aa r X i v : . [ m a t h . GN ] N ov ARHANGELSKII’S α -PRINCIPLES AND SELECTION GAMES STEVEN CLONTZ
Abstract.
Arhangelskii’s properties α and α defined for convergent se-quences may be characterized in terms of Scheeper’s selection principles. Wegeneralize these results to hold for more general collections and consider theseresults in terms of selection games. The following characterizations were given as Definition 1 by Kocinac in [6].
Definition 1.
Arhangelskii’s α -principles α i ( A , B ) are defined as follows for i ∈{ , , , } . Let A n ∈ A for all n < ω ; then there exists B ∈ B such that: α : A n ∩ B is cofinite in A n for all n < ω . α : A n ∩ B is infinite for all n < ω . α : A n ∩ B is infinite for infinitely-many n < ω . α : A n ∩ B is non-empty for infinitely-many n < ω .When ( A , B ) is omitted, it is assumed that A = B is the collection Γ X,x of se-quences converging to some point x ∈ X , as introduced by Arhangelskii in [1]. Pro-vided A only contains infinite sets, it’s easy to see that α n ( A , B ) implies α n +1 ( A , B ).We aim to relate these to the following games. Definition 2.
The selection game G ( A , B ) (resp. G fin ( A , B )) is an ω -lengthgame involving Players I and II. During round n , I chooses A n ∈ A , followedby II choosing a n ∈ A n (resp. F n ∈ [ A n ] < ℵ ). Player II wins in the case that { a n : n < ω } ∈ B (resp. S { F n : n < ω } ∈ B ), and Player I wins otherwise.Such games are well-represented in the literature; see [11] for example. We willalso consider the similarly-defined games G < ( A , B ) (II chooses 0 or 1 points fromeach choice by I) and G cf ( A , B ) (II chooses cofinitely-many points). Definition 3.
Let P be a player in a game G . P has a winning strategy for G ,denoted P ↑ G , if P has a strategy that defeats every possible counterplay bytheir opponent. If a strategy only relies on the round number and ignores themoves of the opponent, the strategy is said to be predetermined ; the existence of apredetermined winning strategy is denoted P ↑ pre G .We briefly note that the statement I pre G ⋆ ( A , B ) is often denoted as the selectionprinciple S ⋆ ( A , B ). Definition 4.
Let Γ
X,x be the collection of non-trivial sequences S ⊆ X convergingto x , that is, infinite subsets of X \ { x } such that for each neighborhood U of x , S ∩ U is cofinite in S . Key words and phrases.
Selection principle, selection game, α i property, convergence. Definition 5.
Let Γ X be the collection of open γ -covers U of X , that is, infiniteopen covers of X such that X
6∈ U and for each x ∈ X , { U ∈ U : x ∈ U } is cofinitein U .The similarity in nomenclature follows from the observation that every non-trivial sequence in C p ( X ) converging to the zero function naturally defines acorresponding γ -cover in X , see e.g. Theorem 4 of [12].The equivalence of α (Γ X,x Γ X,x ) and I pre G (Γ X,x , Γ X,x ) was briefly assertedby Sakai in the introduction of [10]; the similar equivalence of α (Γ X,x Γ X,x ) andI pre G fin (Γ X,x , Γ X,x ) seems to be folklore. In fact, these relationships hold in moregenerality.Note that by these definitions, convergent sequences (resp. γ -covers) may beuncountable, but any infinite subset of either would remain a convergent sequence(resp. γ -cover), in particular, countably infinite subsets. We capture this idea asfollows. Definition 6.
Say a collection A is Γ -like if it satisfies the following for each A ∈ A . • | A | ≥ ℵ . • If A ′ ⊆ A and | A ′ | ≥ ℵ , then A ′ ∈ A .We also require the following. Definition 7.
Say a collection A is almost- Γ -like if for each A ∈ A , there is A ′ ⊆ A such that: • | A ′ | = ℵ . • If A ′′ is a cofinite subset of A ′ , then A ′′ ∈ A .So all Γ-like sets are almost-Γ-like.We are now able to prove a few general equivalences between α -princples andselection games. 1. On α ( A , B ) and G ( A , B ) Theorem 8.
Let A be almost- Γ -like and B be Γ -like. Then α ( A , B ) holds if andonly if I pre G ( A , B ) .Proof. We first assume α ( A , B ) and let A n ∈ A for n < ω define a predeterminedstrategy for I. We may apply α ( A , B ) to choose B ∈ B such that | A n ∩ B | ≥ ℵ .We may then choose a n ∈ ( A n ∩ B ) \ { a i : i < n } for each n < ω . It follows that B ′ = { a n : n < ω } ∈ B since B ′ is an infinite subset of B ∈ B ; therefore A n doesnot define a winning predetermined strategy for I.Now suppose I pre G ( A , B ). Given A n ∈ A for n < ω , first choose A ′ n ∈ A suchthat A ′ n = { a n,j : j < ω } ⊆ A n , j < k implies a n,j = a n,k , and A n,m = { a n,j : m ≤ j < ω } ∈ A . Finally choose some θ : ω → ω such that | θ ← ( n ) | = ℵ for each n < ω .Since playing A θ ( m ) ,m during round m does not define a winning strategy for I in G ( A , B ), II may choose x m ∈ A θ ( m ) ,m such that B = { x m : m < ω } ∈ B . Choose i m < ω for each m < ω such that x m = a θ ( m ) ,i m , noting i m ≥ m . It follows that A n ∩ B ⊇ { a θ ( m ) ,i m : m ∈ θ ← ( n ) } . Since for each m ∈ θ ← ( n ) there exists M ∈ θ ← ( n ) such that m ≤ i m < M ≤ i M , and therefore a θ ( m ) ,i m = a θ ( m ) ,i M = a θ ( M ) ,i M ,we have shown that A n ∩ B is infinite. Thus B witnesses α ( A , B ). (cid:3) RHANGELSKII’S α -PRINCIPLES AND SELECTION GAMES 3 While α ( A , B ) involves infinite intersection and G ( A , B ) involves single selec-tions, the previous result is made more intuitive given the following result, shownfor A = B = Γ X,x by Nogura in [7].
Definition 9. α ′ ( A , B ) is the following claim: if A n ∈ A for all n < ω , then thereexists B ∈ B such that A n ∩ B is nonempty for all n < ω .(Note that α is sometimes used in the literature in place of α ′ .) Proposition 10. If A is almost- Γ -like, then α ( A , B ) is equivalent to α ′ ( A , B ) .Proof. The forward implication is immediate, so we assume α ′ ( A , B ). Given A n ∈A , we apply the almost-Γ-like property to obtain A ′ n = { a n,m : m < ω } ⊆ A n suchthat A n,m = A n \ { a i,j : i, j < m } ∈ A for all m < ω .By applying α ′ ( A , B ) to A n,m , we obtain B ∈ B such that A n,m ∩ B is nonemptyfor all n, m < ω . Since it follows that A n ∩ B is infinite for all n < ω , we haveestablished α ( A , B ). (cid:3) On α ( A , B ) and G fin ( A , B )A similar correspondence exists between α ( A , B ) and G fin ( A , B ). Theorem 11.
Let A be almost- Γ -like and B be Γ -like. Then α ( A , B ) holds if andonly if I pre G < ( A , B ) if and only if I pre G fin ( A , B ) .Proof. We first assume α ( A , B ) and let A n ∈ A for n < ω define a predeterminedstrategy for I in G < ( A , B ). We then may choose A ′ n ∈ A where A ′ n = { a n,j : j <ω } ⊆ A n , j < k implies a n,j = a n,k , and A ′′ n = A ′ n \ { a i,j : i, j < n } ∈ A .By applying α ( A , B ) to A ′′ n , we obtain B ∈ B such that A ′′ n ∩ B = ∅ for infintely-many n < ω . We then let F n = ∅ when A ′′ n ∩ B = ∅ , and F n = { x n } for some x n ∈ A ′′ n ∩ B otherwise. Then we will have that B ′ = S { F n : n < ω } ⊆ B belongsto B once we show that B ′ is infinite. To see this, for m ≤ n < ω note that either F m is empty (and we let j m = 0) or F m = { a m,j m } for some j m ≥ m ; choose N < ω such that j m < N for all m ≤ n and F N = { x N } . Thus F m = F N for all m ≤ n since x N
6∈ { a i,j : i, j < N } . Thus II may defeat the predetermined strategy A n byplaying F n each round.Since I pre G < ( A , B ) immediately implies I pre G fin ( A , B ), we assume the latter.Given A n ∈ A for n < ω , we note this defines a (non-winning) predeterminedstrategy for I, so II may choose F n ∈ [ A n ] < ℵ such that B = S { F n : n < ω } ∈ B .Since B is infinite, we note F n = ∅ for infinitely-many n < ω . Thus B witnesses α ( A , B ) since A n ∩ B ⊇ F n = ∅ for infinitely-many n < ω . (cid:3) This shows that II gains no advantage from picking more than one point perround. This in fact only depends on B being Γ-like, which we formalize in thefollowing results. Theorem 12.
Let B be Γ -like. Then I ↑ pre G < ( A , B ) if and only if I ↑ pre G fin ( A , B ) .Proof. Assume S A is well-ordered. Given a winning predetermined strategy A n for I in G < ( A , B ), consider F n ∈ [ A n ] < ℵ . We set F ∗ n = ( ∅ if F n \ S { F m : m < n } = ∅{ min( F n \ S { F m : m < n } ) } otherwise STEVEN CLONTZ
Since | F ∗ n | <
2, we have that S { F ∗ n : n < ω } 6∈ B . In the case that S { F ∗ n : n < ω } is finite, we immediately see that S { F n : n < ω } is also finite and therefore not in B . Otherwise S { F ∗ n : n < ω } 6∈ B is an infinite subset of S { F n : n < ω } , and thus S { F n : n < ω } 6∈ B too. Therefore A n is a winning predetermined strategy for I in G fin ( A , B ) as well. (cid:3) Theorem 13.
Let B be Γ -like. Then I ↑ G < ( A , B ) if and only if I ↑ G fin ( A , B ) .Proof. Assume S A is well-ordered. Suppose I ↑ G < ( A , B ) is witnessed by thestrategy σ . Let hi ⋆ = hi , and for s ⌢ h F i ∈ ([ S A ] < ℵ ) <ω \ {hi} let( s ⌢ h F i ) ⋆ = ( s ⋆⌢ h∅i if F \ S range( s ) = ∅ s ⋆⌢ h{ min( F \ S range( s )) }i otherwiseWe then define the strategy τ for I in G fin ( A , B ) by τ ( s ) = σ ( s ⋆ ). Then givenany counterattack α ∈ ([ S A ] < ℵ ) ω by II played against τ , we note that α ∗ = S { ( α ↾ n ) ∗ : n < ω } is a counterattack to σ , and thus loses. This means B = S range( α ∗ )
6∈ B .We consider two cases. The first is the case that S range( α ∗ ) is finite. Notingthat α ∗ ( m ) ∩ α ∗ ( n ) = ∅ whenever m = n , there exists N < ω such that α ∗ ( n ) = ∅ for all n > N . As a result, S range( α ) = S range( α ↾ n ), and thus S range( α ) isfinite, and therefore not in B .In the other case, S range( α ∗ )
6∈ B is an infinite subset of S range( α ), andtherefore S range( α )
6∈ B as well. Thus we have shown that τ is a winning strategyfor I in G fin ( A , B ). (cid:3) We note that the above proof technique could be used to establish that perfect-information and limited-information strategies for II in G fin ( A , B ) may be improvedto be valid in G < ( A , B ), provided B is Γ-like. As such, G < ( A , B ) and G fin ( A , B )are effectively equivalent games under this hypothesis, so we will no longer consider G < ( A , B ).3. Perfect information and predetermined strategies
We now demonstrate the following, in the spirit of Pawlikowskii’s celebratedresult that a winning strategy for the first player in the Rothberger game mayalways be improved to a winning predetermined strategy [9].
Theorem 14.
Let A be almost- Γ -like and B be Γ -like. Then • I ↑ G fin ( A , B ) if and only if I ↑ pre G fin ( A , B ) , and • I ↑ G ( A , B ) if and only if I ↑ pre G ( A , B ) .Proof. We assume I ↑ G fin ( A , B ) and let the symbol † mean < ℵ (respectively,I ↑ G ( A , B ) and † = 1, and for convenience we assume II plays singleton subsetsof A rather than elements). As A is almost-Γ-like, there is a winning strategy σ where | σ ( s ) | = ℵ and σ ( s ) ∩ S range( s ) = ∅ (that is, σ never replays the choicesof II) for all partial plays s by II.For each s ∈ ω <ω , suppose F s ↾ m ∈ [ S A ] † is defined for each 0 < m ≤ | s | . Thenlet s ⋆ : | s | → [ S A ] † be defined by s ⋆ ( m ) = F s ↾ m +1 , and define τ ′ : ω <ω → A by τ ′ ( s ) = σ ( s ⋆ ). Finally, set [ σ ( s ⋆ )] † = { F s ⌢ h n i : n < ω } , and for some bijection RHANGELSKII’S α -PRINCIPLES AND SELECTION GAMES 5 b : ω <ω → ω let τ ( n ) = τ ′ ( b ( n )) be a predetermined strategy for I in G fin ( A , B )(resp. G ( A , B )).Suppose α is a counterattack by II against τ , so α ( n ) ∈ [ τ ( n )] † = [ τ ′ ( b ( n ))] † = [ σ ( b ( n ) ⋆ )] † It follows that α ( n ) = F b ( n ) ⌢ h m i for some m < ω . In particular, there is someinfinite subset W ⊆ ω and f ∈ ω ω such that { α ( n ) : n ∈ W } = { F f ↾ n +1 : n < ω } .Note here that ( f ↾ n + 1) ⋆ = ( f ↾ n ) ⋆⌢ h F f ↾ n +1 i . This shows that F f ↾ n +1 ∈ [ σ (( f ↾ n ) ⋆ )] † is an attempt by II to defeat σ , which fails. Thus S { F f ↾ n +1 : n < ω } = S { α ( n ) : n ∈ W } 6∈ B , and since this set is infinite (as σ prevents II from repeatingchoices) we have S { α ( n ) : n < ω } 6∈ B too. Therefore τ is winning. (cid:3) Note that the assumption in Theorem 14 that A be almost-Γ-like cannot beomitted. In [2] an example of a space X ∗ and point ∞ ∈ X ∗ where I ↑ G ( A , B )but I pre G ( A , B ) is given, where A is the set of open neighborhoods of ∞ (whichare all uncountable), and B is the set Γ X ∗ , ∞ of sequences converging to that point.(Note that G ( A , B ) is called Gru
O,P ( X ∗ , ∞ ) in that paper, and an equivalent game Gru
K,P ( X ) is what is directly studied. In fact, more is shown: I has a winningperfect-information strategy, but for any natural number k , any strategy that onlyuses the most recent k moves of II and the round number can be defeated.)While A is often not almost-Γ-like in general, it may satisfy that property incombination with the selection principles being considered. Proposition 15.
Let B be Γ -like, B ⊆ A , and I pre G fin ( A , B ) . Then A is almost- Γ -like.Proof. Let A ∈ A , and for all n < ω let A n = A . Then A n is not a winningpredetermined strategy for I, so II may choose finite sets B n ⊆ A n = A such that A ′ = S { B n : n < ω } ∈ B ⊆ A .It follows that A ′ ⊆ A and | A ′ | = ℵ , and for any infinite subset A ′′ ⊆ A ′ (inparticular, any cofinite subset), A ′′ ∈ B ⊆ A . Thus A is almost-Γ-like. (cid:3) Note that in the previous result, I pre G fin ( A , B ) could be weakened to the choiceprinciple (cid:0) AB (cid:1) : for every member of A , there is some countable subset belonging to B . Corollary 16.
Let B be Γ -like and B ⊆ A . Then • I ↑ G fin ( A , B ) if and only if I ↑ pre G fin ( A , B ) , and • I ↑ G ( A , B ) if and only if I ↑ pre G ( A , B ) .Proof. Assuming I pre G fin ( A , B ), we have I G fin ( A , B ) by Proposition 15 andTheorem 14.Similarly, assuming I pre G ( A , B ) ⇒ I pre G fin ( A , B ), we have I G ( A , B ) byProposition 15 and Theorem 14. (cid:3) This corollary generalizes e.g. Theorems 26 and 30 of [11] Theorem 5 of [5], andCorollary 36 of [3].In summary, using the selection principle notation S ⋆ ( A , B ): STEVEN CLONTZ
Corollary 17.
Let B be Γ -like and B ⊆ A . Then • I G fin ( A , B ) if and only if S fin ( A , B ) if and only if α ( A , B ) , and • I G ( A , B ) if and only if S ( A , B ) if and only if α ( A , B ) . Disjoint selections
In each α i ( A , B ) principle, it is not required for the collection { A n : n < ω } tobe pairwise disjoint. However, in many cases it may as well be. Definition 18.
For i ∈ { , , , } let α i. ( A , B ) denote the claim that α i ( A , B )holds provided the collection { A n : n < ω } is pairwise disjoint.Of course, α i ( A , B ) implies α i. ( A , B ). It’s also immediate that α i. ( A , B ) implies α i. ( A , B ) for the same reason that α i ( A , B ) implies α i +1 ( A , B ).We take advantage of the following lemma. Lemma 19 (Lemma 1.2 of [8]) . Given a family { A n : n < ω } of infinite sets, thereexist infinite subsets A ′ n ⊆ A n such that { A ′ n : n < ω } is pairwise disjoint. Proposition 20.
Let A be Γ -like. For i ∈ { , , } , α i ( A , B ) is equivalent to α i. ( A , B ) .Proof. Assume α i. ( A , B ). Let A n ∈ A . By applying the previous lemma, we have { A ′ n : n < ω } pairwise disjoint with each A ′ n being an infinite subset of A n . Since A is Γ-like, A ′ n ∈ A , so we have a witness B ∈ B such that A ′ n ∩ B satisfies α i. ( A , B )for all n < ω . Since A ′ n ⊆ A n , it follows that A n ∩ B satisfies α i ( A , B ) for all n < ω . (cid:3) It’s also true that α (Γ X,x , Γ X,x ) is equivalent to α . (Γ X,x , Γ X,x ), which is cap-tured by the following theorem.
Theorem 21.
Let A be a Γ -like collection closed under finite unions and A ⊆ B .Then α ( A , B ) is equivalent to α . ( A , B ) .Proof. Let A n ∈ A and assume α . ( A , B ). To apply the assumption, we will definea pairwise disjoint collection { A ′ n : n < ω } . First let 0 ′ = 0 and A ′ = A . Thensuppose m ′ ≥ m and A ′ m ⊆ A m ′ ⊆ S i ≤ m A ′ i are defined for all m ≤ n .If A k \ S m ≤ n A ′ m is finite for k > n ′ , let B = S m ≤ n ′ A m ∈ A ⊆ B . This B thenwitnesses α ( A , B ) since A k \ B is finite for all k < ω .Otherwise pick the minimal ( n + 1) ′ > n where A ′ n +1 = A ( n +1) ′ \ S m ≤ n A ′ m isinfinite. It follows that A ′ n +1 ⊆ A ( n +1) ′ ⊆ S m ≤ n +1 A ′ m . By construction, { A ′ n : n <ω } is a pairwise disjoint collection of members of A , and we may apply α . ( A , B )to obtain B ∈ B where A ′ n \ B is finite for all n < ω .Finally let k < ω . If k = n ′ for some n < ω , then A k \ B = A n ′ \ B ⊆ ( S m ≤ n A ′ m ) \ B is finite. Otherwise, n ′ < k < ( n + 1) ′ for some n < ω . Then( A k \ S m ≤ n A ′ m ) \ B ⊆ A k \ S m ≤ n A ′ m is finite, and ( A k ∩ S m ≤ n A ′ m ) \ B ⊆ ( S m ≤ n A ′ m ) \ B is finite, showing A k \ B is finite. (cid:3) Another fractional version of these α -principles is given as α . in [8], defined ingeneral as follows. Definition 22.
Let α . ( A , B ) be the assertion that when A n ∈ A and { A n : n <ω } is pairwise disjoint, then there exists B ∈ B such that A n ∩ B is cofinite in A n for infinitely-many n < ω . RHANGELSKII’S α -PRINCIPLES AND SELECTION GAMES 7 It’s immediate from their definitions that α . ( A , B ) implies α . ( A , B ), whichimplies α . ( A , B ). Nyikos originally showed that α . (Γ X,x , Γ X,x ) implies α (Γ X,x , Γ X,x );this result generalizes as follows.
Theorem 23.
Let A be a Γ -like collection closed under finite unions. Then α . ( A , B ) implies α ( A , B ) .Proof. We assume α . ( A , B ) and demonstrate α . ( A , B ), which is equivalent to α ( A , B ) by Proposition 20. So let A n ∈ A such that { A n : n < ω } is pairwise-disjoint.We may partition each A n into { A n,m : m < ω } with A n,m ∈ A for all m < ω .Let A ′ n = S { A i,j : i + j = n } ∈ A ; since { A ′ n : n < ω } is pairwise disjoint, we mayapply α . ( A , B ) to obtain B ∈ B where A ′ n ∩ B is cofinite in A ′ n for infinitely-many n < ω .Then for n < ω , choose N ≥ n with A ′ N ∩ B cofinite in A ′ N . Then A n,N − n ⊆ A ′ N ,so A n,N − n ∩ B is cofinite in A n,N − n , in particular, A n,N − n ∩ B is infinite. Therefore A n ∩ B is infinite, and we have shown α . ( A , B ). (cid:3) Corollary 24.
Let A be a Γ -like collection closed under finite unions. Then α x ( A , B ) implies α y ( A , B ) for < x ≤ y . Additionally, if A ⊆ B , then α x ( A , B ) implies α y ( A , B ) for ≤ x ≤ y . For this paragraph we adopt the conventional assumption that Γ
X,x is restrictedto countable sets. Nyikos showed a consistent example where α (Γ X,x , Γ X,x ) failsto imply α . (Γ X,x , Γ X,x ), and a consistent example where α . (Γ X,x , Γ X,x ) failsto imply α (Γ X,x , Γ X,x ) [8]. On the other hand, Dow showed that α (Γ X,x , Γ X,x )implies α (Γ X,x , Γ X,x ) in the Laver model for the Borel conjecture [4]; the authorconjectures that this model (specifically, the fact that every ω -splitting family con-tains an ω -splitting family of size less than b in this model) witnesses an affirmativeanswer to the following question. Definition 25.
A Γ-like collection is strongly- Γ -like if the collection is closed underfinite unions and each member is countable. Question 26.
Let A be strongly- Γ -like. Is it consistent that α ( A , A ) implies α ( A , A ) ? Conclusion
We conclude with the following easy result, and a couple questions.
Proposition 27.
Let B be Γ -like. Then α ( A , B ) holds if and only if I pre G cf ( A , B ) .Proof. We first assume α ( A , B ) and let A n ∈ A for n < ω define a predeterminedstrategy for I. By α ( A , B ), we immediately obtain B ∈ B such that | A n \ B | < ℵ .Thus B n = A n ∩ B is a cofinite choice from A n , and B ′ = S { B n : n < ω } is aninfinite subset of B , so B ′ ∈ B . Thus II may defeat I by choosing B n ⊆ A n eachround, witnessing I pre G cf ( A , B ).On the other hand, let I pre G cf ( A , B ). Given A n ∈ A for n < ω , we note thatII may choose a cofinite subset B n ⊆ A n such that B = S { B n : n < ω } ∈ B . Then B witnesses α ( A , B ) since | A n \ B | ≤ | A n \ B n | ≤ ℵ . (cid:3) STEVEN CLONTZ
Question 28.
Is there a game-theoretic characterization of α ( A , B ) ? Noting that I ↑ G (Γ X , Γ X ) if and only if I ↑ G fin (Γ X , Γ X ) [6], but the same isnot true of G ⋆ (Γ X,x , Γ X,x ) (i.e. there are α spaces that are not α [13]), we alsoask the following. Question 29.
Is there a natural condition on A , B guaranteeing I ↑ G ( A , B ) ⇒ I ↑ G fin ( A , B ) ? Acknowledgements
The author would like to thank Alan Dow, Jared Holshouser, and AlexanderOsipov for various discussions related to this paper.
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Department of Mathematics and Statistics, The University of South Alabama, Mo-bile, AL 36688
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