Selection Games and the Vietoris Space
aa r X i v : . [ m a t h . GN ] F e b SELECTION GAMES AND THE VIETORIS SPACE
CHRISTOPHER CARUVANA AND JARED HOLSHOUSER
Abstract.
We explore the connections between selection games on Hausdorff spaces and theircorresponding Vietoris space of compact subsets. These considerations offer a similar relationshipas the well-known relationship between ω -covers of X and ordinary open covers of the finite powersof X . The primary utility of this method is to establish similar relationships with k -covers and theVietoris space of compact subsets. Particularly, we show that some commonly studied selectionprinciples are equivalent to a related hyperspace being Menger or Rothberger. We then apply theseequivalences to correct a flawed argument in a previous paper which attempted to show that aPawlikowski theorem is true for k -covers. Introduction
Relationships between selection principles on a ground space and the hyperspace of closed subsetswith various topologies has been a growing area of investigation [3, 4, 7, 11, 13, 14]. One of thecommon techniques employed is to translate certain cover types to families of closed sets via thecomplement operation. The resulting relationship is thus between covers of a certain type anddense sets in a related topology given to the space of closed sets. In [1, Theorem 3.22], a moredirect topological relationship is suggested. By restricting our attention to compact sets, we bringto bear relationships between a space X and the space K ( X ) of compact subsets endowed with theVietoris topology in terms of the cover types themselves. In particular, we establish relationshipsbetween ω -covers and open covers on the space of finite subsets of X viewed as a subspace of K ( X )as well as relationships between k -covers and open covers of K ( X ).We also point out an application of these methods, following existing results of [8, 9, 16], to provestrategic equivalence of some Rothberger- and Menger-like games on X with the correspondinggames on the disjoint union X <ω of finite powers of X . Particularly, these classical results establisha relationship between selection principles involving ω -covers on X and open covers on X <ω . Thisis natural since ω -covers are to cover all finite subsets of X and one can code finite subsets of X with tuples.Finally, we also address the following. Steven Clontz pointed out that the without loss of gener-ality claim in the beginning of the proof of [1, Proposition 3.25] (restated in a slightly generalizedversion in [2] as Lemma 7) is flawed. We then noticed a similar flaw at the end of the proof of[1, Proposition 3.27] (restated in a slightly generalized version in [2] as Lemma 8). In this note,we recover the conclusions of those results for k -covers. However, we remain with the followingquestion. Are [2, Lemma 7] and [2, Lemma 8] true as stated?Throughout, we assume that all spaces X considered are Hausdorff, infinite, and, when relevant,non-compact. 2. Preliminaries
Definition 1.
For a topological space X , we let T X denote the collection of all proper, non-emptyopen subsets of X . Date : March 2, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Vietoris topology, Selection principles, Rothberger property, Menger property.
Definition 2.
Generally, for an open cover U of a topological space X , we say that U is non-trivial provided that X U . We let O X denote the collection of all non-trivial open covers of X . Definition 3.
For a space X and a class A of closed proper subsets of X , a non-trivial open cover U is an A -cover if, for every A ∈ A , there exists U ∈ U so that A ⊆ U . We let O ( X, A ) denotethe collection of all A -covers of X . Remark 1.
Note that • if A consists of the finite subsets of X , then O ( X, A ) is the collection of all ω -covers of X ,which will be denoted by Ω X . • if A consists of the compact (proper) subsets of X , then O ( X, A ) is the collection of all k -covers of X , which will be denoted by K X . Definition 4.
Given a set A and another set B , we define the finite selection game G fin ( A , B ) for A and B as follows: I A A A . . . II F F F . . . where A n ∈ A and F n ∈ [ A n ] <ω for all n ∈ ω . We declare Two the winner if S {F n : n ∈ ω } ∈ B .Otherwise, One wins. Definition 5.
Similarly, we define the single selection game G ( A , B ) as follows:I A A A . . . II x x x . . . where each A n ∈ A and x n ∈ A n . We declare Two the winner if { x n : n ∈ ω } ∈ B . Otherwise, Onewins. Definition 6.
We define strategies of various strength below. • A strategy for player One in G ( A , B ) is a function σ : ( S A ) <ω → A . A strategy σ for Oneis called winning if whenever x n ∈ σ h x k : k < n i for all n ∈ ω , { x n : n ∈ ω } 6∈ B . If playerOne has a winning strategy, we write I ↑ G ( A , B ). • A strategy for player Two in G ( A , B ) is a function τ : A <ω → S A . A strategy τ for Twois winning if whenever A n ∈ A for all n ∈ ω , { τ ( A , . . . , A n ) : n ∈ ω } ∈ B . If player Twohas a winning strategy, we write II ↑ G ( A , B ). • A predetermined strategy for One is a strategy which only considers the current turn number.We call this kind of strategy predetermined because One is not reacting to Two’s moves,they are just running through a pre-planned script. Formally it is a function σ : ω → A . IfOne has a winning predetermined strategy, we write I ↑ pre G ( A , B ). • A Markov strategy for Two is a strategy which only considers the most recent move of playerOne and the current turn number. Formally it is a function τ : A × ω → S A . If Two hasa winning Markov strategy, we write II ↑ mark G ( A , B ). • If there is a single element x ∈ A so that the constant function with value x is awinning strategy for One, we say that One has a constant winning strategy , denoted byI ↑ cnst G ( A , B ).These definitions can be extended to G fin ( A , B ) in the obvious way. Definition 7.
The reader may be more familiar with selection principles than selection games. Let A and B be collections. The selection principle S ( A , B ) for a space X is the following property:Given any sequence h A n : n ∈ ω i from A , there exists { x n : n ∈ ω } with x n ∈ A n for each n ∈ ω so that { x n : n ∈ ω } ∈ B . S fin ( A , B ) is similarly defined, but with finite selections instead of singleselections. We will use the notation X | = S (cid:3) ( A , B ) to denote that the selection principle S (cid:3) ( A , B )holds for X . ELECTION GAMES AND THE VIETORIS SPACE 3
Note that • S fin ( O , O ) is the Menger property. • S ( O , O ) is the Rothberger property. Remark 2.
In general, S (cid:3) ( A , B ) holds if and only if I pre G (cid:3) ( A , B ) where (cid:3) ∈ { , fin } . See [5,Prop. 15]. Definition 8.
An even more fundamental type of selection is inspired by the Lindel¨of property.Let A and B be collections. Then (cid:0) AB (cid:1) means that, for every A ∈ A , there exists B ⊆ A so that B ∈ B . Scheepers calls this a Bar-Ilan selection principle in [17].
Remark 3.
Let A and B be collections. We let ctbl( B ) = { B ∈ B : | B | ω } . Then One fails tohave a constant strategy in G ( A , B ) if and only if (cid:0) A ctbl( B ) (cid:1) holds as shown in [5, Prop. 15].In fact, Lemma 1.
For any space X , X | = (cid:18) A ctbl( B ) (cid:19) ⇐⇒ I cnst G ( A , B ) ⇐⇒ I cnst G fin ( A , B ) Proof.
By Remark 3, the only thing to show is the equivalence of the non-existence of a constantstrategy for One in the single selection and finite selection games. This equivalence can be seen bythe fact that any play by Two in the finite selection game can be translated to a play in the singleselection game since a countable collection of finite sets is countable. (cid:3)
Note that, in the language of [10], • (cid:18) O ctbl( O ) (cid:19) is the Lindel¨of property. • (cid:18) Ωctbl(Ω) (cid:19) is the ω -Lindel¨of property, most commonly known as the ǫ -space property. • (cid:18) K ctbl( K ) (cid:19) is the k -Lindel¨of property. Definition 9.
We say that G is a selection game if there exist classes A , B and (cid:3) ∈ { , fin } sothat G = G (cid:3) ( A , B ).2.1. Game-theoretic Tools.Definition 10.
We say that two selection games G and H are equivalent , denoted G ≡ H , if thefollowing hold: • II ↑ mark G ⇐⇒ II ↑ mark H• II ↑ G ⇐⇒ II ↑ H• I
6↑ G ⇐⇒ I
6↑ H• I pre G ⇐⇒ I pre H• I cnst G ⇐⇒ I cnst H Definition 11.
Given selection games G and H , we say that G II H if the following implicationshold: • II ↑ mark G = ⇒ II ↑ mark H• II ↑ G = ⇒ II ↑ H• I
6↑ G = ⇒ I
6↑ H
CHRISTOPHER CARUVANA AND JARED HOLSHOUSER • I pre G = ⇒ I pre H• I cnst G = ⇒ I cnst H Note that II is transitive and that if G II H and H II G , then G ≡ H . We use the subscriptof II since each implication is related to a transference of winning plays by Two. Also, for classes A and B , G ( A , B ) II G fin ( A , B ) . We now recall the Translation Theorems that will be relevant in the sequel.
Theorem 2 ([3]) . Let A , B , C , and D be collections. Suppose there are functions • ← T I ,n : B → A and • → T II ,n : [ S A ] <ω × B → [ S B ] <ω for each n ∈ ω so that(P1) If F ∈ [ ← T I ,n ( B )] <ω , then → T II ,n ( F , B ) ∈ [ B ] <ω (P2) If F n ∈ [ ← T I ,n ( B n )] <ω for each n ∈ ω and S n ∈ ω F n ∈ C , then S n ∈ ω → T II ,n ( F n , B n ) ∈ D .Then G fin ( A , C ) II G fin ( B , D ). Proof.
Most of of the proof of this is in [3, Theorem 16]. The only thing that remains to be provedis the implication I cnst G fin ( A , C ) = ⇒ I cnst G fin ( B , D ) . Suppose One does not have a constant winning strategy in G fin ( A , C ) and let B ∈ B be arbitrary. As ← T I ,n ( B ) ∈ A , there exist F n ∈ (cid:20) ← T I ,n ( B ) (cid:21) <ω so that S n ∈ ω F n ∈ C . Hence, S n ∈ ω → T II ,n ( F n , B ) ∈ D .As B ∈ B was arbitrary, we see that One does not have a constant winning strategy in G fin ( B , D ). (cid:3) Corollary 3 ([3]) . Let A , B , C , and D be collections. Suppose there are functions • ← T I ,n : B → A and • → T II ,n : ( S A ) × B → S B for each n ∈ ω so that the following two properties hold.(Ft1) If x ∈ ← T I ,n ( B ), then → T II ,n ( x, B ) ∈ B .(Ft2) If F n ∈ (cid:20) ← T I ,n ( B n ) (cid:21) <ω and S n ∈ ω F n ∈ C , then S n ∈ ω (cid:26) → T II ,n ( x, B n ) : x ∈ F n (cid:27) ∈ D .Then, for (cid:3) ∈ { , fin } , G (cid:3) ( A , C ) II G (cid:3) ( B , D ). Corollary 4 ([3]) . Let A , B , C , and D be collections. Suppose there is a map ϕ : [ S B ] × ω → S A so that the following two conditions hold. • For all B ∈ B and all n ∈ ω , { ϕ ( y, n ) : y ∈ B } ∈ A . • If G n ∈ [ B n ] <ω where B n ∈ B for each n ∈ ω and S n ∈ ω ϕ [ G n × { n } ] ∈ C , then S n ∈ ω G n ∈ D .Then, for (cid:3) ∈ { , fin } , G (cid:3) ( A , C ) II G (cid:3) ( B , D ). Definition 12.
Consider a class C and a collection C ∈ C . We say that C ′ is an enlargement of C if C ′ ⊆ S C and ( ∀ x ∈ C )( ∃ y ∈ C ′ )[ x ⊆ y ]. Definition 13.
We say that a class C is closed under enlargement if the following property holds:if C ∈ C and C ′ is an enlargement of C , then C ′ ∈ C .Note that O ( X, A ) for any family of closed sets A of a space X is closed under enlargement. ELECTION GAMES AND THE VIETORIS SPACE 5
Definition 14.
Let A and B be classes and ϕ : S B → S A . For A ∈ A , we define the ϕ -refinementof A to be n y ∈ [ B : ( ∃ x ∈ A )[ ϕ ( y ) ⊆ x ] o . Corollary 5.
Suppose A , B , C , and D are classes so that S C ⊆ S A and S D ⊆ S B . Supposethere is a function ϕ : S B → S A so that, for any B ∈ B , ϕ [ B ] ∈ A and, for any E ⊆ S B so that ϕ [ E ] ∈ C , E ∈ D . Then, for (cid:3) ∈ { , fin } , G (cid:3) ( A , C ) II G (cid:3) ( B , D ).If, in addition, • C is closed under enlargement, • for any A ∈ A , the ϕ -refinement of A is an element of B , and • for any E ⊆ S B , E ∈ D implies ϕ [ E ] ∈ C ,then G (cid:3) ( A , C ) ≡ G (cid:3) ( B , D ). Proof.
The first condition of Corollary 4 holds. Now, suppose G n ∈ [ B n ] <ω where B n ∈ B for all n ∈ ω is so that S n ∈ ω ϕ [ G n ] ∈ C . Then S n ∈ ω ϕ [ G n ] = ϕ (cid:2)S n ∈ ω G (cid:3) ∈ C implies that S n ∈ ω G n ∈ D .Hence, G (cid:3) ( A , C ) II G (cid:3) ( B , D ).For the remainder, we use Corollary 3. Define ← T I ,n : A → B to be the ϕ -refinment of A . Thendefine → T II ,n : ( S B ) × A → S A in the following way. If y ∈ S B and A ∈ A are so that thereexists x ∈ A with ϕ ( y ) ⊆ x , let → T II ,n ( y, A ) ∈ A be so that ϕ ( y ) ⊆ → T II ,n ( y, A ). Otherwise, let → T II ,n ( y, A ) = y .By our definitions, if y ∈ ← T I ,n ( A ), then → T II ,n ( y, A ) ∈ A . So suppose, for every n ∈ ω , y ,n , . . . , y k n ,n ∈ ← T I ,n ( A n ) are so that S n ∈ ω { y ,n , . . . , y k n ,n } ∈ D . By the hypotheses, we havethat S n ∈ ω { ϕ ( y ,n ) , . . . , ϕ ( y k n ,n ) } ∈ C . Observe that ϕ ( y j,n ) ⊆ → T II ,n ( y j,n , A n ) for any n ∈ ω and1 j k n which provides that S n ∈ ω { → T II ,n ( y ,n , A n ) , . . . , → T II ,n ( y k n ,n , A n ) } ∈ C since C is closedunder enlargement. (cid:3) As we will see in the applications, Corollary 5 is capturing game equivalence under the conditionthat there is an adequate way to translate between cover types via some translation of open sets.As an introductory application, the translation of winning plays for Two is monotone with respectto closed subspaces, just as one would expect.
Lemma 6.
Let X be a space, A be a family of closed subsets of X , and Y ⊆ X be closed so that Y
6∈ A . Then B := { A ∩ Y : A ∈ A} is a family of closed subsets of Y and, for (cid:3) ∈ { , fin } , G (cid:3) ( O ( X, A ) , O ( X, A )) II G (cid:3) ( O ( Y, B ) , O ( Y, B )) . Proof.
First, for any open V ⊆ Y , let W V be open in X so that W V ∩ Y = V . Then define ϕ : T Y → T X by the rule ϕ ( V ) = W V ∪ ( X \ Y ). If V ∈ O ( Y, B ), we show that ϕ [ V ] ∈ O ( X, A ).Let A ∈ A and find V ∈ V so that A ∩ Y ⊆ V . Then A ⊆ ϕ ( V ).Now, suppose E is a collection of open subsets of Y so that ϕ [ E ] ∈ O ( X, A ). We show that E ∈ O ( Y, B ). Let B ∈ B and A ∈ A be so that B = A ∩ Y . There is some E ∈ E so that A ⊆ ϕ ( E )and so we see that B = A ∩ Y ⊆ ϕ ( E ) ∩ Y = E . So Corollary 5 applies. (cid:3) Notice that Two has a winning Markov strategy in G ( K R , K R ) and since Q is not hemicom-pact, by [1, Theorem 3.22], Two does not have a winning Markov strategy in G ( K Q , K Q ). Thus, G ( K R , K R ) II G ( K Q , K Q ) so the requirement that the subspace be closed is necessary.Similarly, the inequality does not reverse as Two has a winning Markov strategy in G ( O Z , O Z )but does not have a Markov winning strategy in G ( O R , O R ), proving G ( O Z , O Z ) II G ( O R , O R ). CHRISTOPHER CARUVANA AND JARED HOLSHOUSER Applications to Finite Powers
Let X <ω be the disjoint union of all X n for n >
1. Clearly, X <ω is a coding space for all finitesubsets of X so one may anticipate a relationship between open covers of X <ω and ω -covers of X .Indeed, we revisit those well-known connections.The following result concerning ω -covers and how they interact with finite powers can be seenas the real driving force behind the results of this section and, moreover, the inspiration behindLemma 19. Lemma 7 (Adapted from Lemmas 3.2 and 3.3 of [9]) . Let X be a space and n >
1. Then,(a) if U is an ω -cover of X , then { U n : U ∈ U } is an ω -cover of X n .(b) if U is an ω -cover of X n , V = { V ∈ T X : ( ∃ U ∈ U )[ V n ⊆ U ] } is an ω -cover of X . Lemma 8.
Let X be a space and n >
1. For any A ⊆ X , define A n be the disjoint union of A, A , . . . , A n . Then,(a) if U is an ω -cover of X , then { U n : U ∈ U } is an ω -cover of X n .(b) if U is an ω -cover of X n , V = (cid:8) V ∈ T X : ( ∃ U ∈ U ) (cid:2) V n ⊆ U (cid:3)(cid:9) is an ω -cover of X . Proof.
Though the proof here is similar to a proof of Lemma 7, we provide it in full for theconvenience of the reader.(a) Suppose U is an ω -cover of X and let F be any finite subset of X n . Notice that p ( F ) := { x ∈ X : ( ∃ x ∈ F )( ∃ j ∈ ω )[ π j ( x ) = x ] } is a finite subset of X where π j is the usual projection onto the j th coordinate. Then we can find U ∈ U so that p ( F ) ⊆ U . Notice that F ⊆ U n .(b) Now suppose U is an ω -cover of X n and let F = { x , x , . . . , x m } ⊆ X . Certainly, F n isa finite subset of X n so there exists U ∈ U so that F n ⊆ U . For any ~y = ( y , y , . . . , y k ) ∈ F n ,let V ( ~y ) , . . . , V k ( ~y ) be so that ( y , y , . . . , y k ) ∈ k Y j =1 V j ( ~y ) ⊆ U. Observe that V = (cid:8) V j ( ~y ) : ~y ∈ F n , j len( ~y ) (cid:9) is a finite collection of open subsets. So, for 1 ℓ m , define W ℓ = T { V ∈ V : x ℓ ∈ V } and then W = S mℓ =1 W ℓ . Clearly, W is an open subset of X and F ⊆ W .The only thing that remains to be shown is that W n ⊆ U . For 1 k n , consider ( y , . . . , y k ) ∈ W k . Let 1 ℓ , ℓ , . . . , ℓ k m be so that y j ∈ W ℓ j for each 1 j k . We can now note that ~x = ( x ℓ , x ℓ , . . . , x ℓ k ) ∈ k Y j =1 V j ( ~x ) ⊆ U. As W ℓ j ⊆ V j ( ~x ), we see that ( y , . . . , y k ) ∈ k Y j =1 W ℓ j ⊆ k Y j =1 V j ( ~x ) ⊆ U. As k was chosen to be arbitrary, the proof is finished. (cid:3) ELECTION GAMES AND THE VIETORIS SPACE 7
Lemma 9.
For any space X , n >
1, and (cid:3) ∈ { , fin } , G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X n , Ω X n ) ≡ G (cid:3) (Ω X n , Ω X n ) ≡ G (cid:3) (Ω X <ω , Ω X <ω ) . Proof.
For the equivalence G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X n , Ω X n ), we use the map ϕ : T X → T X n definedby ϕ ( U ) = U n . By Lemma 7, we know that ϕ [ U ] ∈ Ω X n given U ∈ Ω X . Moreover, if E is anycollection of open subsets of X so that ϕ [ E ] is an ω -cover of X n , it is clear that E must be an ω -cover of X . Just take x ∈ X to the tuple of length n consisting of x in each coordinate.Observe that Ω X n is closed under enlargement and that, given any ω -cover U of X n , by Lemma7, { V ∈ T X : ( ∃ U ∈ U )[ V n ⊆ U ] } ∈ Ω X . Hence, Corollary 5 applies.The equivalence G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X n , Ω X n ) follows in a similar way, except by using Lemma8. For the equivalence G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X <ω , Ω X <ω ), we first note that G (cid:3) (Ω X <ω , Ω X <ω ) II G (cid:3) (Ω X , Ω X )by Lemma 6. To obtain G (cid:3) (Ω X , Ω X ) II G (cid:3) (Ω X <ω , Ω X <ω ) , we will first need to fix a bijection β : ω → ω . Though the information transfer across the strategytypes is uniform and thus, something similar to one of our translation theorems should apply, wewill prove this without referring to them explicitly.What we will do is describe how Two is to play the game assuming they have a winning play in G (cid:3) (Ω X , Ω X ). Since the statement we wish to prove involves a transferal of winning plays by Two,this will prove what we want. Notice that, for { ( n, m ) : m ∈ ω } , Two can play with their attentiononly on X n . In particular, for each m ∈ ω , in the β ( n, m ) th inning of G (cid:3) (Ω X <ω , Ω X <ω ), givenOne’s play U β ( n,m ) , let Two choose V n,m ⊆ X and U n,m ∈ U β ( n,m ) so that V nn,m ⊆ U n,m in such away that { V n,m : m ∈ ω } is an ω -cover of X . This is possible by Lemma 8 and since G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X n , Ω X n ). Now, the U n,m correspond to a play by Two in the game G (cid:3) (Ω X <ω , Ω X <ω ) andwe need only check that it is a winning play. For any finite subset F of X <ω , there is a maximallength n of any tuple in F . Since { V nn,m : m ∈ ω } is an ω -cover of X n by Lemma 8, we see thatthere must be some m ∈ ω for which F ⊆ V nn,m ⊆ U n,m . This finishes the proof. (cid:3) Lemma 10.
For any space X and an ideal A of compact sets so that X = S A , G fin ( O ( X, A ) , O ( X, A )) II G fin ( O X , O X ) . Proof.
We use Theorem 2. Note that S O ( X, A ) = S O X = T X . Define ← T I ,n : O X → O ( X, A ) bythe rule ← T I ,n ( U ) = n[ F : F ∈ [ U ] <ω o . Observe that ← T I ,n ( U ) ∈ O ( X, A ) as A is compact.Now we define → T II ,n : [ T X ] <ω × O X → [ T X ] <ω in the following way. If V , . . . , V n ∈ T X and U ∈ O X are so that V k = S F k for F k ∈ [ U ] <ω , 1 k n , choose F k, U ∈ [ U ] <ω so that V k = S F k, U . Then we define → T II ,n ( { V , V , . . . , V n } , U ) = n [ k =1 F k, U . Otherwise, let → T II ,n ( { V , V , . . . , V n } , O X ) = { V , V , . . . , V n } . CHRISTOPHER CARUVANA AND JARED HOLSHOUSER
Suppose V , . . . , V n ∈ ← T I ,n ( U ). Then notice that, for 1 k n , F k, U ∈ [ U ] <ω and thus → T II ,n ( { V , V , . . . , V n } , U ) ∈ [ U ] <ω . Now, suppose V n, , . . . , V n,m n ∈ ← T I ,n ( U n ) are so that S n ∈ ω { V n,j : 1 j m n } ∈ O ( X, A ). Thelast thing we need to show is that [ n ∈ ω → T II ,n ( { V n, , . . . , V n,m n } , U n ) ∈ O X . Suppose x ∈ X and notice that O ( X, A ) ⊆ O X . Then there exists n ∈ ω and 1 j m n so that x ∈ V n,j = [ F n,j, U n . Hence, there is an W ∈ F n,j, U n so that x ∈ W . Finally, note that W ∈ → T II ,n ( { V n, , . . . , V n,m n } , U n ). (cid:3) Lemma 10 can be strengthened to single selections when A is the collection of finite subsets of X . However, we have not found a way to apply any of the Translation Theorems in this particularinstance. Lemma 11 (Sakai [16]) . If X | = S (Ω , Ω), then X is Rothberger.The proof of this relies on a bijection ω → ω that ensures that, given a sequence U n of opencovers, single selections from a sequence of ω -covers consisting of a particular kind of closure underfinite unions of the U n form single selections from the U n . The primary reason this creates aproblem for strategic transferal is because the way the open covers are translated to ω -coversrequires the entire sequence up front. Hence, as Two only knows finitely many of One’s moves atany stage in the game, they cannot bring this information to bear.Also, we employ Pawlikowski’s strategy strengthening for the Menger and Rothberger games. Theorem 12 (Pawlikowski [15]) . For any space X and (cid:3) ∈ { , fin } ,I ↑ pre G (cid:3) ( O X , O X ) ⇐⇒ I ↑ G (cid:3) ( O X , O X ) . Lemma 13.
For any space X , G (Ω X , Ω X ) II G ( O X , O X ) . Proof.
The implication I cnst G (Ω X , Ω X ) = ⇒ I cnst G ( O X , O X ) follows from Lemmas 10 and 1and I ↑ pre G ( O X , O X ) = ⇒ I ↑ pre G (Ω X , Ω X ) is the content of Lemma 11.Now, using Theorem 12,I ↑ G ( O X , O X ) = ⇒ I ↑ pre G ( O X , O X ) = ⇒ I ↑ pre G (Ω X , Ω X ) = ⇒ I ↑ G (Ω X , Ω X ) . To finish the proof, we note that [6, Theorem 15] states thatII ↑ G (Ω X , Ω X ) ⇐⇒ II ↑ G ( O X , O X )and that [6, Theorem 17] states thatII ↑ mark G (Ω X , Ω X ) ⇐⇒ II ↑ mark G ( O X , O X ) . (cid:3) Theorem 14.
For any space X , n >
1, and (cid:3) ∈ { , fin } , G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω X n , Ω X n ) ≡ G (cid:3) (Ω X <ω , Ω X <ω ) ≡ G (cid:3) ( O X <ω , O X <ω ) . ELECTION GAMES AND THE VIETORIS SPACE 9
Proof.
Lemmas 10 and 13 obtain G (cid:3) (Ω X <ω , Ω X <ω ) II G (cid:3) ( O X <ω , O X <ω ) . By Lemma 9, to finish the proof, we need only show that G (cid:3) ( O X <ω , O X <ω ) II G (cid:3) (Ω X , Ω X ) . Define ϕ : T X → T X <ω by letting ϕ ( U ) be the disjoint union of all the U n , n >
1. If U is an ω -cover of X , observe that ϕ [ U ] is an open cover of X <ω .Now, suppose E is a collection of open subsets of X so that ϕ [ E ] is an open cover of X <ω .To see that E must indeed be an ω -cover of X , let { x , x , . . . , x m } ⊆ X . Then notice that( x , x , . . . , x m ) ∈ X m so there must be some E ∈ E so that ( x , x , . . . , x m ) ∈ ϕ ( E ). By ourdefinition of ϕ , this means that { x , x , . . . , x n } ⊆ E , so we apply Corollary 5 to obtain what weclaimed. (cid:3) Corollary 15 (Gerlits & Nagy [8]) . Let X be a space. The following are equivalent:(a) X is an ǫ -space,(b) every finite power of X is an ǫ -space,(c) every finite power of X is Lindel¨of, and(d) X <ω is Lindel¨of. Corollary 16 (Just, Miller, and Scheepers [9, Theorem 3.9]) . Let X be a space. The following areequivalent:(a) X | = S fin (Ω , Ω),(b) ( ∀ n ∈ ω )[ X n +1 | = S fin (Ω , Ω)],(c) ( ∀ n ∈ ω )[ X n +1 | = S fin ( O , O )], and(d) X <ω is Menger. Corollary 17 (Sakai [16]) . Let X be a space. The following are equivalent:(a) X | = S (Ω , Ω),(b) ( ∀ n ∈ ω )[ X n +1 | = S (Ω , Ω)],(c) ( ∀ n ∈ ω )[ X n +1 | = S ( O , O )], and(d) X <ω is Rothberger.In the sequel, we will extend Theorem 12 and Corollaries 15, 16, and 17 as much as possible.4. Applications to Ideals of Compact Sets
Definition 15.
For a space X , let K ( X ) be the collection of all non-empty compact subsetsof X endowed with the Vietoris topology; that is, the topology generated by sets of the form { K ∈ K ( X ) : K ⊆ U } and { K ∈ K ( X ) : K ∩ U = ∅} for U ⊆ X open. For U , . . . , U n open in X ,define [ U , . . . , U n ] = K ∈ K ( X ) : K ⊆ n [ j =1 U j and ( ∀ j ) [ K ∩ U j = ∅ ] . These sets form a basis for the topology on K ( X ).For a detailed treatment of the Vietoris topology, see [12]. Definition 16.
We say that an ideal of compact subsets A X of X is closed under A -unions if thereis an ideal A K ( X ) of compact subsets of A := A X as a subspace of K ( X ) so that • K ∈ A X = ⇒ { K ∈ A : K ⊆ K } ∈ A K ( X ) and • K ∈ A K ( X ) = ⇒ S K ∈ A X . Lemma 18.
Let X be a space. • If A X is the ideal of finite subsets of X , then A X is closed under A -unions. • If A X is the ideal of compact subsets of X , then A X is closed under A -unions. Proof.
For the finite sets, notice that, for any finite set F ⊆ X , F := { K ∈ K ( X ) : K ⊆ F } consistsof finite sets. Moreover, F is finite, thus compact. Similarly, if K is a finite set consisting of finitesubsets of X , then S K is a finite subset of X .The second item follows from results of [12]. (cid:3) Colloquially, one may say that compact sets are closed under compact unions when A X is theideal of compact subsets, for example.The following result finds inspiration from Lemma 7 and is the foundation for most of whatfollows. Lemma 19.
Let X be a space and A X be an ideal of compact subsets that is closed under A -unions.(a) If U ∈ O ( X, A X ), then { [ U ] : U ∈ U } ∈ O ( A , A K ( X ) ).(b) If U ∈ O ( A , A K ( X ) ), then V = { V ∈ T X : ∃ U ∈ U ([ V ] ⊆ U ) } ∈ O ( X, A X ) . Proof. (a) Suppose U ∈ O ( X, A X ) and K ∈ A K ( X ) . Since S K ∈ A X , there exists U ∈ U so that S K ⊆ U . Hence, K ⊆ [ U ].(b) Suppose U ∈ O ( A , A K ( X ) ) and let K ∈ A X be arbitrary. Observe that K ∗ := { K ∈ A : K ⊆ K } ∈ A K ( X ) . Let U ∈ U be so that K ∗ ⊆ U . Now, for each K ∈ K ∗ , we can find a basic neighborhood B K so that K ∈ B K ⊆ U . By compactness, there are K , . . . , K n so that K ∗ ⊆ S nj =1 B K j . Let B K j = [ W j, , . . . , W j,m j ] for 1 j n . For x ∈ K , set N x = T { W j,k : x ∈ W j,k } and define V = S x ∈ K N x . Clearly, K ⊆ V and thus K ∈ [ V ], so it suffices to show that [ V ] ⊆ U .So let K ∈ [ V ]; i.e. K ⊆ V . Then we can find x , . . . , x p ∈ K so that K ⊆ S pℓ =1 N x ℓ and K ∩ N x ℓ = ∅ for each 1 ℓ p . Since { x , . . . , x p } is a compact subset of K , it must be anelement of some [ W j, , . . . , W j,m j ].Now, for each 1 ℓ p , there is a q ℓ m j so that x ℓ ∈ W j,q ℓ ; thus N x ℓ ⊆ W j,q ℓ ⊆ S m j q =1 W j,q .So K ⊆ S pℓ =1 N x ℓ ⊆ S m j q =1 W j,q .For each 1 q m j , let 1 ℓ q p be so that x ℓ q ∈ W j,q . As K ∩ N x ℓq = ∅ and N x ℓq ⊆ W j,q ,we see that K ∩ W j,q = ∅ .Hence, we see that K ∈ [ W j, , . . . , W j,m j ] ⊆ U . Therefore [ V ] ⊆ U . (cid:3) Lemma 20.
Let X be a space and A = A X be an ideal of compact subsets that is closed under A -unions where A = A X is viewed as a subspace of K ( X ). Then, for (cid:3) ∈ { , fin } , G (cid:3) ( O A , O A ) II G (cid:3) ( O ( X, A ) , O ( X, A )) . Proof.
Define ϕ : T X → T A by the rule ϕ ( U ) = [ U ]. Certainly, if U ∈ O ( X, A ), then ϕ [ U ] ∈ O A .Moreover, suppose E is a collection of open subsets of X so that ϕ [ E ] ∈ O A . It is clear that E ∈ O ( X, A ). Hence, Corollary 5 applies. (cid:3) Theorem 21.
Let X be a space and A = A X be an ideal of compact subsets that is closed under A -unions where A = A X is viewed as a subspace of K ( X ) and B = A K ( X ) is the correspondingideal of A . Then, for (cid:3) ∈ { , fin } , G (cid:3) ( O ( X, A ) , O ( X, A )) ≡ G (cid:3) ( O ( A , B ) , O ( A , B )) . Proof.
Define ϕ : T X → T A by the rule ϕ ( U ) = [ U ]. If U ∈ O ( X, A ), then, by Lemma 19, ϕ [ U ] ∈ O ( A , B ). If E is a collection of open subsets of X so that ϕ [ E ] ∈ O ( A , B ), then E ∈ O ( X, A ).This is seen by considering any K ∈ A and noticing that K ∈ { F ∈ A : F ⊆ K } ∈ B . ELECTION GAMES AND THE VIETORIS SPACE 11
Certainly, O ( A , B ) is closed under enlargement and, by Lemma 19, we see that the ϕ -refinement of U ∈ O ( A , B ) is an element of O ( X, A ). Therefore, Corollary 5 applies and the proof is complete. (cid:3) The Space of Finite Subsets.
Let P fin ( X ) be the subspace of K ( X ) consisting of the finitesubsets of X . As in Section 3, one would anticipate some relationship between open covers of P fin ( X ) and ω -covers of X . To some degree, P fin ( X ) also avoids all of the unnecessary informationthe finite powers offer such as repetition and order. To begin, we offer an analog to Lemma 7 whichfollows immediately from Lemmas 18 and 19. Corollary 22.
Let X be a space.(a) If U is an ω -cover of X , then { [ U ] : U ∈ U } is an ω -cover of P fin ( X ).(b) If U is an ω -cover of P fin ( X ), then V = { V ∈ T X : ∃ U ∈ U ([ V ] ⊆ U ) } is an ω -cover of X . Theorem 23.
For any space X and (cid:3) ∈ { , fin } , G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω P fin ( X ) , Ω P fin ( X ) ) ≡ G (cid:3) ( O P fin ( X ) , O P fin ( X ) ) . Proof.
The fact that G (cid:3) (Ω X , Ω X ) ≡ G (cid:3) (Ω P fin ( X ) , Ω P fin ( X ) )follows from Theorem 21. By Lemmas 10 and 13, we see that G (cid:3) (Ω P fin ( X ) , Ω P fin ( X ) ) II G (cid:3) ( O P fin ( X ) , O P fin ( X ) ) . Finally, G (cid:3) ( O P fin ( X ) , O P fin ( X ) ) II G (cid:3) (Ω X , Ω X ) . follows from Lemma 20. This finishes the proof. (cid:3) Corollary 24.
For any space X , the following are equivalent:(a) X is an ǫ -space(b) P fin ( X ) is an ǫ -space(c) P fin ( X ) is Lindel¨of. Corollary 25.
For any space X , the following are equivalent:(a) X | = S fin (Ω , Ω)(b) P fin ( X ) | = S fin (Ω , Ω)(c) P fin ( X ) is Menger. Corollary 26.
For any space X , the following are equivalent:(a) X | = S (Ω , Ω)(b) P fin ( X ) | = S (Ω , Ω)(c) P fin ( X ) is Rothberger. Corollary 27 (Scheepers [18]) . For any space X and (cid:3) ∈ { , fin } ,I ↑ pre G (cid:3) (Ω X , Ω X ) ⇐⇒ I ↑ G (cid:3) (Ω X , Ω X ) . Proof.
These follow immediately from Theorems 12 and 23. (cid:3)
The Space of Compact Subsets.
When we move to compact sets, one might expect thatall of the analogous theorems from Sections 3 and 4.1 hold between k -covers of X and open coversof K ( X ). Though we cannot obtain the full scope of those results, we are able recover a significantfragment; namely, everything about finite selection games goes through, and we are able to recovera version of Pawlikowki’s theorem.In a similar spirit to the results of Lemma 7 and Corollary 22, we establish a way to transfer k -cover information between X and K ( X ). It follows immediately from Lemmas 18 and 19. Corollary 28.
Let X be a space.(a) If U is a k -cover of X , then { [ U ] : U ∈ U } is a k -cover of K ( X ).(b) If U is a k -cover of K ( X ), then V = { V ∈ T X : ∃ U ∈ U ([ V ] ⊆ U ) } is a k -cover of X . Corollary 29.
For any space X and (cid:3) ∈ { , fin } , G (cid:3) ( O K ( X ) , O K ( X ) ) II G (cid:3) ( K X , K X ) ≡ G (cid:3) ( K K ( X ) , K K ( X ) ) . Proof.
This follows immediately from Lemma 20 and Theorem 21. (cid:3)
Theorem 30.
For any space X , G fin ( K X , K X ) ≡ G fin ( K K ( X ) , K K ( X ) ) ≡ G fin ( O K ( X ) , O K ( X ) ) . Proof.
This follows immediately from Lemma 10 and Corollary 29. (cid:3)
Corollary 31.
For any space X , the following are equivalent:(a) X is k -Lindel¨of(b) K ( X ) is k -Lindel¨of(c) K ( X ) is Lindel¨of. Corollary 32.
For any space X , the following are equivalent:(a) X | = S fin ( K , K )(b) K ( X ) | = S fin ( K , K )(c) K ( X ) is Menger. Corollary 33.
For any space X ,I ↑ pre G fin ( K X , K X ) ⇐⇒ I ↑ G fin ( K X , K X ) . Proof.
This follows from Theorems 12 and 30. (cid:3)
Like before, single selections present an obstacle. In the context of k -covers, we don’t even obtainan analog to Corollary 26. Example.
In general, S ( K , K ) S ( O , O ). Observe that R | = S ( K , K ) but R = S ( O , O ). If { U n : n ∈ ω } consists of k -covers of R , simply choose U n ∈ U n to be so that [ − n, n ] ⊆ U n . Then { U n : n ∈ ω } is a k -cover of R . On the other hand, consider V n = { B ( q, − n ) : q ∈ Q } and anysequence of selections V n ∈ V n . Notice that the union of the V n has finite Lebesgue measure sothey cannot cover R .Because of this non-example, we cannot obtain a version of Pawlikowski’s result for k -coversas easily as we did for ω -covers. In the next two results we nevertheless prove that there is aPawlikowski style strategy reduction for the k -Rothberger game. The basic idea is to take thegame up to the hyperspace and play with the right kind of open sets to guarantee that Two’s playresults in a k -cover. ELECTION GAMES AND THE VIETORIS SPACE 13
Lemma 34.
Suppose A is any ideal of closed sets that contains all singletons of a space X . Alsosuppose S { F n : n ∈ ω } ∈ O ( X, A ) where F n is a finite collection of open sets for each n ∈ ω .Then, for any A ∈ A , there exists an increasing sequence { α n : n ∈ ω } so that( ∀ n ∈ ω )( ∃ U ∈ F α n ) [ A ⊆ U ] . Proof.
Let A = A ∈ A be arbitrary and, for n >
0, suppose we have A n ∈ A and α n ∈ ω definedso that α n = min { m ∈ ω : ( ∃ U ∈ F m ) [ A n ⊆ U ] } . As U is a proper open set for each U ∈ F α n , we can find x U ∈ X \ U . Notice that A n +1 := A n ∪ { x U : U ∈ F α n } ∈ A since A is an ideal containing singletons and F α n is finite. So then we can set α n +1 = min { m ∈ ω : ( ∃ U ∈ F m ) [ A n +1 ⊆ U ] } . Observe that α n +1 > α n . This finishes the proof. (cid:3) Theorem 35.
For any space X , I ↑ pre G ( K X , K X ) ⇐⇒ I ↑ G ( K X , K X ). Proof.
We need only show I pre G ( K X , K X ) = ⇒ I G ( K X , K X ) . Suppose X | = S ( K X , K X ) and that One is playing according to some fixed strategy in G ( K X , K X ).Any k -cover can be made into a countable k -cover by the selection principle. Hence, we can codeOne’s strategy with { U s : s ∈ ω <ω } with the property that { U s ⌢ k : k ∈ ω } is a k -cover forany s ∈ ω <ω . Using this strategy for One in G ( K X , K X ), we will define a strategy for One in G fin ( K K ( X ) , K K ( X ) ) which will produce a winning counter-play by Two as X | = S ( K X , K X ) = ⇒ K ( X ) | = S ( K K ( X ) , K K ( X ) )= ⇒ K ( X ) | = S fin ( K K ( X ) , K K ( X ) )= ⇒ I G fin ( K K ( X ) , K K ( X ) ) . Moreover, we will show the counter-play produced actually corresponds to a counter-play to One’sstrategy in G ( K X , K X ). We will do this through a sequence of useful claims.The first claim is that( ∀ m > ∀ j > ∀ K ∈ K ( K ( X )))( ∃ s ∈ ω | j m | )( ∀ t ∈ j m )( ∀ L ∈ K )( ∃ k < | j m | ) (cid:2) L ⊆ U t ⌢ ( s ↾ k +1 ) (cid:3) . Fix m > j >
0, let K ⊆ K ( X ) be compact, and consider K := S K , which forms a compactsubset of X . Enumerate j m as { t ℓ : ℓ < | j m |} . Let s (0) ∈ ω so that K ⊆ U t ⌢ s (0) . Then, for n >
0, suppose we have s (0) , . . . , s ( n ) ∈ ω defined so that K ⊆ U t ℓ⌢ s (0) ⌢ ··· ⌢ s ( ℓ ) for each ℓ n . Let s ( n + 1) ∈ ω be so that K ⊆ U t n +1 ⌢ s (0) ⌢ s (1) ⌢ ··· ⌢ s ( n +1) . This defines s : | j m | → ω .Next, fix some t ∈ j m , let L ∈ K be arbitrary, and find k < | j m | so that t = t k . Then, L ⊆ K ⊆ U t k⌢ s (0) ⌢ ··· ⌢ s ( k ) = U t ⌢ s ↾ k +1 . This establishes the claim.The second claim involves defining, for m > j >
0, and s : | j m | → ω , V s ( m, j ) = \ t ∈ j m | j m | [ k =1 [ U t ⌢ ( s ↾ k ) ] . The second claim is that, for fixed m > j > (cid:8) V s ( m, j ) : s ∈ ω | j m | (cid:9) is a k -cover of K ( X ).So let K ⊆ K ( X ) be compact and choose s : | j m | → ω so that( ∀ t ∈ j m )( ∀ L ∈ K )( ∃ k < | j m | ) (cid:2) L ⊆ U t ⌢ ( s ↾ k +1 ) (cid:3) , which is guaranteed by the first claim. Fix t ∈ j m , let L ∈ K , and observe that, for some k < | j m | , L ∈ [ U t ⌢ ( s ↾ k +1 ) ] ⊆ | j m | [ ℓ =1 [ U t ⌢ ( s ↾ ℓ ) ] . Since this is true for any L ∈ K , we see that K ⊆ | j m | [ ℓ =1 [ U t ⌢ ( s ↾ ℓ ) ] . Since t ∈ j m was also taken to be arbitrary, we see that K ⊆ \ t ∈ j m | j m | [ k =1 [ U t ⌢ ( s ↾ k ) ] = V s ( m, j ) . The third claim is that there are increasing functions g, h : ω → ω so that( ∀ K ∈ K ( K ( X )))( ∃ ∞ n ∈ ω )( ∃ s : ( g ( n + 1) − g ( n )) → h ( n + 1)) [ K ⊆ V s ( g ( n ) , h ( n ))] . To accomplish this, we define a particular strategy σ for One in G fin ( K K ( X ) , K K ( X ) ). First, for m > j >
0, define V m,j = n V s ( m, j ) : s ∈ ω | j m | o , which is a k -cover of K ( X ) by the second claim. Also, for m > j >
0, and p >
0, define F m,j,p = n V s ( m, j ) : s ∈ p | j m | o , a finite subset of V m,j . Observe that, for m > j >
0, if 0 < p q , then F m,j,p ⊆ F m,j,q . Infact, (cid:0) ∀ E ∈ [ V m,j ] <ω (cid:1) ( ∃ p >
0) [ E ⊆ F m,j,p ] . To see this, let E ∈ [ V m,j ] <ω and A ∈ (cid:2) ω | j m | (cid:3) <ω be so that E = { V s ( m, j ) : s ∈ A } . Then set p = 1 + max { s ( ℓ ) : ℓ < | j m | , s ∈ A } . It follows that A ⊆ p | j m | which further implies that E ⊆ F m,j,p . Now define p m,j : [ V m,j ] <ω → ω tobe p m,j ( E ) = min { p : E ⊆ F m,j,p } . We next define the strategy σ . Set m = 0, j = 1, and σ ( ∅ ) = V m ,j . For n >
0, suppose { E ℓ : ℓ < n } , { m ℓ : ℓ n } , and { j ℓ : ℓ n } have been defined. Let m n +1 = m n + | j m n n | . Then for E n ∈ [ V m n ,j n ] <ω set j n +1 = max { j n , p m n ,j n ( E n ) } and define σ ( V m ,j , E , . . . , V m n ,j n , E n ) = V m n +1 ,j n +1 This finishes the definition of the strategy σ .As One does not have a winning strategy in G fin ( K K ( X ) , K K ( X ) ), Two can produce a counter-play { E n : n ∈ ω } so that [ { E n : n ∈ ω } is a k -cover of K ( X ). Notice that this provides increasing sequences h m n : n ∈ ω i and h j n : n ∈ ω i .Moreover, as E n ∈ [ V m n ,j n ] <ω and j n +1 > p m n ,j n ( E n ), we have that E n ⊆ F m n ,j n ,j n +1 . That is, [ { F m n ,j n ,j n +1 : n ∈ ω } ELECTION GAMES AND THE VIETORIS SPACE 15 is a k -cover of K ( X ).Define g, h : ω → ω by the rules g ( n ) = m n and h ( n ) = j n and notice that they are increasingfunctions. To verify they are as desired, we first find one n ∈ ω that meets the requisite criterion.Let K ⊆ K ( X ) be compact. Since S { F m n ,j n ,j n +1 : n ∈ ω } is a k -cover of K ( X ), there must besome n ∈ ω and s : | j m n n | → j n +1 so that K ⊆ V s ( m n , j n ). Behold that m n = g ( n ), j n = h ( n ), j n +1 = h ( n + 1), and | j m n n | = m n +1 − m n = g ( n + 1) − g ( n ). The fact that infinitely many such n exist follows from Lemma 34.The final thing to show is that we can actually construct a counter-play against One’s strategyin G ( K X , K X ) with the help of the defined g and h . For n > k < k < · · · < k n , and s i : ( g ( k i + 1) − g ( k i )) → h ( k i + 1), 1 i n , we define W n ( k , . . . , k n ; s , . . . , s n ) = n \ i =1 V s i ( g ( k i ) , h ( k i )) . To assist with notation, we let F k i = h ( k i + 1) ( g ( k i +1) − g ( k i )) . By the third claim, W n := { W n ( k , . . . , k n ; s , . . . , s n ) : ( k < · · · < k n ) and ( ∀ i n ) [ s i ∈ F k i ] } is a k -cover of K ( X ). Since we are assume X | = S ( K X , K X ) and we know that X | = S ( K X , K X ) = ⇒ K ( X ) | = S ( K K ( X ) , K K ( X ) ) , for each n >
1, we can select k n, < . . . < k n,n and s n,i ∈ F k n,i for 1 i n , so that { W n ( k n, , . . . , k n,n ; s n, , . . . , s n,n ) : n ∈ ω } is a k -cover of K ( X ).For each n >
1, choose k n,α n ∈ { k n, , . . . , k n,n } \ { k ℓ,α ℓ : 1 ℓ < n } and consider B n := { g ( k n,α n ) + i : i < g ( k n,α n + 1) − g ( k n,α n ) } . We argue that the sets { B n : n > } are pair-wise disjoint. Suppose m < n and notice that k m,α m = k n,α n by definition. Without loss of generality, suppose k m,α m < k n,α n . Since it followsthat g ( k m,α m ) < g ( k n,α n ), to establish that B m and B n are dsijoint, it suffices to check that g ( k m,α m + 1) g ( k n,α n ). Indeed, k m,α m + 1 k n,α n = ⇒ g ( k m,α m + 1) g ( k n,α n ) . Moreover, for any ℓ ∈ B n , ℓ − g ( k n,α n ) < g ( k n,α n + 1) − g ( k n,α n ). So s n,α n ( ℓ − g ( k n,α n )) is defined.This allows us to define f : ω → ω by the rule f ( ℓ ) = ( s n,α n ( ℓ − g ( k n,α n )) , ℓ ∈ B n , otherwiseWe claim that this f is Two’s desired play. Let K ⊆ X be compact, and notice that { K } is acompact subset of K ( X ). So there exists some n > { K } ⊆ W n ( k n, , . . . , k n,n ; s n, , . . . , s n,n ) = n \ i =1 V s n,i ( g ( k n,i ) , h ( k n,i )) . For ease of notation, let E n = h ( k n,α n ) g ( k n,αn ) and notice that { K } ⊆ V s n,αn ( g ( k n,α n ) , h ( k n,α n )) = \ t ∈ E n | E n | [ ℓ =1 [ U t ⌢ ( s n,αn ↾ ℓ ) ] . We wish to show that f ↾ g ( k n,αn ) : g ( k n,α n ) → h ( k n,α n ). So let ℓ < g ( k n,α n ) be arbitrary. If ℓ B m for any m > f ( ℓ ) = 0 < h ( k n,α n ). Otherwise, ℓ ∈ B m for some m >
1. Then there is some i < g ( k m,α m + 1) − g ( k m,α m ) with ℓ = g ( k m,α m ) + i . Hence, f ( ℓ ) = s m,α m ( ℓ − g ( k m,α m )) = s m,α m ( i ) < h ( k m,α m ) . Also, as g ( k m,α m ) g ( k m,α m ) + i = ℓ < g ( k n,α n ) , we see that k m,α m < k n,α n which provides f ( ℓ ) < h ( k m,α m ) h ( k n,α n ) . Thus, { K } ⊆ | E n | [ ℓ =1 [ U ( f ↾ g ( kn,αn ) ) ⌢ ( s n,αn ↾ ℓ ) ] = ⇒ K ∈ | E n | [ ℓ =1 [ U ( f ↾ g ( kn,αn ) ) ⌢ ( s n,αn ↾ ℓ ) ]which means that for some 1 ℓ | E n | , K ∈ [ U ( f ↾ g ( kn,αn ) ) ⌢ ( s n,αn ↾ ℓ ) ] = ⇒ K ⊆ U ( f ↾ g ( kn,αn ) ) ⌢ ( s n,αn ↾ ℓ ) . Finally, by our definition of f , we note that( f ↾ g ( k n,αn ) ) ⌢ ( s n,α n ↾ ℓ ) = f ↾ g ( k n,αn +1) which means K ⊆ U f ↾ g ( kn,αn +1) . Therefore, if Two plays according to f , Two produces a k -cover of X , finishing the proof. (cid:3) Final Remarks
We end with a couple of other applications of these techniques which relate to the interplaybetween cover types.
Theorem 36.
For any space X and (cid:3) ∈ { , fin } , G (cid:3) ( K K ( X ) , Ω K ( X ) ) II G (cid:3) ( K X , Ω X ) . Proof.
Define ϕ : T X → T K ( X ) by the rule ϕ ( U ) = [ U ]. By Corollary 28, we know that ϕ [ U ] ∈K K ( X ) when U ∈ K X . Now, suppose E ⊆ T X is so that ϕ [ E ] ∈ Ω K ( X ) . Observe that ϕ [ E ] is alsoan ω -cover of P fin ( X ). By Corollary 22, we know that { V ∈ T X : ∃ U ∈ ϕ [ E ] ([ V ] ⊆ U ) } ∈ Ω X which demonstrates that E is an ω -cover of X . Thus, Corollary 5 applies. (cid:3) Theorem 37.
For any space X and (cid:3) ∈ { , fin } , G (cid:3) (Ω X , K X ) II G (cid:3) (Ω K ( X ) , K K ( X ) ) . Proof.
In this case, we apply Corollary 3. Define ← T I ,n : Ω K ( X ) → Ω X by the rule ← T I ,n ( U ) = { V ∈ T X : ∃ U ∈ U ([ V ] ⊆ U ) } . Since every ω -cover of K ( X ) is an ω -cover of P fin ( X ), ← T I ,n is defined by Corollary 22.Now we define → T II ,n : T X × Ω K ( X ) → T K ( X ) in the following way. Let U ∈ Ω K ( X ) and if V ∈ ← T I ,n ( U ), let → T II ,n ( V, U ) be so that [ V ] ⊆ → T II ,n ( V, U ). Otherwise, let → T II ,n ( V, U ) = V . ELECTION GAMES AND THE VIETORIS SPACE 17
Suppose F n ∈ (cid:20) ← T I ,n ( U n ) (cid:21) <ω are so that S n ∈ ω F n ∈ K ( X ). By Corollary 28, we know that [ n ∈ ω { [ V ] : V ∈ F n } ∈ K K ( X ) and, as [ V ] ⊆ → T II ,n ( V, U n ) for each V ∈ F n , we see that [ n ∈ ω (cid:26) → T II ,n ( V, U n ) : V ∈ F n (cid:27) ∈ K K ( X ) . This finishes the proof. (cid:3)
For further work, are [2, Lemma 7] and [2, Lemma 8] true as stated? Additionally, can Theorem36 be used to establish a Pawlikowski style strategy reduction for G (cid:3) ( K , Ω)?
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