuation of the study of bornological opencovers and related selection principles in metric spaces done in (Chandra etal. 2020 [10]) using the idea of strong uniform convergence (Beer and Levi,2009 [6]) on bornology. Here we explore further ramifications, presentingcharacterizations of various selection principles related to certain classes ofbornological covers using the Ramseyan partition relations, interactive resultsbetween the cardinalities of bornological bases and certain selection principlesinvolving bornological covers, producing new observations on the B s -Hurewiczproperty introduced in [10] and several results on the B s -Gerlits-Nagy propertyof X which is introduced here following the seminal work of [13]. In addition,in the finite power X n with the product bornology B n , the ( B n ) s -Hurewiczproperty as well as the ( B n ) s -Gerlits-Nagy property of X n are characterizedin terms of properties of ( C ( X ) , τ s B ) like countable fan tightness, countablestrong fan tightness along with the Reznichenko’s property. Key words and phrases:
Bornology, selection principles, open B s -cover, γ B s -cover, B s -Hurewicz property, B s -Gerlits-Nagy property, Ramseyan partition rela-tion, topology of strong uniform convergence, function space C ( X ).1. Introduction
This article is continuation of our work in [10] and like that paper, here alsowe follow the notations and terminologies of [2, 11, 15, 23]. The primary structurewhere we would work is a bornology. A bornology B on a metric space ( X, d ) is afamily of subsets of X that is closed under taking finite unions, is hereditary andforms a cover of X (see [15]). A base B for a bornology B is a subfamily of B thatis cofinal in B with respect to inclusion i.e. for B ∈ B there is a B ∈ B such that B ⊆ B . A base is called closed (compact) if all of its members are closed (compact)which would be most useful in our endevors. We list a few natural bornologies on X as follows: (1) The family F of all finite subsets of X , the smallest bornologyon X ; (2) The family of all non empty subsets of X , the largest bornology on X ;(3) The family of all non empty d -bounded subsets of X ; (4) The family K of nonempty subsets of X with compact closure (more examples can be seen from [10]).In the bornological investigations, the notion of continuity which turned outto be the most useful is the notion of strong uniform continuity on a bornology(introduced in [6] by Beer and Levi). A mapping f : X → Y where ( Y, ρ ) isanother metric space, is strongly uniformly continuous on a subset B of X if foreach ε > δ > d ( x , x ) < δ and { x , x } ∩ B = ∅ imply Mathematics Subject Classification.
Primary: 54D20; Secondary: 54C35, 54A25 .The third author is thankful to University Grants Commission (UGC), New Delhi-110002,India for granting UGC-NET Junior Research Fellowship (1183/(CSIR-UGC NET DEC.2017))during the tenure of which this work was done. ρ ( f ( x ) , f ( x )) < ε . For a bornology B on X , f is called strongly uniformlycontinuous on B if f is strongly uniformly continuous on B for each B ∈ B . Theyhad also introduced a new topology on Y X the set of all functions from X into Y ,called the topology of strong uniform convergence and studied various propertiesin function spaces. This study has been further continued in [8] .In [26] (see also [16]), M. Scheepers began a systematic study of selection princi-ples in topology and their relations to game theory. Those interested in the long andillustrious history of selection principles and its recent developments can consult thepapers [21,27,30] where many more references can be found. Using the topology ofstrong uniform convergence on a bornology, the study of open covers and relatedselection principles in function spaces had been initiated in [9]. In [10], we car-ried out further advancement in this direction where the main focus was to obtainSchepeers’ like diagrams and study the notion of strong- B -Hurewicz property (or B s -Hurewicz property).This paper intends to complete the line of investigations started in [10] and is or-ganised as follows. In section 2, we present some basic observations on bornologicalcovers and related selection principles under continuous functions. In section 3, wefirst investigate the behaviour of certain selection principles involving bornologicalcovers under the cardinality of a base B and then obtain their characterizationsin terms of Ramseyan partition relations. Section 4 is the most important part ofthis article where we first focus more on the B s -Hurewicz property. Consideringthe product bornology B n on X n , the ( B n ) s -Hurewicz property of X n is shownto be equivalent with the B s -Hurewicz property of X and moreover it is charac-terized Ramsey-theoretically as well as game-theoretically. Later in this section,we introduce B s -Gerlits-Nagy property of X (following the seminal work of [13])and proceed to establish equivalence with ( B n ) s -Gerlits-Nagy property of X n andfurther use Ramseyan partition relations to characterize it. Finally in Section 5,we devote our attention to the function space C ( X ) equipped with the topologyof strong uniform convergence τ s B on B , where our primary objective is to presentcharacterizations of ( B n ) s -Hurwicz property as well as ( B n ) s -Gerlits-Nagy prop-erty of X n in terms of properties of ( C ( X ) , τ s B ) like countable fan tightness, count-able strong fan tightness along with the Reznichneko’s property. In addition, wealso present some characterizations of α i properties using selection principles.2. Preliminaries
We follow the notations and terminologies of [2, 11, 15, 23]. Throughout thepaper (
X, d ) stands for an infinite metric space and N stands for the set of positiveintegers. We first write down two classical selection principles formulated in generalform in [16, 26]. For two nonempty classes of sets A and B of an infinite set S , wedefine S ( A , B ): For each sequence { A n : n ∈ N } of elements of A , there is a sequence { b n : n ∈ N } such that b n ∈ A n for each n and { b n : n ∈ N } ∈ B . S fin ( A , B ): For each sequence { A n : n ∈ N } of elements of A , there is a sequence { B n : n ∈ N } of finite (possibly empty) sets such that B n ⊆ A n for each n and S n ∈ N B n ∈ B .There are infinitely long games corresponding to these selection principles. G ( A , B ) denotes the game for two players, ONE and TWO, who play a round foreach positive integer n . In the n -th round ONE chooses a set A n from A and TWO URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 3 responds by choosing an element b n ∈ A n . TWO wins the play { A , b , . . . , A n , b n , . . . } if { b n : n ∈ N } ∈ B . Otherwise ONE wins. G fin ( A , B ) denotes the game where in the n -th round ONE chooses a set A n from A and TWO responds by choosing a finite (possibly empty) set B n ⊆ A n . TWOwins the play { A , B , . . . , A n , B n , . . . } if S n ∈ N B n ∈ B . Otherwise ONE wins.We will also consider U fin ( A , B ): For each sequence { A n : n ∈ N } of elements of A , there is a sequence { B n : n ∈ N } of finite (possibly empty) sets such that B n ⊆ A n for each n andeither {∪ B n : n ∈ N } ∈ B or for some n , ∪ B n = X (from [16, 26]). CDR sub ( A , B ): For each sequence { A n : n ∈ N } of elements of A there is asequence { B n : n ∈ N } such that for each n , B n ⊆ A n , for m = n , B m ∩ B n = ∅ and each B n is a member of B [26].The following selection principles called α i properties, are defined in [22]. Thesymbol α i ( A , B ) for i = 1 , , , { A n : n ∈ N } ofelements of A , there is a B ∈ B such that α ( A , B ): for each n ∈ N , the set A n \ B is finite. α ( A , B ): for each n ∈ N , the set A n ∩ B is infinite. α ( A , B ): for infinitely many n ∈ N , the set A n ∩ B is infinite. α ( A , B ): for infinitely many n ∈ N , the set A n ∩ B is non empty.Now we recall definitions of some symbols related to Ramseyan partition relation.Let U be an element in A . A function f : [ U ] → { , . . . , k } is said to be acoloring [26] if for each U ∈ U and every V ∈ A with
V ⊆ U , there is a i ∈ { , . . . , k } such that { V ∈ V : f ( { U, V } ) = i } belongs to A .We say that X satisfies the partition relation A → ⌈B⌉ k for k ∈ N if for every U ∈ A and any coloring f : [ U ] → { , . . . , k } there are a i ∈ { , . . . , k } , a set V ∈ B with
V ⊆ U and a finite to one function φ : V → N such that for every V, W ∈ V with φ ( V ) = φ ( W ), f ( { V, W } ) = i [3, 26]. In this case V is said to be eventuallyhomogeneous for f . This symbol is known as BaumgartnerTaylor partition symbol.We say that X satisfies the partition relation A → ( B ) nk for n, k ∈ N if for every U ∈ A and any coloring f : [ U ] n → { , . . . , k } there are a i ∈ { , . . . , k } and a set V ∈ B with
V ⊆ U such that for each V ∈ [ V ] n , f ( V ) = i [24]. Also in this case V is said to be homogeneous for f . This symbol is known as ordinary partition symbol.The following cardinal numbers (see [32] for more details) will be used in thesequel. The eventually dominating order ≤ ∗ on the Baire space N N is defined asfollows. For f, g ∈ N N , we say that f ≤ ∗ g if f ( n ) ≤ g ( n ) for all but finitely many n . Let A be a subset of N N . The set A is bounded if there is a function g ∈ N N such that f ≤ ∗ g for all f ∈ A . The symbol b denotes the minimal cardinalityof an unbounded subset of ( N N , ≤ ∗ ). A subset A of N N is dominating if for eachfunction g ∈ N N there exists a function f ∈ A such that g ≤ ∗ f . The symbol d denotes the minimal cardinality of a dominating subset of ( N N , ≤ ∗ ). Let A be afamily of infinite subsets of N . P ( A ) denotes that there is a subset P of N suchthat for each A ∈ A , P \ A is finite. The symbol p denotes the smallest cardinalnumber k for which the following statement is false: For each family A if any finitesubfamily of A has infinite intersection and |A| ≤ k then P ( A ) holds. The symbol cov ( M ) denotes the smallest cardinal number k such that a family of k first cate-gory subsets of the real line covers the real line. For any family A of subsets of N N D. CHANDRA, P. DAS AND S. DAS with cardinality less than cov ( M ) implies that there is a g ∈ N N such that for every f ∈ A the set { n ∈ N : f ( n ) = g ( n ) } is infinite. The symbol add ( M ) denotes thesmallest cardinal number k such that there is a family of k first category sets of realnumbers whose union is no longer first category. The following relations betweenthe cardinal numbers mentioned above are well known. add ( M ) ≤ cov ( M ) ≤ d , p ≤ b ≤ d , p ≤ cov ( M ) and also add ( M ) = min { b , cov ( M ) } .For x ∈ X , we denote Ω x = { A ⊆ X : x ∈ A \ A } [20]. X is said to have theReznichenko property at x ∈ X if for each countable set A in Ω x there is a partition { A n : n ∈ N } of A into pairwise disjoint finite subsets of A such that for each neigh-bourhood W of x , W ∩ A n = ∅ for all but finitely many n [18, 19]. The collectionof all such countable sets is denoted by Ω gpx . X is said to be Fr´echetUrysohn if foreach subset A of X and each x ∈ A there is a sequence in A converging to x . X isstrictly Fr´echet Urysohn (in short SFU) if S (Ω x , Σ x ) holds for each x ∈ X . X issaid to have countable tightness if for every x ∈ X and A ∈ Ω x there is a countablesubset B of A such that B ∈ Ω x [2]. Also X is said to have countable fan tightness(countable strong fan tightness) at x if X satisfies S fin (Ω x , Ω x ) ( S (Ω x , Ω x )) [2, 25].The symbol Σ x denotes the collection of all sequences that converges to x ∈ X [7].Next we recall some classes of bornological covers of X . Let B be a bornologyon the metric space ( X, d ) with closed base. For B ∈ B and δ >
0, let B δ = S x ∈ B S ( x, δ ), where S ( x, δ ) = { y ∈ X : d ( x, y ) < δ } . It can be easily checked that B δ ⊆ B δ for every B ∈ B and δ >
0. A cover U is said to be a strong- B -cover(in short, B s -cover) [8] if X
6∈ U and for each B ∈ B there exist U ∈ U and δ > B δ ⊆ U . If the members of U are open then U is called an open B s -cover. The collection of all open B s -covers is denoted by O B s . An open cover U = { U n : n ∈ N } is said to be a γ B s -cover [8] (see also [9]) of X , if it is infiniteand for every B ∈ B there exist a n ∈ N and a sequence { δ n : n ≥ n } of positivereal numbers satisfying B δ n ⊆ U n for all n ≥ n . The collection of all γ B s -covers isdenoted by Γ B s . An open cover U of X is said to be B s -groupable [10] if it can beexpressed as a union of countably many finite pairwise disjoint sets U n such thatfor each B ∈ B there exist a n ∈ N and a sequence { δ n : n ≥ n } of positivereal numbers with B δ n ⊆ U for some U ∈ U n for all n ≥ n . X is said to be B s -Lindel¨of [9] if each B s -cover contains a countable B s -subcover.Let { ( X n , d n ) : n ∈ N } be a family of metric spaces and let B n be a bornologyon X n for each n ∈ N . Then the product bornology [15] on Π n ∈ N X n has a baseconsisting of sets of the form B = Π n ∈ N B n where B n ∈ B n for all n ∈ N .For two metric spaces X and Y , Y X ( C ( X, Y )) stands for the set of all functions(continuous functions) from X to Y . The commonly used topologies on C ( X, Y )are the compact-open topology τ k , and the topology of pointwise convergence τ p .The corresponding spaces are, in general, respectively denoted by ( C ( X, Y ) , τ k )(resp. C k ( X ) when Y = R ), and ( C ( X, Y ) , τ p ) (resp. C p ( X ) when Y = R ).Let B be a bornology with a closed base on X . Then the topology of stronguniform convergence τ s B is determined by a uniformity on Y X with a base consistingof all sets of the form[ B, ε ] s = { ( f, g ) : ∃ δ > x ∈ B δ , d ( f ( x ) , g ( x )) < ε } , URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 5 for B ∈ B , ε > τ s B coincides with the topology ofpointwise convergence τ p if B = F .Throughout we use the convention that if B is a bornology on X , then X B .We end this section with some more basic observations about open B s -covers.First note that if f : X → Y is any map and if B is a bornology on X , thenthe collection f ( B ) = { f ( B ) : B ∈ B } is a bornology on f ( X ). Moreover, if f is surjective then f ( B ) is a bornology on Y . Note that if B is a bornology on X with a compact base B and f : X → Y is a continuous function then f ( B ) is abornology on f ( X ) with compact base f ( B ). Lemma 2.1.
Let B be a bornology on X with a compact base B and ( Y, ρ ) bea another metric space. Let f : X → Y be a continuous function on X . If U bean open f ( B ) s -cover ( γ f ( B ) s -cover) of f ( X ) then { f − ( U ) : U ∈ U} is an open B s -cover ( γ B s -cover) of X .Proof. Let B ∈ B . Since U is an open f ( B ) s -cover of X so for f ( B ) ∈ f ( B ) thereis a ε > f ( B ) ε ⊆ U for some U ∈ U . As f is continuous function on B and B is compact, f is strongly uniformly continuous on B [6] i.e. for ε > δ > f ( B δ ) ⊆ f ( B ) ε . Therefore f ( B δ ) ⊆ U i.e. B δ ⊆ f − ( U ). So { f − ( U ) : U ∈ U} is an open B s -cover of X . (cid:3) Proposition 2.1.
Let B be a bornology on X with a compact base B and ( Y, ρ ) be a another metric space. Let f : X → Y be a continuous function on X . Let Π ∈ { S , S fin , U fin } , P , Q ∈ {O , Γ } .If X satisfies Π( P B s , Q B s ) then f ( X ) satisfies Π( P f ( B ) s , Q f ( B ) s ) .Proof. We only show that if X satisfies S ( O B s , Γ B s ) then f ( X ) satisfies S ( O f ( B ) s , Γ f ( B ) s ).Before proceeding with the proof note that if B is a compact and U is an open subsetof X with B ⊆ U then there is a δ > B δ ⊆ U .Let {U n : n ∈ N } be a sequence of open f ( B ) s -covers of f ( X ). By Lemma 2.1, U ′ n = { f − ( U ) : U ∈ U} is an open B s -cover of X for each n . Apply S ( O B s , Γ B s )to the sequence {U ′ n : n ∈ N } to choose a f − ( U n ) ∈ U ′ n for each n such that { f − ( U n ) : n ∈ N } is a γ B s -cover of X . We now show that the sequence { U n : n ∈ N } is a γ f ( B ) s -cover of f ( X ). Let B ′ ∈ f ( B ) and say B ′ = f ( B ) where B ∈ B .Choose a n ∈ N and a sequence { δ n : n ≥ n } of positive real numbers such that B δ n ⊆ f − ( U n ) for n ≥ n i.e. f ( B ) ⊆ U n for n ≥ n . Since f ( B ) is compact, thereis a ε n > f ( B ) ε n ⊆ U n for each n ≥ n . This shows that { U n : n ∈ N } is a γ f ( B ) s -cover of f ( X ) and hence f ( X ) satisfies S ( O f ( B ) s , Γ f ( B ) s ). (cid:3) Proposition 2.2.
Let B be a bornology on X with a compact base B . If X satisfies S (Γ B s , Γ B s ) then every continuous image of X into N N is bounded.Proof. Let ρ be the Baire metric on N N and ϕ : X → N N be continuous. ByProposition 2.1, ϕ ( X ) satisfies S (Γ ϕ ( B ) s , Γ ϕ ( B ) s ). For n, k ∈ N , let U nk = { f ∈ N N : f ( n ) ≤ k } . Consider U n = { U nk : k ∈ N } for each n . Let B ∈ ϕ ( B ). Usingthe compactness of B , we can easily find a k and a sequence { δ k : k ≥ k } ofpositive real numbers such that such that B δ k ⊆ U nk for all k ≥ k . Therefore U n isa γ ϕ ( B ) s -cover of ϕ ( X ). Now apply S (Γ ϕ ( B ) s , Γ ϕ ( B ) s ) to {U n : n ∈ N } to choosea U nk n ∈ U n for each n such that { U nk n : n ∈ N } is an open γ ϕ ( B ) s -cover of ϕ ( X ).Define a function h : N → N by h ( n ) = k n for n ∈ N . Let f ∈ ϕ ( X ). Clearly there D. CHANDRA, P. DAS AND S. DAS is a n ∈ N such that f ∈ U nk n for all n ≥ n i.e. f ( n ) ≤ h ( n ) for all n ≥ n i.e. f ≤ ∗ h . Hence ϕ ( X ) is bounded in N N . (cid:3) Proposition 2.3.
Let B be a bornology on X with a compact base B . If X satisfies S fin ( O B s , O B s ) then for every continuous function ϕ : X → N N , ϕ ( X ) isnot dominating.Proof. By Proposition 2.1, ϕ ( X ) satisfies S fin ( O ϕ ( B ) s , O ϕ ( B ) s ). Consider U n = { U nk : k ∈ N } where U nk = { f ∈ N N : f ( n ) ≤ k } for n, k ∈ N which is a γ ϕ ( B ) s -coverof ϕ ( X ). Apply S fin ( O ϕ ( B ) s , O ϕ ( B ) s ) to {U n : n ∈ N } to choose a finite set V n ⊆ U n for each n such that ∪ n ∈ N V n is an open ϕ ( B ) s -cover of ϕ ( X ). Define a function h : N → N by h ( n ) = max { k ∈ N : U nk ∈ V n } for n ∈ N . Now we show that forany f ∈ ϕ ( X ), f ( n ) ≤ h ( n ) for infinitely many n ∈ N . Let f ∈ ϕ ( X ). Choose a B ∈ ϕ ( B ) such that f ∈ B . Since ∪ n ∈ N V n ∈ O ϕ ( B ) s , in view of [10, Proposition3.1], there are δ n > B δ n ⊆ U nk for some U nk ∈ V n for infinitely many n ∈ N i.e. f ∈ U nk for some U nk ∈ V n for infinitely many n ∈ N i.e. f ( n ) ≤ h ( n ) forinfinitely many n ∈ N . Hence ϕ ( X ) is not dominating. (cid:3) Results related to cardinality and Ramsey Theory
Results concerning cardinality.
Recall that X is called a γ B s -set [10] if X satisfies S ( O B s , Γ B s ). An equivalent condition is that every open B s -cover of X contains a countable set which is a γ B s -cover of X (see [9]). Theorem 3.1.
Let B be a bornology on X with a closed base B and X be B s -Lindel¨of. If | B | < p then X is a γ B s -set.Proof. Let U be an open B s -cover of X . Enumerate U bijectively as { U n : n ∈ N } .By [10, Proposition 3.1], for B ∈ B there are δ n > U n ∈ U such that B δ n ⊆ U n for infinitely many n . Define A B = { n ∈ N : B δ n ⊆ U n } for B ∈ B . Clearly each A B is an infinite subset of N and now consider the family A = { A B : B ∈ B } . Let { A B , . . . , A B k } be any finite subfamily of A . Since B ∪ · · · ∪ B k ∈ B , there are δ n > U n ∈ U such that ( B ∪ · · · ∪ B k ) δ n ⊆ U n for infinitely many n . Thismeans that A B ∩ . . . ∩ A B k must be infinite and consequently we can conclude thatany finite subfamily of A has infinite intersection. Now in view of our assumptionthat | B | < p , we can choose an infinite subset P of N such that for each B ∈ B , P \ A B is finite. Enumerating P as { n k : k ∈ N } , set V = { U n k ∈ U : k ∈ N } . Weshow that V is a γ B s -cover. Let B ∈ B . Using the fact that P \ A B is finite andthe definition of A B , there is a k ∈ N and a sequence { δ n k : k ≥ k } of positivereal numbers such that B δ nk ⊆ U n k for all k ≥ k . This shows that V ⊆ U is a γ B s -cover of X and hence X is a γ B s -set. (cid:3) Theorem 3.2.
Let B be a bornology on X with a closed base B and X be B s -Lindel¨of. If | B | < cov ( M ) then X satisfies S ( O B s , O B s ) .Proof. Let {U n : n ∈ N } be a sequence of open B s -covers of X . Enumerate each U n bijectively as { U nm : m ∈ N } . For B ∈ B , choose a δ > U nm ∈ U n such that B δ ⊆ U nm . Define a function f B : N → N by f B ( n ) = min { m ∈ N : B δ ⊆ U nm forsome δ > } for each n ∈ N . Consider the set { f B : B ∈ B } . Since | B | < cov ( M ),there is a f : N → N such that for each B ∈ B , { n ∈ N : f B ( n ) = f ( n ) } isinfinite [12, Theorem 5] (for our consideration the fact that this particular set is non-empty is the most important fact). We show that { U nf ( n ) : n ∈ N } is the required URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 7 open B s -cover. Let B ∈ B . There is a B ∈ B with B ⊆ B . By definition of f B , B δ ⊆ U nf B ( n ) for some δ >
0. Since the set { n : f B ( n ) = f ( n ) } 6 = ∅ , choosingan appropriate k ∈ N so that f B ( k ) = f ( k ) we observe that B δ ⊆ U kf ( k ) for some δ >
0. So, as claimed, { U nf ( n ) : n ∈ N } is an open B s -cover of X showing that X satisfies S ( O B s , O B s ). (cid:3) Theorem 3.3.
Let B be a bornology on X with a closed base B and X be B s -Lindel¨of. If | B | < d then X satisfies S fin ( O B s , O B s ) .Proof. Let {U n : n ∈ N } be a sequence of open B s -covers of X . Enumerate each U n bijectively as U n = { U nm : m ∈ N } for each n . For B ∈ B , there are δ > U nm ∈ U n such that B δ ⊆ U nm . Define a function f B : N → N by f B ( n ) =min { m ∈ N : B δ ⊆ U nm for some δ > } . Consider the set { f B : B ∈ B } . Since | B | < d , the family { f B : B ∈ B } is not dominating. Therefore there is a function g : N → N such that for any B ∈ B , f B ( n ) < g ( n ) for infinitely many n . Define V n = { U nm : m ≤ g ( n ) } for each n . Clearly V n is a finite subset of U n for each n and ∪ n ∈ N V n is an open B s -cover of X . Hence X satisfies S fin ( O B s , O B s ). (cid:3) Theorem 3.4.
Let B be a bornology on X with a closed base B . If | B | < b then X satisfies S (Γ B s , Γ B s ) .Proof. Let {U n : n ∈ N } be a sequence of γ B s -covers of X , where U n = { U nm : m ∈ N } for each n . For each B ∈ B , there exist a m ∈ N and a sequence { δ m : m ≥ m } of positive real numbers such that B δ m ⊆ U nm for m ≥ m . Define afunction f B : N → N by f B ( n ) = min { k ∈ N : for all m ≥ k, B δ m ⊆ U nm } . Considerthe set { f B : B ∈ B } . Since | B | < b , there is a g : N → N such that f B ≤ ∗ g forall B ∈ B . We now show that { U ng ( n ) : n ∈ N } is a γ B s -cover of X . Let B ∈ B and choose a B ∈ B such that B ⊆ B . As f B ≤ ∗ g , there is a n ∈ N suchthat f B ( n ) ≤ g ( n ) for all n ≥ n . From the definition of f B , one can constructa sequence { ε n : n ≥ n } of positive real numbers such that B ε n ⊆ U ng ( n ) for all n ≥ n and so B ε n ⊆ B ε n ⊆ U ng ( n ) for all n ≥ n . Therefore { U ng ( n ) : n ∈ N } is a γ B s -cover of X which shows that X satisfies S (Γ B s , Γ B s ). (cid:3) Ramsey theoretic results.
We now formulate S and S fin -type selectionprinciples using certain partition relations involving bornological covers. The fol-lowing observation about open B s -covers will be useful in this context. Lemma 3.1.
Let B be a bornology on X with closed base. If an open B s -cover U can be expressed as a union of finite number of subfamilies of U , then at least oneof them must be an open B s -cover of X .Proof. Let U be an open B s -cover and U = ∪ ki =1 U i . If on the contrary, no U i is anopen B s -cover of X , then for each U i there is a B i ∈ B such that for any δ > U ∈ U i , B δi * U . Take B = B ∪ · · · ∪ B k ∈ B . It is clear that for any δ > B δ * U for any U ∈ U . This contradicts that U is an open B s -cover of X .Hence at least one of U i ’s must be an open B s -cover of X . (cid:3) The method of proof of the following theorem is adapted from [26, Theorem 10]and [16, Theorem 6.2] with suitable modifications.
Theorem 3.5.
Let B be a bornology on X with closed base. The following state-ments are equivalent. D. CHANDRA, P. DAS AND S. DAS (1) X satisfies S fin ( O B s , O B s ) . (2) X satisfies O B s → ⌈O B s ⌉ , provided X is B s -Lindel¨of.Proof. (1) ⇒ (2). Let U = { U n : n ∈ N } be an open B s -cover of X and f : [ U ] →{ , } be a coloring. Clearly U can be expressed as T ∪ T , where T = { V ∈ U : f ( { U , V } ) = 1 } and T = { V ∈ U : f ( { U , V } ) = 2 } . By Lemma 3.1, at least oneof T and T must be an open B s -cover of X . Let i ∈ { , } be such that U = T i is an open B s -cover. Inductively, choose sequences {U n : n ∈ N } and { i n : n ∈ N } such that U n = { V ∈ U n − : f ( U n , V ) = i n } is an open B s -cover for n >
1. Nowapply S fin ( O B s , O B s ) to {U n : n ∈ N } to obtain a sequence {V n : n ∈ N } where V n is a finite subset of U n for each n such that ∪ n ∈ N V n ∈ O B s . Further we canassume that V n ’s are pairwise disjoint and there is a i ∈ { , } such that for each U m ∈ ∪ n ∈ N V n , i m = i . Choose k ∈ N large enough so that i ≤ k whenever U i ∈ V . Further choose k > k so that i ≤ k whenever U i ∈ V and so on. Againchoose a sequence l < l < · · · such that ∪ j>l V j ⊆ U k and ∪ j>l m V j ⊆ U k lm for m >
1. Take Z n = ∪ l n ≤ j
Let B be a bornology on X with closed base. The following state-ments are equivalent. URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 9 (1) X satisfies S ( O B s , O B s ) . (2) X satisfies O B s → ( O B s ) , provided X is B s -Lindel¨of. (3) X satisfies O B s → ( O B s ) nk for each n, k ∈ N , provided X is B s -Lindel¨of. Theorem 3.7.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) X satisfies S ( O B s , Γ B s ) . (2) ONE does not have a winning strategy in G ( O B s , Γ B s ) on X . (3) X satisfies O B s → (Γ B s ) nk for each n, k ∈ N , provided X is B s -Lindel¨of.Proof. The equivalence (1) ⇔ (2) is already proved in [10, Theorem 3.7]. We provethe implications (2) ⇒ (3) and (3) ⇒ (1).(2) ⇒ (3). For this proof we follow the line of argument of [26, Theorem 30]. Let U be an open B s -cover of X and let f : [ U ] → { , . . . , k } be a coloring. Enumerate U bijectively as { U ( n ) : n ∈ N } . We define a strategy σ for ONE in G ( O B s , Γ B s )as follows.Define σ ( ∅ ) = U , the first move of ONE. TWO responds by selecting U ( n ) ∈ U .Now U \{ U ( n ) } can be expressed as ∪ ki =1 T i , where T i = { V ∈ U : f ( { U ( n ) , V } ) = i } for i = 1 , . . . , k . By Lemma 3.1, at least one of T i ’s is an open B s -cover of X and assume it to be T i ( n . Clearly T i ( n = { V ∈ σ ( ∅ ) : f ( { U ( n ) , V } ) = i ( n ) } .Enumerate T i ( n bijectively as { U ( n ,m ) : m ∈ N } . Now define σ ( U ( n ) ) = { U ( n ,m ) : m ∈ N } . TWO responds by selecting U ( n ,n ) . Continuing this way, we obtain a i ( n ,...,n r ) ∈ { , . . . , k } and an open B s -cover { V ∈ τ ( U ( n ) , . . . , U ( n ,...,n r − ) ) : f ( { U ( n ,...,n r ) , V } ) = i ( n ,...,n r ) } which is enumerated bijectively as { U ( n ,...,n r ,m ) : m ∈ N } . Then define σ ( U ( n ) , . . . , U ( n ,...,n r ) ) = { U ( n ,...,n r ,m ) : m ∈ N } . TWOresponds by selecting U ( n ,...,n r +1 ) and so on.This defines a strategy σ for ONE in G ( O B s , Γ B s ). By (2), σ is not a winningstrategy for ONE. Consider a play σ ( ∅ ) , U ( n ) , . . . , σ ( U ( n ) , . . . , U ( n ,...,n r ) ) , U ( n ,...,n r +1 ) , . . . which is lost by ONE. Therefore U ( n ) , . . . , U ( n ,...,n r +1 ) , . . . should form a γ B s -cover of X . Since for all r ∈ N , i ( n ,...,n r ) ∈ { , . . . , k } , thereis a i ∈ { , . . . , k } such that for infinitely many r ∈ N , i ( n ,...,n r ) = i . Let V = { U ( n ,...,n r ) : i ( n ,...,n r ) = i } . As every infinite subset of a γ B s -cover is a γ B s -cover, V is a γ B s -cover of X . Finally note that, from the construction of the game, for any r, s ∈ N with r < s we must have f ( { U ( n ,...,n r ) , U ( n ,...,n s ) } ) = i ( n ,...,n r ) . Thereforefor any V, W ∈ V with V = W , f ( { V, W } ) = i . Hence (3) holds.(3) ⇒ (1). Let {U n : n ∈ N } be a sequence of open B s -covers of X . Let U n = { U np : p ∈ N } . Consider the collection V n = { U p ∩ U p ∩ · · · ∩ U np n : U ip i ∈U i , < p < p < · · · < p n } for each n ∈ N . Then V n ’s are open B s -coversof X . For convenience, we write V n = { V nm : m ∈ N } for n ∈ N . Now define V = { V m ∩ V mk : m, k ∈ N } . Clearly V is an open B s -cover of X . Define a coloring f : [ V ] → { , } by f ( { V m ∩ V m k , V m ∩ V m k } ) = ( m = m m = m By (3), there are a
W ∈ Γ B s with W ⊆ V , a function ϕ : W → N and a i ∈ { , } such that for U, V ∈ W , f ( { U, V } ) = i . Clearly i = 2. As i = 1 would imply that every element of W refines V m ,contradicting that W is a γ B s -cover of X .Enumerate W as { V m j ∩ V m j k j : j ∈ N } where m < m < · · · . Consider { V m j k j : j ∈ N } which is a γ B s -cover of X as W is a γ B s -cover of X . The elementsof V n are of the form U p ∩ U p ∩ · · · ∩ U np n where U ip i ∈ U i . For each n ∈ [1 , m ],we choose the n -th coordinate U np n in the representation of V m k and for each j > n ∈ ( m j − , m j ], choose the n th coordinate U np n in the representation of V m j k j .Clearly { U np n : n ∈ N } is a γ B s -cover of X and hence X satisfies S ( O B s , Γ B s ). (cid:3) The B s -Hurewicz property and the B s -Gerlits-Nagy property Some observations on B s -Hurewicz property. In [10], the notion ofstrong B -Hurewicz property (or in short, B s -Hurewicz property) was introducedand several of its basic properties were established. In this section first we againlook back at this very important property and present some more new observations.Recall that X is said to have the B s -Hurewicz property if for each sequence {U n : n ∈ N } of open B s -covers of X , there is a sequence {V n : n ∈ N } where V n isa finite subset of U n for each n ∈ N , such that for every B ∈ B there exist a n ∈ N and a sequence { δ n : n ≥ n } of positive real numbers satisfying B δ n ⊆ U for some U ∈ V n for all n ≥ n . Proposition 4.1.
Let B be a bornology on X with a compact base B and ( Y, ρ ) be a another metric space. Let f : X → Y be a continuous function on X . If X has the B s -Hurewicz property then f ( X ) has the f ( B ) s -Hurewicz property.Proof. Let {U n : n ∈ N } be a sequence of open f ( B ) s -covers of f ( X ). By Lemma2.1, for each n ∈ N , U ′ n = { f − ( U ) : U ∈ U n } is an open B s -cover of X . In view ofthe B s -Hurewicz property of X , we can find a sequence {V ′ n : n ∈ N } of finite setswith V ′ n ⊆ U ′ n for each n , such that for each B ∈ B there exist a n and a sequence { δ n : n ≥ n } of positive reals satisfying B δ n ⊆ f − ( U ) for some f − ( U ) ∈ V ′ n for all n ≥ n . For each n choose V n = { U ∈ U n : f − ( U ) ∈ V ′ n } . We show that {V n : n ∈ N } witnesses the f ( B ) s -Hurewicz property for f ( X ). Let B ′ ∈ f ( B ) andsay B ′ = f ( B ) where B ∈ B . We can find a n ∈ N and a sequence { δ n : n ≥ n } of positive reals satisfying B δ n ⊆ f − ( U ) for some f − ( U ) ∈ V ′ n for all n ≥ n .Subsequently f ( B ) ⊆ U for some U ∈ V n for all n ≥ n . Since f ( B ) is compact, foreach n ≥ n there is a ε n > f ( B ) ε n ⊆ U . This shows that {V n : n ∈ N } witnesses the f ( B ) s -Hurewicz property for f ( X ). (cid:3) Proposition 4.2.
Let B be a bornology on X with a compact base B . If X hasthe B s -Hurewicz property then every continuous image of X into N N is bounded.Proof. Let ρ be the Baire metric on N N and ϕ : X → N N be continuous. ByProposition 4.1, ϕ ( X ) has the ϕ ( B ) s -Hurewicz property. Consider U n = { U nk : k ∈ N } where U nk = { f ∈ N N : f ( n ) ≤ k } for n, k ∈ N which is an open ϕ ( B ) s -cover of ϕ ( X ).Apply ϕ ( B ) s -Hurewicz property to {U n : n ∈ N } to obtain {V n : n ∈ N } whereeach V n is a finite subset of U n such that for each B ∈ ϕ ( B ) there exist a n anda sequence { δ n : n ≥ n } of positive real numbers satisfying B δ n ⊆ U nk for some U nk ∈ V n and for all n ≥ n . Define a function h : N → N by h ( n ) = max { k : U nk ∈V n } . We now show that for any f ∈ ϕ ( X ), f ≤ ∗ h holds. For f ∈ ϕ ( X ) choose a B ∈ ϕ ( B ) such that f ∈ B . Now choose a n and a sequence { δ n : n ≥ n } of URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 11 positive real numbers satisfying B δ n ⊆ U nk for some U nk ∈ V n and for all n ≥ n i.e. f ∈ U nk for some U nk ∈ V n and for all n ≥ n i.e. f ( n ) ≤ h ( n ) for all n ≥ n . Thus f ≤ ∗ h holds and hence ϕ ( X ) is bounded. (cid:3) The following result provides an important insight into the spaces with B s -Hurewicz property in terms of the cardinality of the base of the bornology and thesignificance of the result is due to the fact that several known bornolocical spaceshave closed bases. Theorem 4.1.
Let B be a bornology on X with a closed base B . If | B | < b then X has the B s -Hurewicz property.Proof. Let {U n : n ∈ N } be a sequence of open B s -covers of X . Let U n = { U nm : m ∈ N } for n ∈ N . For B ∈ B there are δ > U nm ∈ U n such that B δ ⊆ U nm . Foreach B ∈ B , define a function f B : N → N by f B ( n ) = min { m ∈ N : B δ ⊆ U nm forsome δ > } . Consider the set { f B : B ∈ B } . Since | B | < b , there is a g : N → N such that f B ≤ ∗ g for all B ∈ B . For each n ∈ N , choose V n = { U nm : m ≤ g ( n ) } .We claim that {V n : n ∈ N } witnesses the B s -Hurewicz property.To see this we will show that for B ∈ B there is a n ∈ N and a sequence { δ n : n ≥ n } of positive real numbers such that B δ n ⊆ U for some U ∈ V n forall n ≥ n . For B ∈ B choose B ∈ B with B ⊆ B . Since f B ≤ ∗ g , there is a n ∈ N such that f B ( n ) ≤ g ( n ) for all n ≥ n . Now by definition of f B , for each n ≥ n there is a δ n > B δ n ⊆ U nf B ( n ) . Clearly U nf B ( n ) ∈ V n for all n ≥ n . So we have a sequence { δ n : n ≥ n } of positive real numbers such that B δ n ⊆ B δ n ⊆ U for some U ∈ V n for all n ≥ n . Hence X has the B s -Hurewiczproperty. (cid:3) Theorem 4.2.
Let B be a bornology on X with closed base. If X has the B s -Hurewicz property then CDR sub ( O B s , O B s ) holds.Proof. Let {U n : n ∈ N } be a sequence of open B s -covers of X . Let { Y n : n ∈ N } be a partition of N into infinite subsets. Since X has the B s -Hurewicz property,ONE has no winning strategy in the B s -Hurewicz game. We define a strategy σ for ONE in the B s -Hurewicz game as follows.Let the first move of ONE be σ ( ∅ ) = U . TWO responds by choosing a finiteset V ⊆ U . Choose a k for which 1 ∈ Y k and define σ ( V ) = U k \ V . TWOresponds by choosing a finite set V ⊆ U k . Suppose that V n has been chosen. Nowdefine σ ( V , V , . . . , V n ) = U m \ {V ∪ V ∪ · · · ∪ V n } whenever n ∈ Y m . TWOresponds by choosing a finite set V n +1 ⊆ U m . This define a strategy σ for ONE inthe B s -Hurewicz game. Since σ is not a winning strategy for ONE, consider a play σ ( ∅ ) , V , σ ( V ) , V , . . . , σ ( V , V , . . . , V n ) , V n +1 , . . . which is lost by ONE. Thus for any B ∈ B there exist a n and a sequence { δ n : n ≥ n } of positive real numbers satisfying B δ n ⊆ U for some U ∈ V n for all n ≥ n . Observe that the collection {V n : n ∈ N } is pairwise disjoint by theconstruction of the strategy. Now for each m ∈ N , let R m = ∪ n ∈ Y m V n +1 . Clearly {R n : n ∈ N } is a collection of pairwise disjoint open B s -covers of X witnessing CDR sub ( O B s , O B s ). (cid:3) Theorem 4.3.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) X has the B s -Hurewicz property. (2) X satisfies S fin ( O B s , O gp B s ) . (3) ONE does not have a winning strategy in G fin ( O B s , O gp B s ) . (4) X satisfies O B s → ⌈O gp B s ⌉ k for each k ∈ N , provided X is B s -Lindel¨of.Proof. The equivalences of (1) , (2) and (3) are due to [10, Theorem 4.2]. We willprove the implications (3) ⇒ (4) and (4) ⇒ (2).(3) ⇒ (4). This proof is inspired by the arguments used in [20, Theorem 3].Let U be an open B s -cover of X and let f : [ U ] → { , . . . , k } be a coloring.Enumerate U bijectively as { U n : n ∈ N } . Let T i = { U j : j > , f ( { U , U j } ) = i } for i ∈ { , . . . , k } . We can write U \ { U } = ∪ ki =1 T i . By Lemma 3.1, there is a i ∈ { , . . . , k } such that T i is an open B s -cover of X . Let U = T i . Applyingthe same argument inductively we obtain sequences {U n : n ∈ N } and { i n : n ∈ N } such that U n +1 ⊆ U n and U n +1 = { U j ∈ U n : j > n + 1 , f ( { U n +1 , U j } ) = i n +1 } for all n ∈ N . Let W i = { U n : i n = i } for each i ∈ { , . . . , k } . Now each U n canbe expressed as ∪ ki =1 ( U n ∩ W i ). Again Lemma 3.1 implies that there must be a i n ∈ { , . . . , k } such that U n ∩ W i n is an open B s -cover of X . Since U n +1 ⊆ U n foreach n ∈ N , we can assume that there is a i ∈ { , . . . , k } such that i n = i for all n ∈ N .Define a strategy τ for ONE in G fin ( O B s , O gp B s ) as follows. Let the first move ofONE be τ ( ∅ ) = U ∩ W i . TWO selects a finite set V ⊆ τ ( ∅ ). Let n = max { n ∈ N : U n ∈ V } . Let the next move of ONE be τ ( V ) = U n ∩ W i . TWO selectsa finite set V ⊆ τ ( V ). Clearly V ∩ V = ∅ . Let n = max { n ∈ N : U n ∈ V } .Again let τ ( V , V ) = U n ∩ W i . TWO selects a finite set V ⊆ τ ( V , V ) and soon. Clearly {V r : r ∈ N } are pairwise disjoint.This defines τ a strategy for ONE in G fin ( O B s , O gp B s ). Since τ is not a winningstrategy for ONE, consider a play τ ( ∅ ) , V , . . . , τ ( V , . . . , V r ) , V r +1 , . . . . which is lost by ONE. So ∪ r ∈ N V r ∈ O gp B s . Let V = ∪ r ∈ N V r . Then V ∈ O gp B s and {V r : r ∈ N } is a partition of V into pairwise disjoint finite sets such that for any V, W ∈ V where V ∈ V r and W ∈ V s with r = s , f ( { V, W } ) = i . This shows that(4) holds.(4) ⇒ (2) Clearly X satisfies O B s → ⌈O B s ⌉ . By Theorem 3.5, X satisfies S fin ( O B s , O B s ). Now we show that every countable open B s -cover of X is B s -groupable. Let U be a countable open B s -cover of X . Let {U n : n ∈ N } be apartition of U into nonempty finite sets. Define a coloring f : [ U ] → { , } by f ( { U, V } ) = 1 if U, V ∈ U n for some n ∈ N and f ( { U, V } ) = 2 otherwise. By(4), there is a V ∈ O gp B s with V ⊆ U . So there is a sequence {V n : n ∈ N } ofpairwise disjoint finite subsets of V witnessing the B s -groupability of V . Since U iscountable, the elements of U \ V can be distributed among V n ’s so that { V n : n ∈ N } witnesses the B s -groupability of U . Hence X satisfies S fin ( O B s , O gp B s ). (cid:3) Our next results concerns with the B s -Hurewicz property of product spaces.For a metric space ( X, d ) one can consider the product space X n endowed with theproduct metric d n defined as d n (( x , . . . , x n ) , ( y , . . . , y n )) = max { d ( x , y ) , . . . , d ( x n , y n ) } . URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 13
Let B be a bornology on X with closed base B . In [15], it has been shownthat the collection { B n : B ∈ B } generates a bornology on X n . We denote thatbornology on X n by B n . Further it can be easily verified that for any n ∈ N and δ >
0, ( B δ ) n = ( B n ) δ which we will need repeatedly. We start with the followingsimple observation followed by a result on selection principles (Theorem 4.4) whichthough not related to the main topic of this section is interesting in its own right. Lemma 4.1.
Let B be a bornology on X with a compact base B . If U is anopen ( B n ) s -cover of X n , then there exists an open B s -cover V of X such that { V n : V ∈ V} is an open ( B n ) s -cover of X n which refines U .Proof. Let U be an open ( B n ) s -cover of X n . Let B ∈ B . Then for B n ∈ B n thereexist a δ > U ∈ U such ( B n ) δ ⊆ U . Since B is a compact subset of X and U is open in X n containing B n , we use Wallace Theorem [11] to find an open set V B in X such that B n ⊆ V nB ⊆ U . Consider the collection V = { V B : B ∈ B } . Weintend to show that V is an open B s -cover of X . For B ∈ B , we have B ⊆ V B .Since V B is open and B is compact with B ⊆ V B , there is a δ > B δ ⊆ V B . Therefore V = { V B : B ∈ B } is an open B s -cover of X .Again for B ∈ B , ( B δ ) n ⊆ V nB i.e. ( B n ) δ ⊆ V nB and V B ⊆ U for some U ∈ U implying that { V n : V ∈ V} is an open ( B n ) s -cover of X n as well as refines U . (cid:3) Theorem 4.4.
Let Π ∈ { S , S fin } and P , Q ∈ {O , Γ } . Let B be a bornology on X with compact base. The following statements are equivalent. (1) X satisfies Π( P B s , Q B s ) . (2) X n satisfies Π( P ( B n ) s , Q ( B n ) s ) for each n ∈ N .Proof. We only present the proof for the case Π = S , P = O and Q = O as othercases follow analogously.(1) ⇒ (2). Let {U m : m ∈ N } be a sequence of open ( B n ) s -covers of X n .By Lemma 4.1, for each U m there exists an open B s -cover V m of X such that { V n : V ∈ V m } refines U m . Apply S ( O B s , O B s ) to the sequence {V m : m ∈ N } to choose V m ∈ V m for each m ∈ N so that { V m : m ∈ N } becomes an open B s -cover of X . Now for each V m we can choose a U m ∈ U m such that V nm ⊆ U m . Wewill show that { U m : m ∈ N } is an open ( B n ) s -cover of X n . Let B n ∈ B n . For B ∈ B there exist a δ > V m such that B δ ⊆ V m . So ( B δ ) n ⊆ V nm i.e.( B n ) δ ⊆ V nm ⊆ U m . Therefore { U m : m ∈ N } is an open ( B n ) s -cover of X n . Hence X n satisfies S ( O ( B n ) s , O ( B n ) s ).(2) ⇒ (1). Let {U m : m ∈ N } be a sequence of open B s -covers of X . For each m ∈ N , let U ′ m = { U n : U ∈ U m } . Clearly U ′ m ’s are open ( B n ) s -cover of X n .Now consider the sequence {U ′ m : m ∈ N } of open ( B n ) s -covers of X n . Since X n satisfies S ( O ( B n ) s , O ( B n ) s ), there is an open ( B n ) s -cover { U nm : m ∈ N } of X n with U nm ∈ U ′ m for each m ∈ N . We will show that { U m : m ∈ N } with U m ∈ U m for m ∈ N is an open B s -cover of X . Let B ∈ B . Then for B n ∈ B n there exist a δ > U nm such that ( B n ) δ ⊆ U nm i.e. ( B δ ) n ⊆ U nm i.e. B δ ⊆ U m . So { U m : m ∈ N } is an open B s -cover of X . Hence X satisfies S ( O B s , O B s ). (cid:3) Theorem 4.5.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) X has the B s -Hurewicz property. (2) X n has the ( B n ) s -Hurewicz property for each n ∈ N . Proof.
We only present proof of (1) ⇒ (2).(1) ⇒ (2). Let {U k : k ∈ N } be a sequence of open ( B n ) s -covers of X n . By Lemma4.1, for each k there exists an open B s -cover V k of X such that { V n : V ∈ V k } refines U k . Consider the sequence {V k : k ∈ N } . By (1), there is a sequence {W k : k ∈ N } of finite sets with W k ⊆ V k for each k such that for B ∈ B thereexist a m ∈ N and a sequence { δ m : m ≥ m } of positive reals satisfying B δ m ⊆ V for some V ∈ W k for all m ≥ m . Now for each k we can choose a finite subset Z k of U k such that for each V ∈ W k there is a U ∈ Z k with V n ⊆ U . We will showthat {Z k : k ∈ N } witnesses the ( B n ) s -Hurwicz property of X n . Let B n ∈ B n .Note that for B ∈ B there already exist a p ∈ N and a sequence { δ p : p ≥ p } ofpositive reals satisfying B δ p ⊆ V for some V ∈ W k for all p ≥ p i.e. ( B n ) δ p ⊆ U for some U ∈ Z k for all p ≥ p . Hence X n has the ( B n ) s -Hurewicz property. (cid:3) Combining Theorem 4.3 and Theorem 4.5, we obtain the following.
Theorem 4.6.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) X n has the ( B n ) s -Hurewicz property for each n ∈ N . (2) X satisfies S fin ( O B s , O gp B s ) . (3) ONE does not have a winning strategy in G fin ( O B s , O gp B s ) on X . (4) X satisfies O B s → ⌈O gp B s ⌉ k for each k ∈ N , provided X is B s -Lindel¨of. The strong B -Gerlits-Nagy property and some observations. It iswell known that the Gerlits-Nagy property was introduced in the seminal paperof the authors [13] as a property stronger than the classical Hurewicz propertyand had been extensively investigated since then (see [28, 31] for example). Inparticular in [20] certain new characterizations of this property were establishedusing the notion of groupability of open covers. In this section we introduce thenotion of strong B -Gerlits-Nagy property (which has not been investigated at all inbornological settings) and while defining the concept, follow the line of [20] whichseems more effective for our purpose. Definition 4.1.
Let B be a bornology on X with closed base. X is said to have thestrong B -Gerlits-Nagy property (in short, B s -Gerlits-Nagy property) if X satisfiesthe selection principle S ( O B s , O gp B s ) . It is clear that S ( O B s , Γ B s ) implies S ( O B s , O gp B s ) which shows that every γ B s -set has the B s -Gerlits-Nagy property. Again S ( O B s , O gp B s ) evidently implies S fin ( O B s , O gp B s ) which in turn assures the B s -Hurewicz property by [10, Theorem4.2]. Therefore as in the classical case, if X has the B s -Gerlits-Nagy propertythen X has the B s -Hurewicz property and further it satisfies S ( O B s , O B s ). Be-low we prove certain results in line of [20] where the proofs are done with suitablemodifications as is necessary for bornological structures.The following example shows that the real line associated with a bornology hasthe B s -Gerlits-Nagy property. Example 4.1.
Consider the real line X = R with the Euclidean metric d and thebornology B generated by { ( − x, x ) : x > } . We show that X has the B s -Gerlits-Nagy property. To see this, let {U n : n ∈ N } be a sequence of open B s -covers of X .Then clearly for each k ∈ N , there is a U ∈ U n such that ( − k, k ) ⊆ U (by choosing ( − k, k ) ∈ B ). URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 15
Consider a sequence of positive integers k < k < · · · . For each n ∈ N , choose a U n ∈ U n such that ( − k n , k n ) ⊆ U n . We show that { U n : n ∈ N } ∈ O gp B s . For this wechoose V n = { U n } for each n ∈ N . Now we show that {V n : n ∈ N } is the sequenceof pairwise disjoint finite sets witnessing the B s -groupability of { U n : n ∈ N } . Let B ∈ B . Clearly U = { ( − n, n ) : n ∈ N } is a γ B s -cover of X . Consequently wecan find a n ∈ N and a sequence of positive real numbers { δ n : n ≥ n } such that B δ n ⊆ ( − n, n ) for all n ≥ n i.e. B δ n ⊆ ( − k n , k n ) ⊆ U n for U n ∈ V n for all n ≥ n . Therefore { U n : n ∈ N } ∈ O gp B s and so X satisfies S ( O B s , O gp B s ) . Hence X has the B s -Gerlits-Nagy property. Theorem 4.7.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) X has the B s -Gerlits-Nagy property. (2) X has the B s -Hurewicz property as well as it satisfies S ( O B s , O B s ) .Proof. (1) ⇒ (2). By (1), X satisfies S ( O B s , O gp B s ) which implies that X satisfies S fin ( O B s , O gp B s ) as well as S ( O B s , O B s ). Again by [10, Theorem 4.2], X has the B s -Hurewicz property. Hence (2) holds.(2) ⇒ (1). Let {U n : n ∈ N } be a sequence of open B s -covers of X . Apply S ( O B s , O B s ) to {U n : n ∈ N } to choose a U n ∈ U n for each n ∈ N such that { U n : n ∈ N } is an open B s -cover of X . Using that X has the B s -Hurewiczproperty and [10, Lemma 4.2], { U n : n ∈ N } is a B s -groupable cover of X . Thus X satisfies S ( O B s , O gp B s ). (cid:3) Proposition 4.3.
Let B be a bornology on X with a closed base B . If | B | < add ( M ) then X has the B s -Gerlits-Nagy property.Proof. Since add ( M ) = min { b , cov ( M ) } and | B | < add ( M ), so | B | < cov ( M )as well as | B | < b . By Theorem 3.2, | B | < cov ( M ) implies that X satisfies S ( O B s , O B s ). Again in view of | B | < b and Theorem 4.1, we can conclude that X has the B s -Hurewicz property. Hence X has the B s -Gerlits-Nagy property. (cid:3) Theorem 4.8.
Let B be bornology on X with closed base. The following statementsare equivalent. (1) X has the B s -Gerlits-Nagy property. (2) X satisfies O B s → ( O gp B s ) nk for each n, k ∈ N .Proof. (1) ⇒ (2). Let U be an open B s -cover of X and f : [ U ] n → { , . . . , k } bea coloring. Since X satisfies S ( O B s , O B s ), X also satisfies O B s → ( O B s ) nk byTheorem 3.6. Consequently there are a V ⊆ U with
V ∈ O B s and a i ∈ { , . . . , k } such that for each V ∈ [ V ] n , f ( V ) = i . By (1), we can find a countable open B s -cover V ′ ⊆ V which is B s -groupable. Thus we have a V ′ ∈ O gp B s and a i ∈ { , . . . , k } such that for each V ∈ [ V ′ ] n , f ( V ) = i holds.(2) ⇒ (1) Clearly X satisfies O B s → ( O B s ) . By Theorem 3.6, X satisfies S ( O B s , O B s ). Now we show that every countable open B s -cover of X is B s -groupable. Let U be a countable open B s -cover of X . Let {U n : n ∈ N } be apartition of U into nonempty finite sets. Define a coloring f : [ U ] → { , } by f ( { U, V } ) = 1 if U, V ∈ U n for some n ∈ N and f ( { U, V } ) = 2 otherwise. By (2),there is a V ∈ O gp B s with V ⊆ U . Therefore U is B s -groupable. Hence X satisfies S ( O B s , O gp B s ). (cid:3) Theorem 4.9.
Let B be bornology on X with compact base. The following state-ments are equivalent. (1) X has the B s -Gerlits-Nagy property. (2) X n has the ( B n ) s -Gerlits-Nagy property for each n ∈ N .Proof. We only prove (1) ⇒ (2).Let {U k : k ∈ N } be a sequence of open ( B n ) s -covers of X n . By Lemma 4.1, foreach U k there exists an open B s -cover V k of X such that { V n : V ∈ V k } refines U k .Apply S ( O B s , O gp B s ) to {V k : k ∈ N } to choose a V k ∈ V k for each k ∈ N such that { V k : k ∈ N } is a B s -groupable cover of X . Now for each V k there is a U k ∈ U k such that V nk ⊆ U k . We will show that { U k : k ∈ N } is a ( B n ) s -groupable cover of X n .As { V k : k ∈ N } ∈ O gp B s , there is a sequence {W p : p ∈ N } of pairwise disjointfinite sets such that for B ∈ B there exist a p ∈ N and a sequence { δ p : p ≥ p } of positive reals satisfying B δ p ⊆ V k for some V k ∈ W p for all p ≥ p . Let Z p for p ∈ N be a finite subset of { U k : k ∈ N } such that for each V k ∈ W p , V nk ⊆ U k for some U k ∈ Z p . Now as before we will have ( B n ) δ p = ( B δ p ) n ⊆ V nk ⊆ U k forsome U k ∈ Z p for all p ≥ p . Therefore { U k : k ∈ N } is a ( B n ) s -groupable coverof X n witnessing S ( O ( B n ) s , O gp ( B n ) s ) implying the ( B n ) s -Gerlits-Nagy property of X n . (cid:3) Observation on product spaces X × Y . We end Section 4 with two moreobservations regarding both the properties in product spaces. Let B be a bornologyon X and ( Y, ρ ) be another metric space. There is a natural bornology b B on( X × Y, d × ρ ) induced by B defined as b B = { C ⊆ X × Y : π X ( C ) ∈ B } [6], where π X : X × Y → X is the projection map. A base for b B is { B × Y : B ∈ B } . Lemma 4.2.
Let B be a bornology on X with compact base and ( Y, ρ ) be anothercompact metric space. If U is an open ( b B ) s -cover of X × Y , then there exists anopen B s -cover V of X such that { V × Y : V ∈ V} refines U . The proof is analogous to the proof of Lemma 4.1.
Theorem 4.10.
Let B be a bornology on X with compact base and ( Y, ρ ) be anothercompact metric space. The following statements hold. (1) X satisfies S ( O B s , Γ B s ) if and only if X × Y satisfies S ( O ( b B ) s , Γ ( b B ) s ) . (2) X has the B s -Hurewicz property if and only if X × Y has the ( b B ) s -Hurewiczproperty. (3) X has the B s -Gerlits-Nagy property if and only if X × Y has the ( b B ) s -Gerlits-Nagy property. Let (
Y, d Y ) be a subspace of ( X, d ) where d Y is the induced metric on Y ⊆ X .Let B be a bornology on X . It can be easily checked that B Y = { B ∩ Y : B ∈ B } is a bornology on Y . If Y is closed and B has a compact base, then B Y has acompact base. The following result needs no further explanations. Lemma 4.3.
Let B be a bornology on X with compact base and Y be a closedsubset of X . The following statements are true. (1) If X satisfies S ( O B s , Γ B s ) , then Y satisfies S ( O B sY , Γ B sY ) . (2) If X has the B s -Hurewicz property, then Y has the B sY -Hurewicz property. (3) If X has the B s -Gerlits-Nagy property, then Y has the B sY -Gerlits-Nagyproperty. URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 17 Results in function spaces
Further observations regarding selection principles and C ( X ) . In thissection we continue our investigation of selection principles in function spaces whichwe had started in [10] and establish some new results which have not been discussedin earlier articles.As before, let B be a bornology on ( X, d ) with closed base and (
Y, ρ ) be anothermetric space. For f ∈ C ( X, Y ), the neighbourhood of f with respect to the topology τ s B of strong uniform convergence is denoted by[ B, ε ] s ( f ) = { g ∈ C ( X, Y ) : ∃ δ > , ρ ( f ( x ) , g ( x )) < ε, ∀ x ∈ B δ } , for B ∈ B , ε > C ( X ) , τ s B ). The space ( C ( X ) , τ s B ) ishomogeneous and so it is enough to concentrate at the point 0 when dealing withlocal properties of this function space.We start with recollecting the following result from [9] which will be useful inour context. Lemma 5.1. ( [9, Lemma 2.2]) Let B be a bornology on the metric space ( X, d ) with closed base. The following statements hold. ( a ) Let U be an open B s -cover of X . If A = { f ∈ C ( X ) : ∃ U ∈ U , f ( x ) =1 for all x ∈ X \ U } . Then ∈ A \ A in ( C ( X ) , τ s B ) . ( b ) Let A ⊆ ( C ( X ) , τ s B ) and let U = { f − ( − n , n ) : f ∈ A } , where n ∈ N . If ∈ A and X / ∈ U , then U is an open B s -cover of X . The following exemplary observation about γ B s -covers is also useful. Lemma 5.2.
Let B be a bornology on X with closed base. Let A = { f n : n ∈ N } be a sequence of functions in ( C ( X ) , τ s B ) . If A ∈ Σ then for any neighbourhood U of in the real line, { f − n ( U ) : n ∈ N } is a γ B s -cover of X .Proof. As U is a neighbourhood of 0, we can find a ε > − ε, ε ) ⊆ U .Let B ∈ B . Consider the neighbourhood [ B, ε ] s (0) of 0. Since { f n : n ∈ N } converges to 0 with respect to τ s B , we can choose a n ∈ N such that f n ∈ [ B, ε ] s (0)for all n ≥ n . This means that for each n ≥ n there exists a δ n > B δ n ⊆ f − n ( − ε, ε ) ⊆ f − n ( U ) i.e. for the sequence { δ n : n ≥ n } of positive realnumbers we have B δ n ⊆ f − n ( U ) for all n ≥ n . Hence { f − n ( U ) : n ∈ N } is a γ B s -cover of X . (cid:3) Proposition 5.1.
Let B be a bornology on X with closed base. If X satisfies S (Γ B s , Γ B s ) then ( C ( X ) , τ s B ) satisfies S (Σ , Σ ) .Proof. Let { A n : n ∈ N } be a sequence of elements in Σ , where A n = { f n,k : k ∈ N } for n ∈ N . Consider the set U n = { f − n,k ( − n , n ) : k ∈ N } . By Lemma 5.2, each U n is a γ B s -cover of X . Now apply S (Γ B s , Γ B s ) to {U n : n ∈ N } to choose a f − n,k n ( − n , n ) ∈ U n for each n ∈ N such that { f − n,k n ( − n , n ) : n ∈ N } ∈ Γ B s .We claim that { f n,k n : n ∈ N } converges to 0. Let [ B, ε ] s (0) be neighbourhoodof 0 where B ∈ B and ε >
0. First choose n ∈ N such that n < ε . Since { f − n,k n ( − n , n ) : n ∈ N } is a γ B s -cover, there exist a n and a sequence { δ n : n ≥ n } of positive real numbers such that B δ n ⊆ f − n,k n ( − n , n ) for all n ≥ n . Clearly B δ n ⊆ f − n,k n ( − ε, ε ) for all n ≥ n , where n = max { n , n } . This shows that { f n,k n : n ∈ N } converges to 0 with respect to τ s B . Therefore ( C ( X ) , τ s B ) satisfies S (Σ , Σ ). (cid:3) Theorem 5.1.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) ONE has no winning strategy in G ( O B s , Γ B s ) on X . (2) ONE has no winning strategy in G (Ω , Σ ) on ( C ( X ) , τ s B ) .Proof. Let σ be a strategy for ONE in G (Ω , Σ ) on ( C ( X ) , τ s B ). We now use σ to define a strategy ψ for ONE in G ( O B s , Γ B s ) on X as follows.The first move of ONE in G (Ω , Σ ) is σ ( ∅ ). Let U = { f − ( − ,
1) : f ∈ σ ( ∅ ) } and assume that X
6∈ U . By Lemma 5.1, U is an open B s -cover of X . Define ψ ( ∅ ) = U , the first move of ONE in G ( O B s , Γ B s ). TWO responds with U = f − ( − , G (Ω , Σ ) be f . To define ψ ( U ), we lookat the move σ ( f ) of ONE in G (Ω , Σ ). Consider U = { f − ( − , ) : f ∈ σ ( f ) } ,which is again an open B s -cover of X by Lemma 5.1. Define ψ ( U ) = U . TWOresponds with U = f − ( − , ). Let TWO’s move in G (Ω , Σ ) be f and so on.This defines a strategy ψ for ONE in G ( O B s , Γ B s ). Since ψ is not a winningstrategy, consider a ψ -play ψ ( ∅ ) , U , ψ ( U ) , U , . . . which is lost by ONE in G ( O B s , Γ B s ). Thus { U n : n ∈ N } is a γ B s -cover of X ,where U n = f − n ( − n , n ) for each n ∈ N . Consequently { f n : n ∈ N } is convergesto 0 with respect to τ s B .Now correspond to the ψ -play there is also a σ -play σ ( ∅ ) , f , σ ( f ) , f , . . . in G (Ω , Σ ) and { f n : n ∈ N } ∈ Σ . Hence σ is not a winning strategy for ONEin G (Ω , Σ ).(2) ⇒ (1). Let ψ be a strategy for ONE in G ( O B s , Γ B s ) on X . We define astrategy σ for ONE in G (Ω , Σ ) on ( C ( X ) , τ s B ) as follows.The first move of ONE in G ( O B s , Γ B s ) is ψ ( ∅ ) = U (say). Since U ∈ O B s , for B ∈ B there are a δ > U ∈ U such that B δ ⊆ U . Let U ,B = { U ∈ U : B δ ⊆ U } . For each U ∈ U ,B choose a f B,U ∈ C ( X ) such that f B,U ( B δ ) = { } and f B,U ( X \ U ) = { } . Consider the collection A = { f B,U : B ∈ B , U ∈ U ,B } .Clearly A ∈ Ω . Now define σ ( ∅ ) = A , the first move of ONE in G (Ω , Σ ). TWOresponds by choosing f B ,U ∈ A . Let U be the TWO’s response in G ( O B s , Γ B s ).Now let ψ ( U ) = U . We similarly construct A = { f B,U : B ∈ B , U ∈ U ,B } whichis in Ω . Define σ ( f B ,U ) = A . TWO responds by choosing f B ,U ∈ A . Let U be the TWO’s response in G ( O B s , Γ B s ) and so on.This defines a strategy σ for ONE in G (Ω , Σ ). Since σ is not a winningstrategy for ONE, consider a σ -play σ ( ∅ ) , f B ,U , σ ( f B ,U ) , f B ,U , . . . which is lost by ONE. So { f n : n ∈ N } ∈ Σ , where f n = f B n ,U n , n ∈ N . Consider { U n : n ∈ N } . Since f − n (1 , ⊆ U n for each n ∈ N , by Lemma 5.2, { U n : n ∈ N } )is a γ B s -cover of X .The corresponding ψ -play in G ( O B s , Γ B s ) is ψ ( ∅ ) , U , ψ ( U ) , U , . . . . URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 19
Since { U n : n ∈ N } ∈ Γ B s , ψ is not a winning strategy for ONE in G ( O B s , Γ B s ). (cid:3) Next we present some applications of well known α i -properties with respect tothe topology of strong uniform convergence on B . In the following results we makeuse of the fact that a sequence { f n : n ∈ N } in ( C ( X ) , τ s B ) converges to 0 withrespect to τ s B if and only if {| f n | : n ∈ N } converges to 0 with respect to τ s B . Theorem 5.2.
Let B be a bornology on X with closed base and X is B s -Lindel¨of.The following statements are equivalent. (1) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (2) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (3) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (4) X satisfies S ( O B s , Γ B s ) .Proof. We only prove (3) ⇒ (4) and (4) ⇒ (1).(3) ⇒ (4). Let {U n : n ∈ N } be a sequence of open B s -covers of X . Enumerateeach U n bijectively as { U nk : k ∈ N } . For each n the collection V n = { U k ∩ U k ∩· · · ∩ U nk n : 1 < k < k < · · · < k n } is an open B s -cover of X . Now define A n = { f ∈ C ( X ) : there is a V ∈ V n with f ( X \ V ) = { }} for each n . By Lemma5.1, A n ∈ Ω . Now apply (3) to { A n : n ∈ N } to obtain a sequence n < n < · · · and a A ∈ Σ such that A n i ∩ A = ∅ for each i ∈ N . Let f n i ∈ A n i ∩ A . Clearly { f n i : i ∈ N } ∈ Σ . Now there is a V n i l i ∈ V n i such that f n i ( X \ V n i l i ) = { } foreach i . We claim that { V n i l i : i ∈ N } is a γ B s -cover of X . Let B ∈ B . Since { f n i : i ∈ N } ∈ Σ , for the neighbourhood [ B, s (0) of 0 there is a i such that f n i ∈ [ B, s (0) for all i ≥ i i.e. there is a δ i > | f n i ( x ) | < x ∈ B δ i and i ≥ i i.e. B δ i ⊆ f − n i ( − , ⊆ V n i l i for all i ≥ i . This shows that { V n i l i : i ∈ N } is a γ B s -cover of X . Now for each n with 1 ≤ n ≤ n let U n ∈ U n be the n -th component in the representation of V n l . Also for each i > n ∈ ( n i − , n i ] let U n ∈ U n be the n -th component in the representation of V n i l i .Then it is easy to verify that { U n : n ∈ N } is a γ B s -cover of X . Hence X satisfies S ( O B s , Γ B s ).(4) ⇒ (1). Let { A n : n ∈ N } be a sequence of elements in Ω . Since X is B s -Lindel¨of, ( C ( X ) , τ s B ) has countable tightness. Therefore we can assume that A n ’s are countable. Say A n = { f n,k : k ∈ N } for each n . We can also assume that f n,k ≥ n and k . By [10, Theorem 3.7] and Theorem 5.1, we can saythat ONE has no winning strategy in G (Ω , Σ ) on ( C ( X ) , τ s B ). We now define astrategy σ for ONE in G (Ω , Σ ) as follows.Define σ ( ∅ ) = A , the first move of ONE. TWO responds with f ,k i ∈ A . Let D = { f ,k + f ,k : k i < k < k } . It is easy to see that D ∈ Ω . Define σ ( f ,k i ) = D . TWO responds with f ,k i + f ,k i . Again consider D = { f ,k + f ,k + f ,k : k i < k < k < k } ∈ Ω and define σ ( f ,k i , f ,k i + f ,k i ) = D .TWO responds with f ,k i + f ,k i + f ,k i and so on.This defines a strategy σ for ONE in G (Ω , Σ ). Since σ is not a winningstrategy, consider a play σ ( ∅ ) , f ,k i , σ ( f ,k i ) , f ,k i + f ,k i , σ ( f ,k i , f ,k i + f ,k i ) , f ,k i + f ,k i + f ,k i , . . . which is lost by ONE. Therefore f ,k i , f ,k i + f ,k i , f ,k i + f ,k i + f ,k i , . . . converges to 0 with respect to τ s B . Since f n,m ≥ n, m ∈ N , the sequence f ,k i , f ,k i , f ,k i , f ,k i , f ,k i , f ,k i , . . . also converges to 0 with respect to τ s B , which contains infinitely many elementsfrom each A n . Hence (1) holds. (cid:3) Summarizing Theorem 3.7, Theorem 5.1 and Theorem 5.2 we obtain the follow-ing.
Corollary 5.1.
Let B be a bornology on X with closed base and X is B s -Lindel¨of.The following statements are equivalent. (1) ONE has no winning strategy in G (Ω , Σ ) on ( C ( X ) , τ s B ) . (2) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (3) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (4) ( C ( X ) , τ s B ) satisfies α (Ω , Σ ) . (5) X satisfies S ( O B s , Γ B s ) . (6) ONE has no winning strategy in G ( O B s , Γ B s ) on X . (7) X satisfies O B s → (Γ B s ) nk for each n, k ∈ N , provided X is B s -Lindel¨of. Remark 5.1.
Moreover using [9, Corollary 2.10], the following equivalent condi-tions can also be added to Corollary 5.1. (8) ( C ( X ) , τ s B ) is Fr´echet-Urysohn. (9) ( C ( X ) , τ s B ) is strictly Fr´echet-Urysohn. (10) Every open B s -cover of X contains a countable set which is a γ B s -cover of X . Theorem 5.3.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) ( C ( X ) , τ s B ) satisfies α (Σ , Σ ) . (2) ( C ( X ) , τ s B ) satisfies α (Σ , Σ ) . (3) ( C ( X ) , τ s B ) satisfies α (Σ , Σ ) . (4) ( C ( X ) , τ s B ) satisfies S (Σ , Σ ) . (5) ONE has no winning strategy in G (Σ , Σ ) .Proof. We only give proof of the implications (3) ⇒ (4), (4) ⇒ (5) and (4) ⇒ (1).(3) ⇒ (4). Let { S n : n ∈ N } be a sequence of elements in Σ where S n = { f n,m : m ∈ N } . We assume that f n,m ≥ n, m ∈ N . Fix a n ∈ N , we constructa new sequence g n,m = f ,m + f ,m + · · · + f n,m , m ∈ N . Then the sequence { g n,m : m ∈ N } converges to 0 with respect to τ s B . Applying (3) to the sequence { T n : n ∈ N } , where T n = { g n,m : m ∈ N } for each n ∈ N , we obtain an increasingsequence 1 = n < n < n < · · · of positive integers such that { g n i ,m i : i ∈ N } converges to 0 and g n i ,m i ∈ T n i for each i . For each integer k ≥ j ∈ N with n k < j ≤ n k +1 , note that g n k +1 ,m k +1 = f ,m k +1 + · · · + f n k +1 ,m k +1 , and now choose h j = f j,m k +1 . We will show that { h j : j ∈ N } where h j ∈ S j witnesses S (Σ , Σ ).For this we need to show that the sequence { h j : j ∈ N } converges to 0 with respectto τ s B .Let [ B, ε ] s (0) ( B ∈ B , ε >
0) be a neighbourhood of 0. There exists a i ∈ N such that g n i +1 ,m i +1 ∈ [ B, ε ] s (0) for all i ≥ i . This means that there is a δ i > | f ,m i +1 ( x ) + · · · + f n i +1 ,m i +1 ( x ) | < ε for all x ∈ B δ i and for all i ≥ i i.e. | f j,m i +1 ( x ) | < ε for all x ∈ B δ i and for all j = 1 , . . . , n i +1 , i ≥ i i.e. f j,m i +1 ∈ [ B, ε ] s (0) for all n i < j ≤ n i +1 and for all i ≥ i i.e. h j ∈ [ B, ε ] s (0) URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 21 for all j > n i . Therefore { h j : j ∈ N } converges to 0 with respect to τ s B . Hence(4) holds.(4) ⇒ (5). Let σ be a strategy for ONE in G (Σ , Σ ). The first move of ONE is σ ( ∅ ). Enumerating σ ( ∅ ) bijectively as { f ( n ) : n ∈ N } . Assume that for each finitesequence of natural numbers τ of length at most m , U τ is already defined. Nowdefine { f ( n ,...,n m ,k ) : k ∈ N } to be σ ( f ( n ) , . . . , f ( n ,...,n m ) ) \ { f ( n ) , . . . , f ( n ,...,n m ) } .We have for each finite sequence of natural numbers τ , { f τ⌢n : n ∈ N } ∈ Σ .Apply S (Σ , Σ ) to choose a n τ such that { U τ⌢n τ : τ is a finite sequence ofnatural numbers } converges to 0 with respect to τ s B . Inductively define a sequence n = n ∅ , n k +1 = n ( n ,...,n k ) for k ≥
1. Then the sequence U n , U ( n ,n ) , . . . , U ( n ,...,n k ) , . . . converges to 0 with respect to τ s B . Since it is actually a sequence of moves of TWOduring a play in G (Σ , Σ ), σ is not a winning strategy for ONE.(4) ⇒ (1). Let { S n : n ∈ N } be a sequence of elements in Σ . From each S n wecan easily make new sequences S n,m which are pairwise disjoint and each of whichconverges to 0. Consider the collection { S n,m : n, m ∈ N } . Now apply S (Σ , Σ )to choose a f n,m ∈ S n,m for n, m ∈ N such that { f n,m : n, m ∈ N } converges to0. Clearly the sequence { f n,m : n, m ∈ N } contains infinitely many elements fromeach S n . Hence (1) holds. (cid:3) Some observations on X n and ( C ( X ) , τ s B ) .Theorem 5.4. Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) ( C ( X ) , τ s B ) has countable tightness. (2) X n is ( B n ) s -Lindel¨of for each n ∈ N .Proof. (1) ⇒ (2). Let U be an open ( B n ) s -cover of X n . By Lemma 4.1, there is anopen B s -cover V of X such that { V n : V ∈ V} refines U . Now for B ∈ B there exista δ > V ∈ V such that B δ ⊆ V . Let V B = { V ∈ V : B δ ⊆ V } . For each V ∈ V B choose a f B,V ∈ C ( X ) such that f B,V ( B δ ) = { } and f B,V ( X \ V ) = { } .Consider the set F = { f B,V : B ∈ B , V ∈ V B } . Clearly 0 ∈ F . By (1), there is acountable subset F ′ of F such that 0 ∈ F ′ . Let W = { V : f B,V ∈ F ′ } . Then W is acountable subset of V . Now for V ∈ W choose a U ∈ U such that V n ⊆ U . Considerthe set Z = { U : V n ⊆ U for V ∈ W} . Clearly Z is a countable subset of U . Weshow that Z is an open ( B n ) s -cover of X n . Let B n ∈ B n . As [ B, s (0) ∩ F ′ = ∅ ,there is a f B ,V ∈ F ′ with f B ,V ∈ [ B, s (0) i.e. there exists a δ > B δ ⊆ f − B ,V ( − , ⊆ V i.e. ( B δ ) n ⊆ V n ⊆ U for some U ∈ Z i.e. ( B n ) δ ⊆ U .So Z is a countable ( B n ) s -subcover of U .(2) ⇒ (1). Let A ⊆ ( C ( X ) , τ s B s ) with 0 ∈ A . Let U m = { g − ( − m , m ) : g ∈ A } for m ∈ N . Then by Lemma 5.1 U m is an open B s -cover of X for each m ∈ N .Let V m = { U n : U ∈ U m } . Clearly V m is an open ( B n ) s -cover of X n . By (2), foreach m ∈ N there is a countable ( B n ) s -subcover W m = { U nk,m : k ∈ N } of V m ,where U k,m = g − k,m ( − m , m ) for k ∈ N . Choose A ′ = { g k,m : k, m ∈ N } , a countablesubset of A . We show that 0 ∈ A ′ . Let [ B, ε ] s (0) be a neighbourhood of 0, where B ∈ B and ε >
0. Choose a m ∈ N with m < ε . Now for B n ∈ B n there exist a δ > U nk,m ∈ W m such that ( B n ) δ ⊆ U nk,m i.e. B δ ⊆ U k,m = g − k,m ( − m , m )i.e. g k,m ∈ [ B, ε ] s (0) i.e. [ B, ε ] s (0) ∩ A ′ = ∅ . So 0 ∈ A ′ . Hence (1) holds. (cid:3) The next three results can be obtained by applying Theorem 4.4 to each [9,Theorem 2.3], [9, Theorem 2.5] and [9, Theorem 2.7] respectively.
Theorem 5.5.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) ( C ( X ) , τ s B ) has countable strong fan tightness. (2) X n satisfies S ( O ( B n ) s , O ( B n ) s ) for each n ∈ N . Theorem 5.6.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) ( C ( X ) , τ s B ) has countable fan tightness. (2) X n satisfies S fin ( O ( B n ) s , O ( B n ) s ) for each n ∈ N . Theorem 5.7.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) ( C ( X ) , τ s B ) is strictly Fr´echet-Urysohn. (2) X n satisfies S ( O ( B n ) s , Γ ( B n ) s ) for each n ∈ N . In line of [20, Theorem 21], we can obtain a bornological version which presentssimilar characterization using the topology of strong uniform convergence τ s B on B . Theorem 5.8.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) X has the B s -Hurewicz property. (2) ( C ( X ) , τ s B ) has countable fan tightness and Reznichenko’s property i.e. itsatisfies S fin (Ω , Ω gp ) . (3) ONE does not have a winning strategy in G fin (Ω , Ω gp ) on ( C ( X ) , τ s B ) .Proof. The equivalence (1) ⇔ (2) is due to [10, Theorem 5.8]. The implication(3) ⇒ (2) is easily followed. We prove the implication (1) ⇒ (3).(1) ⇒ (3). X has the B s -Hurewicz property implies that ONE does not havea winning strategy in the B s -Hurewicz game on X . Let σ be a strategy for ONEin the game G fin (Ω , Ω gp ). We use σ to define a strategy ψ for ONE in the B s -Hurewicz game as follows.The first move of ONE in G fin (Ω , Ω gp ) is σ ( ∅ ). Let U = { f − ( − ,
1) : f ∈ σ ( ∅ ) } and assume that X
6∈ U . By Lemma 5.1, U is an open B s -cover of X . Define ψ ( ∅ ) = U , the first move of ONE the B s -Hurewicz game. TWO responds bychoosing a finite subset V of ψ ( ∅ ). Let C ⊆ σ ( ∅ ) be a finite collection of functionssuch that V = { f − ( − ,
1) : f ∈ C } . Then C is a legitimate move of TWO in G fin (Ω , Ω gp ).To define ψ ( V ), look at the move σ ( C ) of ONE in G fin (Ω , Ω gp ). Let A = σ ( C ) \ C . Clearly A ∈ Ω and U = { f − ( − , ) : f ∈ A } is an open B s -cover of X again by Lemma 5.1. Now define ψ ( V ) = U . TWO responds bychoosing a finite subset V of ψ ( V ). Let C be the finite subset of A with V = { f − ( − , ) : f ∈ C } . Thus C is a legitimate move of TWO in G fin (Ω , Ω gp ).Again look at σ ( C , C ) and let A = σ ( C , C ) \ { C , C } which is in Ω . U = { f − ( − , ) : f ∈ A } is an open B s -cover of X . Define ψ ( V , V ) = U . TWOresponds by choosing a finite set V ⊆ ψ ( V , V ). Let C ⊆ A be a finite set with V = { f − ( − , ) : f ∈ C } and so on.This defines a strategy ψ for ONE in the B s -Hurewicz game. Since ψ is not awinning strategy, consider a ψ -play ψ ( ∅ ) , V , ψ ( V ) , V , ψ ( V , V ) , . . . URTHER OBSERVATIONS ON BORNOLOGICAL COVERING PROPERTIES 23 which is lost by ONE. Therefore for B ∈ B there exist a n ∈ N and a sequence { δ n : n ≥ n } of positive real numbers satisfying B δ n ⊆ U for some U ∈ V n for all n ≥ n .Now corresponding to the ψ -play there is a σ -play in G fin (Ω , Ω gp ) σ ( ∅ ) , C , σ ( C ) , C , σ ( C , C ) , C , . . . We will show that ∪ n ∈ N C n ∈ Ω gp .It is clear from the construction of the game that C n ’s are pairwise disjoint andfor U ∈ V n , U = f − ( − n , n ) for some f ∈ C n . Let [ B, ε ] s (0) be a neighbourhoodof 0 where B ∈ B and ε >
0. For B ∈ B there is a n ∈ N and a sequence { δ n : n ≥ n } of positive real numbers satisfying B δ n ⊆ U for some U ∈ V n forall n ≥ n . Choose a n ∈ N such that n < ε . Choose n = max { n , n } . Then B δ n ⊆ f − ( − n , n ) ⊆ f − ( − ε, ε ) for some f ∈ C n for all n ≥ n i.e. f ∈ [ B, ε ] s (0)for some f ∈ C n for all n ≥ n i.e. [ B, ε ] s (0) ∩ C n = ∅ for all n ≥ n . Therefore { C n : n ∈ N } witnesses the groupability of ∪ n ∈ N C n . Hence σ is not a winningstrategy for ONE in G fin (Ω , Ω gp ). (cid:3) Theorem 5.8 together with Theorem 4.5 gives the following.
Corollary 5.2.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) X n has the ( B n ) s -Hurewicz property for each n ∈ N . (2) ( C ( X ) , τ s B ) has countable fan tightness and Reznichenko’s property i.e. itsatisfies S fin (Ω , Ω gp ) . (3) ONE does not have a winning strategy in G fin (Ω , Ω gp ) on ( C ( X ) , τ s B ) . Theorem 5.9.
Let B be a bornology on X with closed base. The following state-ments are equivalent. (1) X has the B s -Gerlits-Nagy property. (2) ( C ( X ) , τ s B ) has countable strong fan tightness and Reznichenko’s property.Proof. (1) ⇒ (2). By Theorem 4.7, X has the B s -Hurewicz property as wellas it satisfies S ( O B s , O B s ). Also from [10, Theorem 5.8], ( C ( X ) , τ s B ) has theReznichenko’s property. Again X satisfies S ( O B s , O B s ) implies that ( C ( X ) , τ s B )has countable strong fan tightness by [9, Theorem 2.3]. Hence (2) holds.(2) ⇒ (1). Using [10, Theorem 5.8] and [9, Theorem 2.3], we have X has the B s -Hurewicz property as well as it satisfies S ( O B s , O B s ). Hence X the B s -Gerlits-Nagy property. (cid:3) Corollary 5.3.
Let B be a bornology on X with compact base. The followingstatements are equivalent. (1) X n has the ( B n ) s -Gerlits-Nagy property for each n ∈ N . (2) ( C ( X ) , τ s B ) has countable strong fan tightness and Reznichenko’s property. Theorem 5.10.
Let B , B be bornologies on X and X respectively with closedbases. If ( C ( X ) , τ s B ) and ( C ( X ) , τ s B ) are homeomorphic, then the followingstatements are true. (1) X has the B s -Hurewicz property if and only if X has the B s -Hurewiczproperty. (2) X has the B s -Gerlits-Nagy property if and only if X has the B s -Gerlits-Nagy property. Proof.
We only prove (1) as the proof of (2) is analogous.(1). Let ψ : ( C ( X ) , τ s B ) → ( C ( X ) , τ s B ) be a homeomorphism. Suppose that X has the B s -Hurewicz property. By [10, Theorem 5.8], ( C ( X ) , τ s B ) satisfies S fin (Ω , Ω gp ). We will show that ( C ( X ) , τ s B ) satisfies S fin (Ω , Ω gp ). For this wefirst show that S fin (Ω , Ω ) holds and then prove that every countable element inΩ is groupable. (0 , 0 are zero elements in C ( X ) and C ( X ) respectively).Let { A n : n ∈ N } be a sequence of elements in Ω . It is easy to observe thatif A n ∈ Ω then ψ − ( A n ) ∈ Ω . Apply S fin (Ω , Ω ) to { ψ − ( A n ) : n ∈ N } tochoose a finite subset B n of ψ − ( A n ) for each n ∈ N such that ∪ n ∈ N B n ∈ Ω . Now ψ ( B n ) is a finite subset of A n for each n ∈ N . Take a neighbourhood [ B, ε ] s (0 ) of0 where B ∈ B , ε >
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E-mail address : [email protected] * Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal,India E-mail address ::